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Deliverable: D7.1Deliverable Title: Artficial Chemistry / Chemical Organization Theory coarse-grainingAuthors: Peter Dittrich, Richard Henze, Jan Huwald, Bashar Ibrahim, Pietro Speroni di FenizioDate: 30.10.2015

In WP7 we are developing an artificial chemistry application scenario and study how chemical organization can be applied for coarse-graining chemistry-like systems. We progressed in developing a more generalized concept for organizations in a finite particle systems (see also D4.2, D5.1, and Kreyssig et al. Bioinformatics, 30(17), i475-i481, 2014.) and on an idea how to obtain organizations from measurements without needing to know the exact reaction network. This deliverable contains a novel algorithm (including its implementation) to compute the set of all organizations by exploiting the algebraic lattice property of this set.

Speroni di Fenizio, P. (2015). The Lattice of Chemical Organisations. In: P. Andrews et al. (Eds.), Proceedings of the European Conference on Artificial Life 2015, MIT Press, Boston, MA, pp. 242-248, doi: http://dx.doi.org/10.7551/978-0-262-33027-5-ch048

The software for computing the lattice of organizations:https://github.com/pietrosperoni/LatticeOfChemicalOrganisations

Furthermore, we made progress in exploring the envisaged biological application scenario: the mitotic checkpoint(s) (spindle assembly checkpoint, SAC) and nucleus organization, i.e., in particular the kinetochore, which is important for the SAC. The same method would be also applicable to PML nuclear bodies, a scenario in which coarse-graining would be very beneficial. The results of our study so far are described by the following publications contained in this deliverable.

R. Henze, H. Huwald, N. Mostajo, P. Dittrich, B. Ibrahim (2014). Structural analysis of in silico mutant experiments of human inner-kinetochore structure, Biosystems, 127: 47-59. doi: 10.1016/j.biosystems.2014.11.004, 2014

B. Ibrhaim, R. Henze (2014). Active Transport Can Greatly Enhance Cdc20:Mad2 Formation, Int. J. Mol. Sci., 15(10):19074-19091; doi:10.3390/ijms151019074

The Lattice of Chemical Organisations

Pietro Speroni di Fenizio 1

1 Bio Systems Analysis Group, Jena Centre for Bioinformatics, Friedrich Schiller-University Jena, D-07737 Jena [email protected]

Abstract The paper describes how, in an Artificial Chemistry under flow conditions, the set of organisations form a lattice. The consequences of this are described, in particular how a series of theorems, valid for lattices, can be applied to more easily discover the complete set of organisations. An algorithm is then developed that uses such theorems to explore such lattice. The algorithm is applied first to the NTop Artificial Chemistry and then to an extension of it. Due to its complexity this system is also suggested as a benchmark case to test new Artificial Chemistries’ algorithms.

Introduction In 1994 Walter Fontana and Leo Buss introduced the concept of Organisation to represent the fixed points in Constructive Dynamical Systems (Fontana and Buss, 1994a, 1994b, 1996). Looking at Dynamical Systems they observed how they could predict only quantitative dynamics, and no qualitative development. True novelty could never appear through Ordinary Differential Equations. But novelty is inherent in this world. And it is unpredictability is what makes the world so interesting. So, to describe how a system could transform the qualitative space of possibilities, they presented the concept of Constructive Dynamical Systems. Those were a molecular based system, where new molecules could be generated through the interaction of existing ones. Such system was thought as a model for the macro-molecules inside a cell, but this was just one of the possible examples, of such a general theory. Using standard dynamical systems as a metaphor, they then went on describing what would be the equivalent fixed points in their system; points where no novelty would appear, and called such cases Organisations. A set of molecules would be an organisation if, and only if, for each molecule inside the set, there was a reaction among the molecules in the set that would produce it; and, given any reaction among the molecules of the set, the result would always be a molecule of the set. The first property was called Self-Maintenance and the second Closure. A set satisfying both of those properties would be called an Organisation. Later (Dittrich, Speroni di Fenizio, 2007) Organisations were then renamed Semi-Organisations when it became clear that those properties were not enough to permit to those sets to be dynamically stability. Organisations were so re-defined as special Semi-Organisations where it is possible for all the reactions among molecules inside them to be active, and have no molecule type diminish. Studying Organisations and Semi-Organisations, it was soon found that under quite common assumptions, those structures would form a Lattice. It should be noted that this it is not

always true for every reaction system, but it is true if we assume that every molecule has a certain probability to disappear (i.e. every molecule would have an out-flux greater than zero). Those systems were every molecule has an out flux were called Flow Systems to distinguish them from the more general General Reaction Systems (Dittrich, Speroni, 2007). It should be noted how all of Fontana and Buss’ models had an out-flux applied to each molecule that would destroy it at the same speed at which others were produced. So those models were Flow Systems, and the set of organisations would form a Lattice. Other Flow Systems are also possible; for example if we consider cells, and thus cell growth, although it is not true that each molecule has a certain probability higher than zero of being excreted, it is true that as the cell grows the relative concentration of each molecule that is not generates diminishes, having the same effective result as if each molecule was subject to an out-flux. Then when the cell reproduces the average amount of that molecule halves, eventually reaching zero after enough reproductions. Thus the set of molecules inside a cell form a Flow System, and the set of possible Organisations in a living system form a lattice. Vice-versa, the reaction system in the atmosphere of a planet is not a Flow System, but a General Reaction System, and the Organisations do not form a lattice but just a Partially Ordered Set. Flow Systems and General Reaction Systems are not the only possible type of Artificial Chemistries: a more detailed analysis can distinguish also Catalytic Flow Systems as a specific type of Flow System where each molecule reacts in a catalytic way. In other words they are not used up in the

Catalytic Flow Systems

Flow Systems General Reaction Systems

All molecules in each reaction are catalytic; all molecules have an out flux

All molecules have an out-flux

No requirements on the out-flux nor on which molecules are catalytic

each semi-organisation is an organisations

some semi-organisations are organisations

some semi-organisations are organisations

both organisations and semi-organisations form a lattice

both organisations and semi-organisations form a lattice

neither organisations nor semi-organisations form a lattice

Table 1: Different type of Artificial Chemistries produce sets of organisations with different properties.

Pietro Speroni di Fenizio (2015) The Lattice of Chemical Organisations. Proceedings of the European Conference onArtificial Life 2015, pp. 242-248

DOI: http://dx.doi.org/10.7551/978-0-262-33027-5-ch0��

reaction, while the prime material that generates the result is supposedly coming from a substrate of basic material floating around. Substrate to which each molecule eventually decays through the out-flux. Catalytic Flow Systems, Flow Systems, and Reaction Systems are quite different in terms of their relative properties (see Table 1). This paper aims to investigate the consequences of the fact that under flow conditions, the set of Organisations form a lattice.

What does it mean to understand an Artificial Chemistry In this context understanding an Artificial Chemistry means having a list of all the possible organisations; for each organisation know what are the organisations directly above and below it. With the organisations A being directly above (directly below) the organisation B we mean that A contains B (B contains A), A > B, (A < B) and there is no other organisation C that contain B and is contained by A (that contain A and is contained by B), ∄ C such that A > C > B (∄ C such that A < C <B). Thus being able to predict, given a starting condition, where the system will likely evolve, and if we mix two different systems what the result of it will be. I am speaking here of broad qualitative prediction, not precise quantitative ones. Knowing what molecules will be present, while ignoring the relative quantities, is a satisfactory result. Suppose you have two test tubes; each with a different experiment of the same Artificial Chemistry (AC); each having a small out-flux, thus that we can be sure that the set of organisations of this AC is a lattice. If we consider the molecules inside, eventually they will express an organisation, as only the molecules which can be generated will be present and all the others will be lost through the out-flux. What will happen if we join those two test tubes? Because the organisations in a Flow System form a lattice, we can immediately say that such action will generate the organisation union of the two organisations. And this exist and is unique because in a lattice the union of two elements is always present and unique. Of course if the organisations present in the two test tubes are one inside the other, then the union is trivially the biggest one. More interestingly we can ask: suppose we have two test tubes, where two organisations are expressed, A and B; suppose we want to add some molecules to both A and B without changing the organisation inside; what can we add? And the answer is: we can add anything from the organisation C, where C is the organisation intersection between A and B. Again we know that C uniquely exists because the set of organisations form a lattice. From this appears obvious that we do not only need to know the set of organisations, but also know the content of the two tables that given any two organisations A and B would tell us their union (A ∪ B) and their intersection (A ∩ B). Calculating those tables is also something that becomes more efficient by the fact that those organisations form a lattice and thus we can use theorems from lattice theory to calculate some of the results. Thus understanding the set of organisations gives us some clear, practical abilities to understand what happens in an Artificial Chemistry. And when this is a lattice, it all becomes easier.

Basic Known Definitions from Previous Work In a reaction system a closed set C is a set of molecules such that given any two molecules their reaction is still contained in C. A semi-self-maintaining set S is a set such that given any molecule m that is consumed in S, there exist a reaction inside S such that m is produced. Any set that is both semi-self-maintaining and closed is a semi-organisation. A self-maintaining set S is a semi-self-maintaining set such that there exist a reaction speed for all reactions among the molecules in S, with each reaction speed higher than zero, and such that each molecule has a production rate higher or equal to zero. In laymen’s terms, a semi-self-maintaining set is a set where all molecules can be produced, and a self-maintaining one is one where this is globally possible. In Catalytic Flow systems the two coincide as every semi-self-maintaining set is self-maintaining [Speroni di Fenizio, 2007]. Given any set F, we can generate its closure as the smallest closed set containing F. This always exists (albeit it can be infinite in size, if the AC contains an infinite diversity of molecules) and it is unique; we will indicate this as GC(F). Given two closed sets, we can define the closed union (⨆C) as the closed set generate by the union of the two sets. Thus ∀A, B, closed sets; A ⨆C B ≡ GC ( A∪B ). Similarly we can define, A ⨅C B ≡ GC ( A∩B ). And then if C is the set of Closed Sets, <C, ⨆C, ⨅C> will be a Lattice. Given any set F we can generate its semi-self-maintenance subset GsSM(F) as the biggest semi-self-maintaining subset contained in F. Again this exists and is unique. In a Flow System (but not in always in a reaction system), if we take a closed set C and we apply the operator GsSM(C) the result is still closed. So given any set F we can associate a semi-organisation to it as GsO (F) = GsSM (GC (F) ). This can be expanded by defining the operator Generate Self Maintaining set: GSM (T) as the operator that returns the biggest Self Maintaining Set contained in T. Applying the equivalent definition of union and intersection among semi-self-maintaining sets, self maintaining sets, semi-organisations, and organisations, we obtain that in Flow Systems the set of semi-self maintaining sets, self maintaining sets, semi-organisations and organisations are all lattices with their respective union and intersection. Thus if O is the set of organisations in a Flow System <O, ⨆O, ⨅O> is a lattice. But not necessarily in a reaction system. We shall now explore the consequences of this. Note: In the rest of the paper we shall use ⨆ to mean ⨆O and ⨅ to mean ⨅O.

Useful Theorems There are several theorems from Lattice theory which can help in mapping the lattice of organisations, and thus understanding better an Artificial Chemistry. Generally speaking if we have two organisations A, B then A ⨆O B = GSM (GsSM (GC (A ∪ B))). Those calculations can be long, so if we can shortcut the calculations just by working on the knowledge we already have this speeds up sensibly the work. In passing we note that in a Flow System for all sets A, although GSM (GsSM (A)) = GSM (A), it is usually faster to calculate it as GSM (GsSM (A) ) as |GsSM (A)| < |A|, and GsSM being an algebraic operator is usually faster to calculate than GSM.


Theorem 1: If A, B and C are elements in a Lattice <L, �, �>, then we know that (A � B) � C = A � (B � C). Now suppose that we have two organisations, S, C, and we are interested in calculating T = S � C. If we can express S as S = A � B then T = S � C = (A � B) � C =A � (B � C). And if we know the value of B � C (for example, B � C = R) then T = S � C = A � R. Similarly if we need to calculate A � R. Since R = B � C then S � C = A � R. So calculating the union of one of those relations will solve also the other. Same reasoning can be applied for the intersection.

Theorem 2: Suppose we have three organisations, A, B and C with A contained into B and B contained into C (thus A < B < C ). And let R be a fourth organisation. If we want to calculate B � R; and we know that A � R = C � R then B � R = A � R. In other words if we have two organisations, A and B and, when interacting with an external organisation R both will give a particular result S (that is S = A � R = C � R), then every other organisation in between A and C if united with R will be equal to S.

Theorem 3: Suppose we have two organisations, A, S with A < S then A � S = S then we can consider that given any other organisations B, such that A � B = S, for every organisation C such that A < C < S then C � B = S. Symmetrically for every D such that B < D < S, A � D = S. But since C � B = C � S, and B < D < S then C � D = S. So once we calculate a single union we can fill in several entries of the Union Table. The same symmetric reasoning, with exactly the same dual theorem, can be applied for the intersection. We will now use those theorems to find the lattice of all organisations.

Finding the Lattice of Organisations Let M be a set of Molecules, with a reaction * such that � a, b � M: a*b � M � �. Let <M, *> be an Artificial Chemistry. Let <O, �, �> be the lattice of Organisations over <M, *>. We need to find all the organisations in O. Let us start by assuming that we have two basic organisations, the empty set

�, and M: �, M � O. Those will be the top and the bottom organisations in the lattice. Every lattice with a finite number of elements has always a top element (the union of all the elements, also called the 1) and a bottom element (the intersection of all the elements, also called the 0). So we start by taking the bottom element as the organisation generated by the empty set, and the top element as the organisation generated by the set of all molecules. One obvious solution to find all organisations is to check every single subset of M, which means checking 2|M| subsets. If M is big this could be impractical or simply impossible. Let’s look at ways in which we can exclude some sets without testing them. In fact without even listing them. To reach O we are going to build a chain of sub-lattices of organisations, N0 < N1 < … < Nm = O, with N0 the sub-lattice generated by the empty set and by the set of all molecules: N0=<{G0(�),GO(M)},�,�>. Note that given a set of organisations P, it is trivial to build a sub-lattice out of it just by taking the closure respect to � and �. Thus once we have the sub-lattice Ni, and we find another organisation H, we can calculate Ni+1 as GC�� (Ni � H). Where with GC�� (P) we indicate the set of all organisations that can be generated by recursively applying the organisation union, �, and the organisation intersection, �, between organisations in P and the organisations generated in this process. When we are doing this we also store two tables, the Table of Unions of Ni: T�i, and the Table of Intersections of Ni: T�i. In general for every A, B in Ni, T�i will store A � B and T�i will store A � B. And being Ni a sub-lattice, we know that both A � B, A � B � Ni. Thus we know that if the tables T�i T�iare complete with every union and interaction calculated, and the result is an organisation known we do not have to proceed further, and we have found a sub-lattice. It is important that this process of calculating all the organisation union and intersection is made as efficient as possible. In this we are helped by the fact that O is a lattice, and thus we can apply the theorems listed above.

Adding one molecule at a time Let us start by assuming we have a sub-lattice of organisations Ni, then � B � Ni, we need to test all the molecules that are not in B, so � e � M \ B, we need to study Be . Calculating 1

Be leads to 3 possible outcomes: 1) Be = B; 2) Be = C with e � C and B < C; 3) Be = C with e � C and B < C.

In the first case we will say that e, in the context of B, goes down (case 1); goes by the side (case 2); or goes up (case 3). So we go through all the molecules and we store for each molecule e, if in the context of B, e goes upward, downward, or sideward. We also store the generated organisation Be. For each new organisation H that we find, we expand Ni into Ni+1 by adding the organisation to the list of known organisations, and calculating the union and intersection closure. Of course, if B � {e} = C, with C already a known organisation, we don’t need to test Be at all, and we know we are in case 3.

Note: In this paper we will indicate briefly GO({e}�B) as Be, for every molecule e, and similarly if we need to test a set adding 1

two molecules, e, f, we will indicate GO({e}�{f}�B) as Bef.

BA

S

C D

Figure 1: Theorem 3. If A � B = S, any organisation C between A and S united with any organisation D between B and S will always produce S. We do not need to calculate C � D.


Once we have finished exploring what happens by adding one molecule to a specific organisation, we continue with the next organisation, still maintaining the memory of all the molecules that added to B go up, on the side, or down. Also note that if Be goes sidewards generating C, then Ce will go downward, so we do not need to test it. Once we have finished exploring for each known organisations (which included the new ones we might have found in the meantime), what happens when we add one molecule, we need to consider what happens when we add two molecules. This is where some shortcuts will be possible.

Adding more than one molecule at a time Let us start by saying that if we have two sets of molecules x, y, and two organisations B and C, with B < C, we do not need to test Bx if we can prove that Bx = Cy. Because we will already consider the organisation generated by Bx, when we consider the organisation generated by Cy. Let us suppose we take an organisation B, and two molecules e, f in M\B. Do we need to test Bef? We need now to look at the combinations depending if e or f of upward, downward or sideward: here we find a symmetric matrix with 6 different cases (see table 2): If e goes up, than Be = C thus for every f, Bef = Cf. So any organisation that would be found through Bef will be found through Cf. So if either e or f goes upward, we do not need to test Bef. This clarifies cases 1, 2 and 3. We will immediately state that we are going to test Bef in case that both e, f go downward. And this clarifies case 6.

We need now to explore case 4 and 5. Let us suppose that e goes sideward, while f goes downward. So we are in case 5; do we need to test Bef? Since Be goes sideward, there exist an organisation C such that Be = C with e ∉ C and B < C. So Bef will be equivalent to test Cef. And since C > B than we will simply test this as part of testing C. Finally let us consider the case where both e and f go sideward. In this case Be = C; Bf = D. Thus there exist an organisation H such that C ⨆ D = H. Now Bef = Hef, but B < H, so we do not need to test it as part of B. And the result is that we only need to test sets of molecules such that all the molecules in this set go downward (table 3).

To summarise: So far the algorithm proceeds as follows:

Let L0 be the starting sub lattice of organisations; let O be the set of organisations known; let Up, Down, Side be empty dictionaries; for each organisation B in O: for each m not in B ∪ Down[B] ∪ Up[B] ∪ Side[B]: calculate Bm; if Bm = B: Down[B].add(m); continue; if Bm not in O: calculate Li+1 from Li, Bm, using the theorems above; expand O with Li+1 \ Li; if m not in Bm: for each organisation in Li+1, C between B and Bm: Up[C].add(m) Down[Bm].add(m) Side[B].add(m) else: Up[B].add(m)

At this point a sub lattice Ln will be discovered with a general structure with all the organisations that can be reached by adding one molecule at a time, and applying the union and intersection operations. This is not the complete lattice yet. For each organisation O we need to add the organisations generated by adding to O to every possible set S made up with molecules from Down[O]. In this case we can distinguish in four possible results (one more than before):

1) downward case: OS = B; 2) upward case: S ⊂ OS; 3) sideward case: S ⊄ OS, S ∩ OS = ∅; 4) diagonal case: S ⊄ OS, S ∩ OS ≠ ∅.

The fourth case, the diagonal case, is the new one and happens when S is partially contained in OS. Again, as we build the sets S from the smallest to the biggest we do not need to check the organisation generated by any set that has a subset T such that O < OT. The proof follows the exact same structure as the proof above. So given any organisation discovered we need to test it by trying to expand it with the molecules not in it, one by one. And with every subset of the “downward” molecules. While we do not need to test a set of “downward” molecules if it contains any subset which is upward, sideward, or diagonal. So the obvious thing to do, is to start with the smaller organisations, with the smaller subsets, and slowly build our way up.

Table 2: Will adding two molecules at the same time to an organisation produce novelty? Depends on what each molecule does by itself. Six cases are possible.

casese goes

Upward Sideward Downward

f goes

Upward 1 2 3

Sideward 2 4 5

Downward 3 5 6

Table 3: If we consider two molecules at the same time and add them to an organisation, this can produce novelty that would not be found through other ways only in one case.

casese goes

Upward Sideward Downward

f goes

Upward ✓ ✓ ✓

Sideward ✓ ✓ ✓

Downward ✓ ✓ to test


The “No Organisation Left Behind” theorem. We need now to prove that once the algorithm is followed, every organisation in the lattice of organisations will be found. This can easily be shown. Suppose by contradiction that C is an organisation in the Lattice that has not been found with the algorithm. Suppose that B is the maximal organisation contained in C found by the algorithm. Such organisation is unique because if there were multiple maximal organisations, the union of them would also be contained in C and would be known because (as part of the algorithm) we are studying the union of all the known organisations. Similarly we assume that C is the minimal organisation unknown above B. So B < C, A maximal organisation known under C, C minimal organisation unknown above B. Now we need to show that following the algorithm we would discover C, against the hypothesis. Let S be the set of molecules in C, but not in B; S = C\B. Thus obviously BS = C. If we consider any subset T of S, B < BT < C. Since B is self-maintaining and C is closed, B � GO (BT) � C. Now, as part of the algorithm we explore one by one the subsets of S starting with the smaller one. For each subset T: either T goes upward letting C be discovered; or T goes sidewards or diagonal. But in this case there would exist an organisation D = BT, with B < D < C. Now if D is known, it would be against the hypothesis that B is Maximal, and if it is unknown, it would be against the hypothesis that C is minimal. So there cannot be a D such that D = BT. Thus for all T, T must go downward, eventually letting us test S and discovering C. Against the hypothesis. So this algorithm explores the full lattice of organisations. A preliminary version of the code is available at ( https://github.com/pietrosperoni/LatticeOfChemicalOrganisations )

Testing the Algorithm on the NTop To test out the system it was used the NTop Artificial Chemistry (Banzhaf, 1993, 1994). This artificial Chemistry uses boolean vectors of size 4 as molecules, which are then folded into 2x2 matrixes, to react. This results are 16 reacting molecules. One of those acts as an algebraic zero (0 * a = a * 0 = 0, for every a) and it is usually eliminated. With the 15 remaining molecules it is possible to obtain 54 organisations out of a space of 32.768 possible subsets. The Brute Force algorithm tests all those subsets. Instead the algorithm

developed above was applied. The first step is to take two trivial organisations and the top, o1, and the bottom, o2, were taken. Then the bottom one was expanded, by adding one by one the 15 molecules. The first molecules led to o3. But the 3 organisations (o1, o2, and o3) formed a sub-lattice so it was not possible to expand this further. Second, and third molecules also generated o3. The fourth molecule, added to o2, generated o4. But now it was possible to find o5 as o5 = o3 � o4. Again o1-o5 formed a sub-lattice. The next molecule generated o6 and permitted to find o7, o8 and o9. o10 lead to o11-o14; o15 all the way to o23. So each organisation found would usually bring others with it, easily calculated. Once the o1-o23 sub-lattice of organisations was found, all the organisations, that could be found by adding a single molecule to o2 had been discovered. Also each molecule tested was divided into downward, upward and sideward, thus simplifying the tests to do later on. Then the algorithm started expanding on those organisations. Expanding o3 did not discover any new organisation. As did o4, o5, …, o9. expanding o10 lead to o24 (and nothing else). Expanding o15 lead to o27, then to o32 and finally to o42. And then nothing else. At this point a sub-lattice was found where adding a single molecule to each of the known 42 existing organisation would always lead back to a known organisation. o1-o42 was not just any sub-lattice, but a sub lattice that could not be pierced by adding a single molecule at a time. Then the algorithm started adding 2 molecules at a time. Expanding o6 it was possible to find o43 which then interacting with the other organisations generated o44 to o54. It was important to follow the algorithm, not just to understand it better, but because it showed a number of informations about the lattice. First and foremost the fact that the lattice has indeed a number of sub-lattices. Each new organisation found permit us to expand the space of the known organisations to the next sub-lattice in a chain that leads to the complete lattice. Although only 10 basic organisations calculated were necessary to generate the whole lattice, nothing tells us how to find those generators. Indeed finding a minimum number of generators, or just even any set of generators of organisations, is an open problem. It was also interesting to count how many relations among organisations needed to be calculated, and how many could be derived from the theorems. The results (figure 2) suggests that as the number of organisations grows the number of organisations that needs to be calculated drops following a

0

10

20

30

40

50

60

Expansions Steps0 1 2 3 4 5 6 7 8 9 10

Number of Organisations DiscoveredNumber of Extra Organisations Obtained

Figure 2: NTop Original: Number of organisations in each subsequent sub-lattice; the complete lattice, 54 organisations, was found in 10 subsequent expansions.

1%

10%

100%

Number of Organisations in the Sublattice

1 10 100

Relations calculated by hand

Figure 3: NTop Original: Percentage of Union or Intersection calculated by hand as opposed to extrapolated with the theorems.


power law (making a straight line on a log-log plot). While the remaining relations can all be derived theoretically.

Testing the Algorithm on the Expanded NTop

To try the algorithm on a more challenging artificial chemistry, it was applied to an expansion of the NTop. This time molecules of size 9 were used, which then were folded as 3x3 molecules. This gave 512 molecules. The folding can be done in 9! ways, and this both for the molecule on the right side and on the left side, thus producing 9! * 9! possible Artificial Chemistries. Then the resulting matrix will contain numbers between 0 and 3, and this will be mapped onto the set {0, 1}. This mapping can be done in 24 possible ways. The resulting boolean 3x3 matrix must then be unfolded, and this also can be done in 9! ways. So in total there are 24 * (9!)3 =1.9 * 1017 possible Artificial Chemistries. Many of those chemistries produce only a trivial lattice of organisation. For example a lattice that would only contain few organisations, or where every set was an organisation. In our case, if we consider the molecule �a = [a1, a2, a3, a4, a5, a6, a7, a8, a9], the molecules was folded as

[a1, a4, a7] [a2, a5, a8] [a3, a6, a9]

when it would react on the left side, and as [a1, a2, a3] [a4, a5, a6] [a7, a8, a9]

when it would react on the right side of the reaction. The result is then transformed according to the map: f(x) = {0�0; 1�1; 2�1; 3�0} and the resulting boolean matrix was unfolded so that from the matrix: [a1,1, a1,2, a1,3]

[a2,1, a2,2, a2,3] [a3,1, a3,2, a3,3]

the vector [a1,1, a1,2, a1,3, a2,1, a2,2, a2,3, a3,1, a3,2, a3,3] was produced. As with the NTop, the algebraic zero, 0 = [0,0,0,0,0,0,0,0,0] was excluded. And in this case also the molecule 1 = [1,1,1,1,1,1,1,1,1] was excluded. The result is an AC with 510 possible molecules, 2510 possible sets of molecules; more than 10153 sets to test. Obviously too many to directly test them all.

Results The algorithm described in this paper was applied. For now it was not possible to find all organisations. As with the NTop the algorithm started with the lattice which just included the empty and the complete organisations. Then it expanded the set of organisations going through 29 sub-lattices of respectively of 3, 5, 9, 17, 33, 84, 107, 133, 173, 238, 365, 672, 1604, 1612, 1703, 1978, 2066, 3284, 3522, 3557, 4711, 4713, 9377, 9641, 10090, 10196, and 10288 organisations (figure 4). After which no new organisation was found expanding the empty organisation. So 10288 organisations were found just by adding 29 times a single molecule to the empty organisation. This created also two symmetric tables with all the intersection and unions. As the algorithm went on those tables were completed more and more using only the theorems. Again as the number of known organisations grew the percentage of organisations that needed to be calculated by hand decreased following a power law (figure 5). As such the more the lattice is known, the more powerful those theorems are to find the remaining organisations. Of the 510 molecules 29 permit us to find all the organisations that could be generated by the empty set. Then those organisations started to be expanded themselves. After the 10288 organisations sub lattice was found the algorithm tested one by one each of those organisations, and for the first 83 organisations, adding a single molecule would keep generating well known organisations. Then on the 84th organisation a molecule was added that expanded the sub lattice, from 10288 to over 69000. And then at 69779 the system could not handle the data using more than 150 gigabyte of RAM and crashed. It should be noted that in a space of 2510 molecules it was realistically impossible to map the whole space of all the organisations.

Testing the Algorithm against the Brute Force As a final test the Brute Force Algorithm was applied to this Artificial Chemistry. On a home laptop it only found 263 Organisations before crashing. And the average time to find each organisation was 42 seconds. While the home laptop could find 3000 organisations keeping an average of 0.2 seconds per organisation. In figure 6 the two averages are compared for the first 263 organisations. It should nevertheless be noted that both averages were growing

0.01%

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1 10 100 1,000 10,000 100,000

Relations calculated by hand

Figure 5: Percentage of Union or Intersection calculated by hand as opposed to extrapolated with the theorems.

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Expansions Steps0 5 10 15 20 25 30

Number of Organisations DiscoveredNumber of Extra Organizations Obtained

Figure 4: Number of organisations in each subsequent sub lattice.


although the difference kept increasing. Also it is important to remember that the Brute Force only returns a set of organisations. No informations about the relative relations between the organisations, is returned. What organisations are above or below which others; what is the union or intersections of two organisations, is an information which is equally missing. All information that the Lattice Algorithm can easily return as it needs them to compute the lattice. So not only is the Brute Force much slower and generally un-efficient algorithm. But it also is insufficient to really let us know an artificial chemistry.

Consequences and Conclusions Reaction networks appear everywhere. And in their exploration the study of organisations, and their lattice is a necessary step to really understand their global behaviour as constructive dynamical system. The fact that organisations form a Lattice permit us to compute them faster. Although several results claim to be able to find all the organisations of a reaction network (Centler at al 2008, Centler at al 2010) they never used the algebraic properties of the Artificial Chemistry (namely that it is a lattice). Another important aspects that was uncovered in this work is the concept of sub-lattice, as a subset of organisations such that each union and intersection is an organisation inside the sub-lattice. From mathematics we know that if we consider the space of all sub-lattices of a lattice, they form a lattice, too: the lattice of sub-lattices of the lattice of organisations. Such lattice is not explored here, instead we merely extract a chain inside such lattice of sub-lattices and use this to build the lattice of organisations. Interestingly on the NTop system we could identify a sub lattice of organisations that was closed respect to the operation of adding a single molecule to any organisation. This shows that if we must find all the organisations, we cannot simply consider adding 1 molecule. But it also means that in some situations we are not interested in the complete lattice of organisations, but only in a sub-lattice since the the artificial chemistry will only explore that. For example in a Flow System both organisations and semi organisations form a lattice. But while each organisation is a semi-organisation the opposite is generally not true. Thus the lattice of organisations is a sub-lattice of the lattice of semi-organisation. And this fact could be used to map it. Similarly, Artificial Chemistries are not the only case of lattice present in the fields of Bioinformatics, Artificial Life and

Systems Biology. Researchers looking at autocatalytic cycles and closed sets also are looking at sets of molecules that form lattices. Thus the same procedures that were exposed here, and the same theorems, can be applied over there, with comparable results. Every research clarifies some elements, while opening new questions. - In particular it is still unclear what is the most effective way to apply the lattice theorems to study the lattice of organisations. Yes, theorems can shortcut the calculations, and we could see that the algorithm was at least three orders of magnitude faster than the Brute Force algorithm (and the difference was increasing), but finding which theorems can be applied can be time consuming as well. So a smart strategy might need to be applied to chose when to try to apply the theorems, as a further improvement of the algorithm. - Also it is unclear why the number of times the theorems are not applicable follows so closely a power law. This might be related to the nature of the Lattice of Organisations as a graph. But the details are still missing. - The roles of sub-lattices, what is the sub-lattice of sub-lattices, and how the sub-lattices of the lattice or organisations can be used to study an ecology of different experiments on one artificial chemistry is also an open question. - And finally the artificial chemistry presented here is very

vast and exploring it all is at the moment impossible. It could as such be used as a benchmark for future work.

Acknowledgement: The author acknowledges support from the European Union through funding under FP7–ICT–2011–8 project HIERATIC (316705). Also thanks Peter Dittrich for personal remarks and the two anonymous reviewer for fundamental comments.

References Banzhaf W. (1993) Self-replicating sequences of binary numbers.

Foundations II: Strings of length N- 4 Biol. Cybern, 69, pg 275-281 Banzhaf W. (1994) Self-replicating sequences of binary numbers.

Foundations I: The Built Up of Complexity. Complex Systems, 8, pg 215-225

Centler, F, Kaleta C., Speroni di Fenizio, P; Dittrich, P (2008) Computing chemical organizations in biological networks. Bioinformatics 24 (14): 1611-1618.

Centler, F.; Kaleta C.; Speroni di Fenizio, P.; Dittrich, P. (2010) A parallel algorithm to compute chemical organizations in biological networks. Bioinformatics Applications Note 26 (14), pg 1788 - 1789

Dittrich, P., Speroni di Fenizio, P (2007). Chemical organization theory. Bull. Math. Biol., 69(4): pg 1199-1231

Fontana, W. and Buss, L. W. (1994). ’The arrival of the fittest': toward a theory of biological organization. Bull. Math. Biol., 56: pg 1-64

Fontana, W. and Buss, L. W. (1994). What would be conserved if ‘the tape were played twice’?. Proc. Natl. Acad. Sci. USA, 91: pg 757–761

Speroni di Fenizio, P.; (2007). chemical organizations theory. Doctoral thesis, Chair of Systems Analysis, Department of Computer Science, University of Jena, Germany, 2007

Fontana, W. and Buss, L. W. (1996). The barrier of objects: from dynamical systems to bounded organizations. Casti J. and Karlqvist A., editors, Boundaries and Barriers, pg 56–116, Addison-Wesley

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Figure 6: Average time needed to find an organisation, for the first 263 organisations as the number of organisations grows.


Please cite this article in press as: Henze, R., et al., Structural analysis of in silico mutant experiments of human inner-kinetochorestructure. BioSystems (2014), http://dx.doi.org/10.1016/j.biosystems.2014.11.004

ARTICLE IN PRESSG ModelBIO 3530 1–13

BioSystems xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

BioSystems

jo ur nal home p age: www.elsev ier .com/ locate /b iosystems

Structural analysis of in silico mutant experiments of humaninner-kinetochore structure

Richard Henzea,1, Jan Huwalda,1, Nelly Mostajoa, Peter Dittricha,!, Bashar Ibrahima,b,c,!!Q1

a Bio Systems Analysis Group, Institute of Computer Science, Jena Centre for Bioinformatics and Friedrich Schiller University Jena, 07743 Jena, Germanyb Umm Al-Qura University, 1109 Makkah Al-Mukarramah, Saudi Arabiac Al-Qunfudah Center for Scientific Research (QCSR), 21912 Al-Qunfudah, Saudi Arabia

a r t i c l e i n f o

Article history:Received 26 March 2014Received in revised form 20 August 2014Accepted 4 November 2014Available online xxx

Keywords:Rule-basedModelingSimulationInner-kinetochore structureBond mutationStructural analysis

a b s t r a c t

Large multi-molecular complexes like the kinetochore are lacking of suitable methods to determine theirspatial structure. Here, we use and evaluate a novel modeling approach that combines rule-bases reactionnetwork models with spatial molecular geometries. In particular, we introduce a method that allows tostudy in silico the influence of single interactions (e.g. bonds) on the spatial organization of large multi-molecular complexes and apply this method to an extended model of the human inner-kinetochore. Ourcomputational analysis method encompasses determination of bond frequency, geometrical distances,statistical moments, and inter-dependencies between bonds using mutual information. For the analy-sis we have extend our previously reported human inner-kinetochore model by adding 13 new proteininteractions and three protein geometry details. The model is validated by comparing the results of insilico with reported in vitro single protein deletion experiments. Our studies revealed that most simula-tions mimic the in vitro behavior of the kinetochore complex as expected. To identify the most importantbonds in this model, we have created 39 mutants in silico by selectively disabling single protein interac-tions. In a total of 11,800 simulation runs we have compared the resulting structures to the wild-type.In particular, this allowed us to identify the interaction Cenp-W-H3 and Cenp-S-Cenp-X as having thestrongest influence on the inner-kinetochore’s structure. We conclude that our approach can become auseful tool for the in silico dynamical study of large, multi-molecular complexes.

© 2014 Elsevier Ireland Ltd. All rights reserved.

1. Introduction

During cell-cycle, accurate DNA segregation is mediated bythe kinetochore, a multi-protein-complex that assembles at thecentromere of each sister chromatid. Its dysfunction can causeaneuploidy and may lead to the development of cancer (Ciminiand Degrassi, 2005; Suijkerbuijk and Kops, 2008). A single kineto-chore complex contains over 100 proteins (Chan et al., 2005). Theseproteins can be grouped into two regions: the inner-kinetochore,which is tightly associated with the centromere DNA, and theouter-kinetochore, which interacts with microtubules. Kinetochore

! Corresponding author. Tel.: +49 3641 9 46460; fax: +49 3641 9 46302.!! Corresponding author at: Bio Systems Analysis Group, Institute of ComputerScience, Jena Center for Bioinformatics and Friedrich Schiller University Jena, 07743Jena, Germany. Tel.: +49 3641 9 46460; fax: +49 3641 9 46302.

E-mail addresses: [email protected] (R. Henze), [email protected](J. Huwald), [email protected] (N. Mostajo), [email protected](P. Dittrich), [email protected] (B. Ibrahim).

1 These authors contributed equally to this work.

structure and function change during the cell cycle (Varma et al.,2013). Thus it is necessary to learn about the coherence of itsstructure and function. Furthermore, the structure itself and its sta-bility are of special interest. The outer-kinetochore is thought tobe structurally unstable and formed in early mitosis (Cheesemanand Desai, 2008; Maiato et al., 2004) while the inner-kinetochoreis more stable and present during the entire cell cycle (Black andCleveland, 2011; Dalal and Bui, 2010). The inner-kinetochore com-plex includes a centromeric H3, as well as its variant Cenp-A (Cse4in budding yeast, Cnp1 in fission yeast, and CID/CenH3 in fruit flies),and 17 CCAN proteins (Cenp-B, Cenp-C, Cenp-H, Cenp-I, Cenp-Kto Cenp-U, Cenp-W and Cenp-X) (Okada et al., 2006). These 19proteins build a bridge between two histone octamers (cf. Fig. 1)(Tschernyschkow et al., 2013). The DNA wraps around the histoneoctamer forming a nucleosome.

In our study we are interested in the spatial behavior ofthe bridges that form. Studying the 3D structure of the inner-kinetochore is challenging: experimentally it is difficult becausethe average diameter of a CCAN protein is 40 A and connec-tions between them are not visible through a microscope. Also,

http://dx.doi.org/10.1016/j.biosystems.2014.11.0040303-2647/© 2014 Elsevier Ireland Ltd. All rights reserved.

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Fig. 1. Interaction graph of centromere proteins. The graph shows all centromere proteins (round vertices), the nucleosomes (square vertices) and all their possible interactionsQ2(edges) according to the presented model. The graph topology is derived from the literature (cf. Table 1). The edge annotations display the importance of each interaction ascomputed in this paper: Thin lines denote bonds considered insignificant: their deletion did not influence the assembly of the inner-kinetochore. Thick lines denote the mostinfluencing interaction for every protein. Considering the graph with only thick lines, the blue lines denote terminating bonds while the red lines are the non-terminatingbonds. This implies that the red lines are the most important ones for the formation of the inner-kinetochore. These results are discussed detailed in Section 2.3. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

theoretical analysis methods are hindered by the combinatorialexplosion of the amount of intermediate complexes and proteinassembly states (Görlich et al., 2013; Tschernyschkow et al.,2013). Classical in silico modeling approaches based on explicitrepresentations of all intermediate complexes, such as differentialequations, cannot cope with that combinatorial explosion. Arecent S-Phase inner-kinetochore model by Tschernyschkow et al.(2013) overcomes this hurdle using an implicit representation,combining a rule-based language with a particle based simula-tion approach (Gruenert et al., 2010). This model incorporatesmolecular geometric information, in contrast to classical modelsapplied so far to cell-cycle mechanisms (e.g. Caydasi et al., 2012;Doncic et al., 2005; Ibrahim et al., 2008b, 2009, 2008a, 2007;Kreyssig et al., 2012, 2014; Lohel et al., 2009; Rohn et al., 2008).The spatial rule-based approach with an application to mitotickinetochore has been recently discussed in detail by Ibrahim et al.(2013). In this study, we extend the S-Phase inner-kinetochoremodel by Tschernyschkow et al. (2013). We include 13 additionalinteraction data from the literature and molecular geometries forCenp-A, Cenp-C, and Cenp-T. While Tschernyschkows’ model isbuilt mainly on FRET (Fluorescence Resonance Energy Transfer)and F3H (Fluorescent Three Hybrid) measured interactions (cf.Eskat et al., 2012), our new literature data also contain exemplaryNMRS (Nuclear Magnetic Resonance Spectroscopy, cf. Kato et al.,2013) measurements. All CCAN protein interactions or proximities,measured with any mentioned method (e.g. FRET, F3H, NMRS), areincluded in our simulation. This implies that any kind of source canbe translated to rules for our simulation software, which results ina model defining the particles dynamics (cf. Section 4.1).

The conjectured kinetochore structure is obtained by simulat-ing the aggregation of all 19 randomly distributed CCAN proteinsfor a sufficiently long time. With all the interactions included thisresults in our wild-type structure. Therefore, given different ran-dom initial conditions, the model generates not one but a set ofstructures. For analyzing these structures we build upon the tech-niques detailed in Tschernyschkow et al. (2013) and Ibrahim et al.

(2013). To identify them as (dis)similar we used: bond frequency,relative particle positions, and moments of each particles posi-tion. In addition, mutual information to identify protein clusters,forming the same bonds in every simulation run. See Section 4 fordetails.

Assuming kinetochores loss of function occurs only if the struc-ture changes significantly, we showed that all generated wild-typesstructures are highly similar. Also, for validating our model, wecompared it with in vitro mutation experiments. Therefore, wegenerate mutant simulations for all single-protein deletions. Foreach mutation we check if it generates the same structures as thewild-type model. We show that the mutation-function mappingobtained by this procedure is largely in accord with in vitro experi-ments. Many inner-kinetochore proteins participate in a relativelylarge number of reactions: Cenp-U has 9 interaction partners andCenp-R has 6 (cf. Table 1). We want to identify the relevance ofindividual bonds to gain a functional understanding of the kine-tochore. Protein deletion in our model is simulated through theremoval of its interactions. This reveals protein effect on the com-plex structure, but not which interaction affects it. Thus, we addnovel mutation experiments in which we disable single interac-tions and compare the resulting structure with the wild-type.

In the following, we discuss the congruence between our insilico protein deletion experiments and the results from the liter-ature. Subsequently the simulated bond deletion experiments arepresented. Also shown are isolated interactions with the highestinfluence on the structure of the inner-kinetochore (cf. Fig. 1).

2. Results and discussion

The analysis performed was to study the inner-kinetochoreduring S-phase in order to accomplish: (1) wild-type behaviorto determine a reliable structure and function while includingthe new data available; (2) protein deletion for comparing within vitro results and validating the model; (3) bond deletion fora better understanding of the function of each interaction in

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Table 1Interactions of the inner-kinetochore components.

Protein Basic model (Tschernyschkow et al., 2013) Additional reactions

Cenp-A Cenp-B (Orthaus et al., 2008), Cenp-N (Hellwiget al., 2011)

Cenp-C (Carroll et al., 2010; Kato et al., 2013; Przewloka et al., 2011)

Cenp-B Cenp-C (Suzuki et al., 2004), Cenp-W, -Q (Eskatet al., 2012) Cenp-U (Hellwig et al., 2009)

Cenp-C H3 (Ando et al., 2002; Hori et al., 2008a; Obuse et al., 2004)Cenp-H Cenp-I, -K, -L, -M, -N (Hori et al., 2008a)Cenp-I Cenp-U (Hellwig et al., 2009)Cenp-K Cenp-N (Eskat et al., 2012), Cenp-O, -R, -U

(Tschernyschkow et al., 2013)Cenp-L Cenp-R, -U (Eskat et al., 2012)Cenp-M Cenp-S, -U, H3 (Tschernyschkow et al., 2013)Cenp-N Cenp-R (Eskat et al., 2012)Cenp-O Cenp-P, -U, -Q (Eskat et al., 2012) Cenp-R (Eskat et al., 2012; Hori et al., 2008b)Cenp-P Cenp-Q, -U (Tschernyschkow et al., 2013)Cenp-Q Cenp-U (Tschernyschkow et al., 2013) Cenp-R (Eskat et al., 2012; Hori et al., 2008b)Cenp-R Cenp-U (Tschernyschkow et al., 2013)Cenp-S Cenp-T (Nishino et al., 2012), H3, Cenp-X

(Amano et al., 2009)Cenp-T H3 (Hori et al., 2008a) Cenp-W (Hori et al., 2008a; Nishino et al., 2012)Cenp-UCenp-W H3 (Nishino et al., 2012)Cenp-XH3

The table contains all interactions of the inner-kinetochore components. The interactions in the second column are taken from the basic inner-kinetochore model(Tschernyschkow et al., 2013) while the new interactions are shown in the third column.

the complex; and (4) mutual information for determining thecorrelation between a pair of bonds.

These four gave the results detailed below. The model, structureanalysis methods and mutual information analysis is elaborated inSection 4.

2.1. Wild-type behavior

Extending the existing model (Tschernyschkow et al., 2013)with additional interaction rules and a detailed structure of Cenp-A, Cenp-C and Cenp-T (cf. Fig. 2) did not affect its functionality,meaning that a bridge between the two nucleosomes was formed(cf. Fig. 3). The phylogram of all wild-type structures revealed twoclusters (see Fig. 8, Panel A), each representing another orienta-tion of Cenp-C (switched N and C termini). This is in contrast tothe published model (Tschernyschkow et al., 2013) with three dif-ferent families of structures, where each misses one of the threebonds Cenp-A-Cenp-B, Cenp-B-Cenp-W or Cenp-W-H3. In our newmodel all of these three bonds got realized in at least 140 out of 200simulation runs.

2.2. Protein deletion

We generated all single protein deletion mutants for each inner-kinetochore protein (except histone proteins H2A, H2B and H4).To keep the particle count constant during the simulation, theremoval of the protein is implemented by disabling all its pos-sible reactions, turning it into a freely moving particle. For eachmutant we simulated 200 instances and analyzed the final stateof the simulations using the techniques detailed in Section 4.3.Known protein-mutation-phenotypes from the literature allowedus to compare and eventually validate the in silico with the in vitroresults. We consider a protein as stabilized when the variance of itsposition decreases in the mutated structure, in comparison to thewild-type. This is a measure for decreased motility within the com-plex. Loosening a protein denotes the reversed effect. The resultsare summarized in Table 2.

Cenp-A: The kinetochores formation is dependent on the pres-ence of the centromere-specific nucleosome that contains the H3variant, Cenp-A (Bergmann et al., 2011; Howman et al., 2000;

Obuse et al., 2004; Stoler et al., 1995; Sullivan et al., 1994; Toppet al., 2004). Also, bridges between nucleosomes require Cenp-Ato be formed (Tschernyschkow et al., 2013). During S-phase theCenp-A-nucleosome is considered to be “the sole epigenetic markof centromere” (for review see Perpelescu and Fukagawa (2011),Quenet and Dalal (2012)). Deleting Cenp-A in our model loosensthe majority of the inner-kinetochore components (Cenp-B, -C, -H,-I, -K, -L, -M, -N, -O, -P, -Q, -R, -S, and -U). These results are in con-cert with mentioned experimental findings (Bergmann et al., 2011;Howman et al., 2000; Obuse et al., 2004; Stoler et al., 1995; Sullivanet al., 1994; Topp et al., 2004). Also, we found that the location ofCenp-W to the H3-nucleosome gets stabilized (cf. Fig. 3).

Cenp-B: Cenp-B binds specifically to the major DNA componentof human centromere hetero chromatin. It has been suggested thatCenp-B plays a role in the organization of the chromatin structure(Muro et al., 1992; Yang et al., 1996). In our model, deleting Cenp-B destabilizes the localization of many inner-kinetochore proteins,namely: Cenp-I, -L, -M, -O, -P, -Q, -R, and -U. In accordance withits strong connection to hetero chromatin, the results mean thatCenp-B may be a nexus to inner-kinetochore proteins.

Cenp-C: Cenp-C is an evolutionarily conserved crucial kineto-chore protein, because its depletion influences the level of Cenp-Hand -I complex and also the assembly of the outer-kinetochore(Fukagawa and Brown, 1997; Kalitsis et al., 1998). Our analysisshows that deleting Cenp-C stabilizes Cenp-H and not Cenp-I. Addi-tionally, the complex formation between Cenp-H and Cenp-I isnot affected: the distance changes from 56 to 57 A. The standard-derivation within the wild-type simulations of 17 A implies nosignificant alteration. We additionally found that Cenp-C destabi-lized the localization of both Cenp-B and Cenp-W. Meaning Cenp-Cmight not be just strongly needed to form the outer-kinetochore,but also to assemble Cenp-B and Cenp-W to the complex.

Cenp-K: Cenp-K influences many outer-kinetochore compo-nents like Ndc80 and KNL1 (Cheeseman et al., 2008) but not theinner-kinetochore. Deleting Cenp-K in our model results solely instabilizing Cenp-N. This could be due to the reason that Cenp-Nis an anchor for the outer-kinetochore proteins Ndc80 and Knl1(Hellwig et al., 2011).

Cenp-H/I: Cenp-H and Cenp-I form a sub-complex in the inner-kinetochore (Perpelescu and Fukagawa, 2011). Depleting either

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Fig. 2. Sub-particle configuration for chain proteins Cenp-A, -C, and -T. The proteins Cenp-A, Cenp-C, and Cenp-T are modeled as chains of bonded sub-particles. Sub-particlesare of equal size, except for the heads of Cenp-A and Cenp-T. The graph is drawn to scale. Particles are annotated with their radii.

Fig. 3. Inner-kinetochore structure in 3D, completely assembled and mutated. Left: Wild-type simulation, where all inner-kinetochore proteins assemble between the twonucleosomes (cube shaped templates, consisting of eight histone molecules, A-Nucl and H3-Nucl). A bridge forms in all simulation runs, nevertheless not all bonds gotrealized. Right: Simulation with deleted protein CENP-A (removed all its interactions). No bridge is formed in any of the 200 runs while all CCAN proteins assemble aroundthe H3 containing nucleosome. Colors of the protein correlate with the introduced colors in Fig. 1. Cenp-A, Cenp-C and Cenp-T chains are visible (rose, orange and purple).CCAN proteins which cannot be seen due the sterical restrictions are without an arrow. (For interpretation of the references to color in this figure legend, the reader is referredto the web version of the article.)

Table 2Results of protein deletion experiments (in vitro and in silico).

Protein In vitro effects In silico effects

On specific proteins DKL DE (10"2) Split cophylogram

Cenp-A Complete destabilization ofinner-kinetochore

Cenp-B, -C, -H, -I, -K, -L, -M, -N,-O, -P, -Q, -R, -S, -U and -W

"0.71 8.43 Yes

Cenp-B Organization of chromatinstructure

Cenp-I, -L, -M, -N, -O, -P, -Q, -R,-U and -W

"0.46 7.87 Yes

Cenp-C Assembly of outer-kinetochore Cenp-B, -H, -K, -L, -M, -N, -R -Uand -W

"0.67 11.08 Yes

Cenp-H No effects on Cenp-A Cenp-I, -N and -U "0.49 6.69 NoCenp-I No effects on Cenp-A "0.20 4.74 NoCenp-K Outer-kinetochore

components, notinner-kinetochore

Cenp-N "0.57 5.06 No

Cenp-L Outer-kinetochore/spindle "0.43 5.83 NoCenp-M Decreases Cenp-H/I level Cenp-H, -I, -K, -L, -N, -O, -P, -Q

and -U"0.65 9.12 Yes

Cenp-N Kinetochore assembly defects Cenp-B, -H, -K, and -R "0.19 6.49 NoCenp-O Recovery of spindle damage Cenp-L "0.29 4.98 NoCenp-P Recovery of spindle damage "0.21 4.69 NoCenp-Q Recovery of spindle damage Cenp-B, -M and -W "0.53 5.37 NoCenp-R Recovery of spindle damage Cenp-M "0.30 6.85 NoCenp-S Decreasing kinetochores

volumeCenp-H, -I, M, -U and -X 0.35 11.05 Yes

Cenp-T Cenp-M "0.11 6.49 NoCenp-U Stable

kinetochore-microtubuleattachment, stabilize Cenp-A

Cenp-B, -H, -I, -L, -M, -N, -P, -Q,and -W

"0.22 8.36 Yes

Cenp-W Cenp-B "0.04 7.82 YesCenp-X Decreasing kinetochores

volume"0.37 5.50 Yes

H3 Cenp-C, -H, -I, -K, -L, -M, -N, -O,-P, -Q, -R, -S, -T, -U, -W and -X

"2.39 8.50 Yes

Outcome of protein deletion experiments listed per protein. Column 2 lists effects of in vitro experiments from the literature. The proteins listed in column three changed theirpositional variance significantly (more than 30% deviation) under the mutation. Columns 4 and 5 present the Kullback–Leibler divergence (DKL , cf. Eq. (4)) and the Euclidean

distance (DE =pPn

i=1#pi " qi#, where p and q are any two given vectors), between the mutation and the wild-type. The Last column shows whether the cophylogram is

mixed or separates mutant and the wild-type (human decision). We refer to the Supplementary Tables for more details.




Table 3Results of bond deletion experiments (in silico).

Protein In silico effects

On specific proteins DKL DE (10"2) Split cophylogram

Cenp-A-Cenp-B Cenp-B, -M and -W 0.07 1.41 NoCenp-A-Cenp-N Cenp-B, -H, -K, -M, -N, -S, -U and -W 0.01 1.42 NoCenp-A-Cenp-C "0.18 1.38 NoCenp-B-Cenp-C Cenp-C "0.15 1.45 NoCenp-B-Cenp-Q Cenp-B, -L, -N, -O, -Q, -R, -S and -W "0.64 1.69 NoCenp-B-Cenp-U Cenp-B, -H, -I, -M, -U and -W "0.04 1.69 NoCenp-B-Cenp-W Cenp-B, -H, -I, -M, -O, -T, -U and -W "0.49 1.73 NoCenp-C-H3 Cenp-B, -C, -N and -U "0.81 1.36 NoCenp-H-Cenp-I Cenp-I, -M and -S "0.14 1.67 NoCenp-H-Cenp-K Cenp-N, -S and -T "0.19 1.43 NoCenp-H-Cenp-L Cenp-B, -H and -W "0.05 1.46 NoCenp-H-Cenp-M Cenp-B, -H, -K, -L, -N, -R and -W "0.48 1.38 NoCenp-H-Cenp-N Cenp-B, -K, -L, -M, -N, -R and -W "0.60 1.64 NoCenp-I-Cenp-U Cenp-H and -I 0.10 1.55 NoCenp-K-Cenp-O Cenp-M and -W "0.11 1.14 NoCenp-K-Cenp-U Cenp-H and -M "0.12 1.71 NoCenp-K-Cenp-N Cenp-B, -H, -K, -M, -N, -S, -U and -W "0.25 1.29 NoCenp-K-Cenp-R Cenp-W "0.13 1.67 NoCenp-L-Cenp-R Cenp-L,-N -Q and -R "0.40 1.68 NoCenp-L-Cenp-U Cenp-L 0.03 1.26 NoCenp-M-Cenp-S Cenp-H, -I, -M, -N and -T "0.37 1.16 NoCenp-M-Cenp-U Cenp-B, -H, -I, -M, -P, -Q and -U "0.17 1.98 NoCenp-M-H3 Cenp-l, -M, -N, -P, -Q, -S and -U "0.37 1.26 NoCenp-N-Cenp-R Cenp-B, -H, -L, -M, -N and -R 0.38 1.94 NoCenp-O-Cenp-P Cenp-H, -M, -N and -S 0.03 1.16 NoCenp-O-Cenp-Q Cenp-B and -N "0.10 1.33 NoCenp-O-Cenp-R 0.09 1.17 NoCenp-O-Cenp-U Cenp-H, -M, -S, -U and -W "0.15 1.57 NoCenp-P-Cenp-Q Cenp-W 0.12 1.15 NoCenp-P-Cenp-U Cenp-M and -P "0.25 1.43 NoCenp-Q-Cenp-U Cenp-M and -S 0.04 1.10 NoCenp-Q-Cenp-R Cenp-B, -H, -L, -M, -N, -Q, -R, -S, -U, -W and -X 0.22 1.32 NoCenp-R-Cenp-U Cenp-R and -W "0.34 1.76 NoCenp-S-Cenp-T 0.10 1.55 NoCenp-S-Cenp-X Cenp-X "0.16 5.39 YesCenp-S-H3 Cenp-M, -S, -W and -X "0.24 1.41 NoCenp-T-Cenp-W Cenp-B, -M and -W 0.03 1.37 NoCenp-T-H3 Cenp-S 0.30 1.82 NoCenp-W-H3 Cenp-B, -M, -T and -W "1.03 1.35 Yes

Outcome for all applied analysis techniques (Bond frequency analysis, Distance Distribution, Phylogram and Statistical moments). For every bond deletion are effects of ourin silico experiments shown (column 2–5). The proteins listed in column 2 changed their variance significantly (more than 30% deviation) under the mutation. Column 3 and

4 present the Kullback-Leibler divergence (DKL , cf. Eq. (4)) and the Euclidean distance (DE =pPn

i=1#pi " qi#, where p and q are any two given vectors), between the bond

deletion and the wild-type (DE). Last column shows the human made decision whether the phylogram separates into the mutation and the wild-type ore not. We refer tothe Supplementary Tables for more details.

Cenp-H or Cenp-I does not affect the amount of Cenp-A (Okadaet al., 2006). This is irrelevant for our analysis, due to our simu-lations are restricted to one copy of every CCAN protein. Anotherstudy showed that removing any of Cenp-H/I/K has several effectsin the assembly of inner-kinetochore and chromosome alignment(Cheeseman et al., 2008; Okada et al., 2006). This is not repro-ducible in our model due the spindle assembly checkpoint signalingpathway is not included. Our analysis shows that deleting Cenp-Hinfluences Cenp-I and -U; while deleting Cenp-I has no influenceon the inner-kinetochore structure. This might be, because Cenp-H has more interactions with other CCAN proteins than Cenp-I (cf.Table 1).

Cenp-L: Cenp-L phenotype mechanism is still unclear, albeit ithas been demonstrated that Cenp-L depletion by small-interferingRNA (siRNA) induced monopolar spindles in most mitotic cells(McClelland et al., 2007). Our model does not show any effects tothe inner-kinetochore, which is in accordance with experimentalfindings.

Cenp-M: Cenp-M is another essential component of the kineto-chore because it is directly correlated with aneuploidy (Foltz et al.,2006; Izuta et al., 2006; Okada et al., 2006). It has been shown thatits depletion by siRNA affects Cenp-H and Cenp-I levels at cen-tromeres (Izuta et al., 2006). Deleting Cenp-M in our in silico model

resulted in a loosened localization of the components Cenp-H, -I, -P, -Q, -R and -U. Furthermore, our model shows that Cenp-Mdeletion is stabilizing Cenp-K, -L, and -N. Our model could mimicthe literature results from Izuta et al. (2006) and furthermore sug-gests that Cenp-M has a huge influence to the localization of someinner-kinetochore proteins.

Cenp-N: Depletion of Cenp-N leads to kinetochore assemblydefects, such as a reduced recruitment of Cenp-H, -I and -K. (Carrollet al., 2009; Foltz et al., 2006). Deleting Cenp-N in our model affectsthe localization of Cenp-H, -K, which is in concert with the experi-mental findings. Additionally, we found the stabilization of Cenp-Band -R.

Cenp-S/X: Cells lacking Cenp-S or Cenp-X show reduction in thesize of the kinetochore outer plate (Amano et al., 2009). Addition-ally, Cenp-S and Cenp-X forms stable complexes (Nishino et al.,2012; Takeuchi et al., 2013). Our model indicates that Cenp-Xlocalization depends on Cenp-S but not vice versa. Also, dele-tion of Cenp-S influences the localization of Cenp-H, -I, -M, and-U while Cenp-X deletion has no effect to any component of theinner-kinetochore. The size of the kinetochore has reduced with-out Cenp-S, because it is the only connection to Cenp-X. Thisimplies that the removal of Cenp-S also includes the removal ofCenp-X.

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A B

DC

Fig. 4. Influence of bond deletions to the mean and variance of every molecule. Shown is the distribution of the relative alteration from mean and variance (cf. Eqs. (8)–(11)).The alteration is regarding the wild-type results while the distribution is over all bond mutations. Green bars denote the median of the distribution (50th percentile), bluebars the minimum and maximum (0th and 100th percentile) and red bars present the 10th and 90th percentile. (A) Position on the rotation axis (x), and (C) distance to therotation axis (r). For Cenp-T and Cenp-C all particles of their chain are treated individually. Considering values above 1.3 and below 0.7 (relative change greater than 30%)reveals only one protein, fulfilling this criteria for the position x: Cenp-X, under the bond deletion Cenp-S-Cenp-X. For the distance r exceeds no mutation this threshold.The 10th and 90th percentile all are between 1.1 and 0.9 (1.02 and 0.96, respectively). Most of the bond deletions do not affect the mean significantly. Furthermore, thedistribution of the logarithmic relative alteration of variance for every molecules (B) position on the rotation axis (x), and (D) its distance to the rotation axis (r). For allCenp-C and Cenp-T molecules the 10th and 90th percentiles are close to the median, which is nearly unchanged. This can be interpreted as robustness of those proteins:only few bond deletions influence the variance of them. Also the minima and maxima are relatively low, compared with other proteins, e.g. Cenp-M or Cenp-W. For everyprotein minima or maxima (in all panels) deviate considerably from 1 (which corresponds to the wild-type value), meaning for every protein there exist bond deletions,decreasing and increasing the positional variance significantly. The alteration of the mean is not remarkable. This implies that no bond deletion exists, which may changethe overall structure, due to every protein has a nearly fixed position. Considering the variance as mobility of the molecules, suggests that some deletions relax the complex,while others brace it. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Cenp-U: Cenp-U (also known as MLF1IP or Cenp-50) is a compo-nent of CCAN due to its co-localization with Cenp-A throughout thecell cycle in human cells (Cheeseman and Desai, 2008; Hanissianet al., 2004; Okada et al., 2006). Cenp-U is required for stablekinetochore-microtubule attachment (Hua et al., 2011). Its deple-tion can cause a mitotic defect in chromosome attachment andchromosome alignment, but not affecting the spindle assemblycheckpoint (Foltz et al., 2006). Effects on the inner-kinetochore arenot known in literature yet. Intriguingly, we found that Cenp-U haslarge effects on many other protein’s localizations like Cenp-B, -H, -I, -L, -M, -P, -Q, and -W. This huge effect might influence theassembly of inner-kinetochore and thus, the proper attachment tomicrotubules.

Cenp-W/T: Cenp-W is a small protein (ca. 26.2 A diameter)which forms a complex with Cenp-T (Hori et al., 2008a). In con-trast, a recent finding shows the interdependency of localizationbetween Cenp-T and Cenp-W (Takeuchi et al., 2013). As these find-ings are for mitosis and not for S-phase, we could not obtain themduring our simulations. Instead we found that only Cenp-B local-ization is loosened by Cenp-W, while Cenp-M is stabilized under a

depleted Cenp-T. With the in silico findings of the Cenp-C mutation,this suggest an interdependency between Cenp-B and Cenp-W.

Cenp-O/P/Q/R: Cenp-O, -P, -Q, and -R group are important for therecovery of spindle damage (Hori et al., 2008b). Our model shows noeffects for the deletion of Cenp-P and minor effects for the deletionof either Cenp-O (destabilizing Cenp-L), Cenp-Q (stabilizing Cenp-B, -M and -W), and Cenp-R (stabilizing Cenp-M). Therefore one cansay that the single protein mutations do not influence the formationof the QPOR group.

This comparison of in vitro with in silico mutants shows thatour model is consistent with most of the currently known behav-ior of the inner-kinetochore. Thus, it can be used to approach andsimulate the real inner-kinetochore system.

2.3. Bond deletion

In the previous section, we identified the proteins Cenp-A, -B, -C,-M, -U, -W and H3 (cf. Table 2) as essential for the assembly of theinner-kinetochore and validated our model in accordance to real

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0

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1 P-O H3-CenpW

B-AC-AC-BI-H K-HL-HM

-HN-AN-HN-KO-KP-OQ-BQ-OQ-PR-KR-LR-NR-OR-QS-MT-SU-BU-IU-KU-LU-MU-OU-PU-QU-RW

-BW

-TX-SH3-CenpCH3-CenpMH3-CenpSH3-CenpTH3-CenpWAArm

-BAArm

-CAArm

-NTArm

-STArm

-WCenpTArm

-H3CArm

-ACArm

-BCArm

-CCenpCArm

-H3CArm

-AArmRe

lative

freq

uenc

y ove

r all s

imua

tlion

runs

Possible bond formations

Fig. 5. Bond realization frequency cb for wild-type and two mutants. Bar plot of the bond realization frequency cb for all bonds of the wild-type (black) and the mutants withdeleted Cenp-P-Cenp-O reaction (gray) and deleted H3-Cenp-W reaction (blue). In all cases, the frequency is sampled over all 200 simulations. In the shown example thedeletion of Cenp-P-Cenp-O has no significant influence to any other bond. However, the deletion of H3-Cenp-W changes eight other bonds significantly: Cenp-S-Cenp-M,Cenp-T-Cenp-S, Cenp-W-Cenp-B, Cenp-W-Cenp-T, Cenp-X-Cenp-S, H3-Cenp-C, H3-Cenp-S and H3-Cenp-T. This technique only reveals if the deleted bond influences therealization of other bonds, but makes no prediction about the actual spatial structure of the complex. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of the article.)

data. Now we identify those interactions that affect the assemblyof the inner-kinetochore the most.

For this, we created all single interaction mutants by remov-ing single bonds from the model, referred as a bond deletion. Thisexperimental setup has not been done in vitro yet. For each mutant200 instances has been analyzed and the final state compared withthe wild-type.

The cophylogram, a combined phylogram of a wild-type anda mutant, highlights changes to the overall structure induced bya bond deletion. Cophylogram analysis shows that there are onlytwo bonds whose deletion influences the overall structure of theinner-kinetochore significantly. In other words, the complex isrobust under 37 out of 39 single bond deletions (cf. Table 3).These structure-affecting bonds are Cenp-S-Cenp-X and Cenp-W-H3. Cenp-S-Cenp-X is the only interaction of Cenp-X with otherCCAN proteins. Removing this bond causes Cenp-X to move freelyin space, obviously a strong alteration of the overall structure.

To identify smaller (not visible in cophylogram) effects for indi-vidual proteins, we calculated the mean and the variance of thefinal position of each protein in every bond mutation. The meanof all protein did not change significantly under any mutation (cf.Fig. 4). This proves once more the stability of kinetochores compo-nents, while they are disturbed. For this reason we will concentrateon the variance analysis (cf. Table 3):

Removing the Cenp-Q-Cenp-R bond highly influences proteinlocalizations, by increasing significantly the variance of the major-ity of protein positions. Considering the variance as measure forthe stiffness of the complex, this implies that the bond betweenCenp-Q and Cenp-R is most important for the stability of the inner-kinetochore. In contrast, the deletion of either Cenp-H-Cenp-M orCenp-H-Cenp-N stabilizes the complex by decreasing the positions

variance of Cenp-B, -K, -L, -N, -R and -W. These interactions stressthe inner-kinetochore the most.

Furthermore, we identified from bond deletions, the least andmost affected proteins (cf. Table 3). The least affected proteins areCenp-O and Cenp-X because both are stable localized and theirvariance is only affected significantly by only two different bondmutations. On the contrary, Cenp-M localization is unstable in theinner-kinetochore structure, where 21 from 39 different bond dele-tions influence its positions variance. This is in accordance with ourstatement on Cenp-M protein mutation.

Next, we tried to identify the most critical bonds. In detail,we were looking for bonds that change the kinetochore structurein a similar way like the deletion of bonded proteins (e.g. thebond mutation Cenp-A-Cenp-B has similar effects like the proteindeletion Cenp-B). We reduced the set of candidates by removingthose with the weakest magnitude of influence to kinetochoresstructure. That way we reduce the interaction graph to the bondsthat are most necessary for the ensemble of the inner-kinetochore.From the 39 possible interactions 21 seem necessary to form thestable complex. The results are displayed in Fig. 1 and Table 4.

The bond frequency analysis reveals if a deleted bond promotesor prohibits the formation of any other bond. Deleting Cenp-W-H3promotes four other bonds significantly: Cenp-S-Cenp-T, Cenp-W-Cenp-T, Cenp-C-H3 and Cenp-T-H3. In addition, it represses therealization of Cenp-S-H3 and Cenp-S-Cenp-M. Due to the over-crowding around the H3 nucleosome, deleting Cenp-W-H3 reducesthe competition for space near the nucleosome. Thus all other inter-action partners of H3 (except Cenp-S) bind more frequently to thenucleosome. Other bond mutations had no significant influence tothe frequency of other bond formations. An example of this analysisis shown in Fig. 5.

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Table 4Most influential bond for every protein.

Protein Bond with the most influence

Cenp-A Cenp-A-Cenp-NCenp-B Cenp-B-Cenp-Q, -WCenp-C Cenp-C-H3Cenp-H Cenp-H-Cenp-N, -MCenp-I Cenp-I-Cenp-HCenp-K Cenp-K-Cenp-NCenp-L Cenp-L-Cenp-RCenp-M Cenp-M-Cenp-HCenp-N Cenp-N-Cenp-HCenp-O Cenp-O-Cenp-KCenp-P Cenp-P-Cenp-UCenp-Q Cenp-Q-Cenp-B, -RCenp-R Cenp-R-Cenp-N, -QCenp-S Cenp-S-H3, Cenp-MCenp-T Cenp-T-H3Cenp-U Cenp-U-Cenp-B, -MCenp-W Cenp-W- Cenp-BCenp-X Cenp-X-Cenp-SH3 H3- Cenp-M, -W

Every protein is able to interact with several other proteins (cf. Table 1). Our analysisrevealed the most influential bond for every molecule, according to the analysisoutcome (cf. Table 3). Whereby most influential means that its deletion affects inner-kinetochores structure most, compared with the other possible bonds.

These results show a high stability of the kinetochore structureunder single bond deletion. This is in accordance with the impor-tance of the complex for living systems.

2.4. Mutual information

In our inner-kinetochore model, we measured the uncertaintyof the formation of particular bonds by Shannon entropy and iden-tified the incidence-relationship of different bonds by computingtheir mutual information (cf. Section 4.3.5). We applied this anal-ysis to the wild-type and all bond mutations, but not for proteinmutations, as there are no in vitro data to compare with.

Fig. 6. Mutual information of protein interactions in the wild-type. The amountof information (in bits) that the realization of one interaction (bonded or non-bonded) reveals about the realization of another. Sampled over 200 simulations.Interactions that have zero mutual information with all other interactions are omit-ted. The highest information for non-chain molecules could be measured betweenCenpA-CenpC:CenpC:H3 and CenpS-CenpT:CenpT-H3. Chained molecules have anincreased mutual information due to their forced proximity.

The information theoretic analysis of our wild-type identifiedthe highest probability of bond formation between the pairs ofbonds Cenp-A-Cenp-C:Cenp-C-H3 (mutual information 0.97 bit)and Cenp-S-Cenp-T:Cenp-T-H3 (mutual information 0.82 bit witha maximum of 1.00 bit, cf. Fig. 6). The mutual information betweenCenp-A-Cenp-C and Cenp-C-H3 did not alter significantly in anyof the bond mutated simulations. In contrast, deleting eitherCenp-S-H3 or Cenp-W-H3 lowers the mutual information of Cenp-S-Cenp-T:Cenp-T-H3 significantly. This outcome suggests a highcorrelation of the interactions around the two nucleosomes. ExceptCenp-M-H3 the removal of any H3 interaction lowers this infor-mation. This implies that the cluster around the H3 nucleosomeis essential for the formation of the inner-kinetochore and robustagainst any other bond deletion.

Concluding all the results suggests Cenp-S-H3, Cenp-W-H3,Cenp-N-Cenp-R and Cenp-Q-Cenp-R as the bonds of the inner-kinetochore with the most influence on the structure, which hasbeen seen in the data from all of our methods (cf. Table 3).

3. Conclusion

Deriving the spatial structure of a protein complex from het-erogeneous experimental knowledge is an open challenge. Solvingit would accelerate data-driven cell biology by allowing to con-struct and validate in silico models from a database of unconnectedexperimental results.

This paper moves toward that goal. We improved the exist-ing inner-kinetochore model by incorporating 16 new data pointsfrom six publications. In total the model now relies on 12 publica-tions and uses interpretations obtained from FRET, M2H and NMRSexperimental measurements. In addition to new interactions rules,the crystal structure of the proteins Cenp-A, -C and -T are incor-porated by mimicking them as chains of spheres. We created thewild-type structure for our new inner-kinetochore model duringS-phase and validated it with experimental findings from litera-ture. Assuming the structure-function equivalence, we show thatour model reproduces most effects described in the literature (seeTable 2 for an overview).

We demonstrate that the incorporated wet-lab results are con-sistent: they can be integrated in a single model without conflictingeach other. We have to emphasize that our model is constructedto avoid selection bias. All applicable results from the literatureknown to us were incorporated and the model was implementedin the most canonical way: we only used a single model, instead ofchoosing the best fitting among a set of different models. Instead ofrelying on atomic-scale particles and forces, we used coarser par-ticles and a rule-based, stateful bonding mechanism. Our modelentities are of the same scale as the measured entities of the wet-lab experiments: each protein is represented as a spherical particle,with regard to its diameter and mass, or a complex of sphericalparticles.

The wild-type structure of this refined model produces a singlephenotype instead of three phenotypes, as Tschernyschkows modelproduced. Removing any of the CCAN proteins from the simulationeffected the structure in accordance to the literature. Our modelsuggests Cenp-A, -B, -C, -M, -U, -W, and H3 as the crucial proteinsfor the stable formation of the inner-kinetochore. Bond deletionsexhibit a highly varying impact. Our simulation data show thatCenp-Q-Cenp-R, Cenp-H-Cenp-M, and Cenp-H-Cenp-N influencethe stability of inner-kinetochore. Deletion of the Cenp-Q-Cenp-Rbond looses most of the kinetochore proteins while removing eitherCenp-H-Cenp-M or Cenp-H-Cenp-N stabilizes kinetochore proteinsmore. With respect to the overall inner-kinetochore structure, wedetermine the region around the H3 nucleosome as especially sen-sitive to small changes. We found that the most influencing bond is

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the Cenp-W-H3 interaction. Clearly, Cenp-S-Cenp-X has also highinfluence on the structure because it is freely moving in our model.These in silico predictions can guide further in vivo experiments.

This success hints at the possibility of an impedance match ofin silico model and in vivo experiment for protein structure for-mation. The primary technical challenge addressed in this paperis the discrimination of phenotypes (sets of structures) as equalor not. We employed several methods. Cophylograms made theoverlap of two sets of structures visible to the human eye, but arehard to analyze in an automatized fashion. Distance distributionsare heuristics: totally different structures can have the same distri-bution by chance, but are amendable to automatization. They arethus appropriate to filter out potentially equal phenotypes prior tomanual inspection. Analysis of the mutual information and statisti-cal techniques, like mean or variance of proteins, allows a detailedinsight into the microscopical behavior of the inner-kinetochore.

4. Materials and methods

4.1. Model description

The inner-kinetochore consists of centromere bridges betweennucleosomes. We developed a model for building one of thesebridges. The nucleosomes the bridge attaches to are modeled asparticles with fixed position. The model encompasses one copy ofeach CCAN protein. They are initially randomly distributed in spaceand form the bridge during the simulation.

The model is formulated as rule-based spatial chemistry asimplemented by the SRSim simulator (Gruenert et al., 2010). Pro-teins are modeled as soft spheres with a mass-dependent radius:r$ 3%m. Space is implemented with a spherical soft boundary (radius170 A) and a cubic hard boundary (edge size 400 A). The sphericalsoft boundary keeps all proteins (even the free diffusing ones) rela-tive close to the kinetochore. Thus, the statistics are not influencedby the box size of the reactor. As a mutated, free diffusing particlehas a random position in space it influences the statistics quantita-tively but not qualitatively. Brownian motion is implemented usingLangevin dynamics. Bonds form irreversibly with an interactionrate of 0.015 between two proteins when their surfaces are closerthan 1 A. For the list of possible bonds, see Table 1. The moleculesCenp-A, Cenp-C, and Cenp-T are known to be chains (cf. Nishinoet al., 2012, Fig. 2). To incorporate the geometric knowledge, wemodeled each using 3, 10 and 22 sub-particles, respectively. Thosesub-particles are arranged in chains using the bond mechanism.Their radii are shown in Fig. 2. The protein mass is distributedover the sub-particles in proportion to their volume. Interactionsbetween Cenp-A, -C, and -T with other proteins can happen at anyof the sub-particles.

4.2. Simulation with SRSim

We used our freely available SRSIM software Gruenert et al.(2010), which combines the BioNetGen language (BNGL) for rule-based reaction systems Blinov et al. (2004), Faeder et al. (2005)with a three-dimensional coarse-grained simulation building uponthe LAMMPS molecular dynamics (MD) simulator Plimpton (1995).SRSim fills a gap located in between the fine-grained MD simulationmodels, which do not allow the formulation of reaction networks,and spatial graph drawing software tools, which do not include anypossibility for dynamical simulation. For a spatial simulation withSRSim, the description of the reaction dynamics, reaction vesselas well as a simulation protocol can be specified. In particular,the size and shape of the reactor, its boundary conditions, theinitial state of the reactor and the length of the time-steps. Thesoftware SRSim needs four initialization files: (i) a BNGL file with

Table 5Overview of mathematical symbols.

Symbol Value Meaning

N 19 All moleculesS 200 Total simulation countM 50 Number of bondsi, j 1, . . ., N Molecule indicesk, l 1, . . ., S Simulation run indicesb 1, . . ., M Bond indexd 1, . . ., 3 Dimension indexr R+

0 Indicates a distance

Meaning of the variables and constants used repeatedly in this paper.

the description of the proximity network in BNGL format; (ii) afile with geometry parameters of each molecule like molecularmass, radius and attachment sites in spherical coordinates. Alsoit contains general parameters about forces, binding angles andtemperature in the reactor; (iii) a file for predefined geometries,like cubes, which can be placed in the reactor; (iv) the last file isthe LAMMPS init file, it initializes the LAMMPS simulator, here thesize, form, units and more parameters of the reactor are set. It ispossible to record various information to files during and after thesimulation. Regularly, snapshots of the position of each particleare written, for the later structural analysis and visualization aswell as for tracking and analyzing the simulation process.

In this article, the simulation is executed for 5 & 106 steps. Theparticle configuration attained after that time is considered as akinetochore structure resulting from the model description. As onesimulation depends on the random initial particle configuration,we executed 200 simulations with different starting conditions forthe wild-type and every mutant. This resulted in a total of 11,800simulation runs. Also, a typical exemplary setup can be found in theSupplementary materials. For more details on simulation protocoloptions we refer to Gruenert et al. (2010), Ibrahim et al. (2013).

4.3. Structure analysis

We aim to detect whether a reaction or a whole protein issignificant to the formation of the kinetochore. To this end we sim-ulate 200 instances of wild-type and mutants for a fixed time andcompare the final simulation state structure using four differentmethods: (1) comparison of the bond frequency; (2) comparison ofgeometrical structure using distance measures based on the indi-vidual protein positions; (3) comparison of statistical moments ofparticle position; and (4) analysis of mutual information betweenpairs of bonds. All used indices are described in Table 5.

4.3.1. Bond frequency analysisThe set of realized bonds is a crude approximation of the kineto-

chore structure. To include intraspecies variance, some bonds maybe established by chance. We consider the realization frequency cbof each bond, sampled over all simulation runs:

c = (c1, . . ., cM), (1)

cb = 1S

SX

k=1

#b,k, (2)

#b,k =

8<

:

1 if bond b is realized in simulation k,

0 otherwise. (3)

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The M-dimensional vector c of all bond frequencies is computedfor the wild-type and all mutants. It can be compared visually as inFig. 5 or by estimating the Kullback–Leibler divergence (KLD):

DKL(b|W) =MX

i=1

ln

✓cb

i

cWi

◆cb

i , (4)

where cb refers to the frequency vector of a certain bond mutationand cW to the bond frequency of the wild-type. Those values areshown in Tables 2 and 3.

4.3.2. Distance distributionWe want to decide whether the molecular structure of the

mutant is identical to the wild-type. To this end, assume we hada distance measure ! to compare the final states of two simula-tion runs S, S'. We are still left with two problems. First, we donot have a single wild-type structure, but 200 of them and theymay have a high structural diversity. And second, we do not knowthe distance threshold beyond which a structure is not consideredidentical anymore.

We overcome those problems by considering the distancebetween wild-type and mutants as a random variable. We estimatethe distribution of those distances from all the pairs from two setsof simulations, for example all wild-type and all mutant simula-tions. We can then compare the intraspecies distance distributionof the wild-type (WW) with the interspecies distance distributionof the wild-type and a mutant (MW). If the mutant has the samestructure as the wild-type, those two distributions will be indis-tinguishable. Thus from a difference of MW and WW a differenceof mutant and wild-type structure can be concluded. If MW hasan additional mode, but otherwise conforms to WW, this hints ata reduced fidelity: the mutant can produce the wild-type struc-ture, but depending on chance also produces a mutated structure.To introduce no statistical anomalies we estimate the intraspecies

distance distribution of the wild-type, not by comparing a set of 200wild-type structures with itself, but with a separate set of wild-typestructures of identical size.

We break up the interval I = [0, max(!(W, W))] into 100 bins anddedicate every value of ! to its bin for estimating the distributionof !. To compare two distributions interpret them as vectors andcompute their Euclidean distance (cf. Tables 2 and 3). One candidatefor the distance function ! is the average difference of the distancesof all molecules in one structure:

!(S, S') = 1N(N " 1)

vuutNX

i=1

NX

j=1

(ri,j " r'i,j)

2, (5)

where ri,j and r'i,j are the Euclidean distance of molecule i and j in the

structure simulation S and S', respectively. Note that this measureis possible at all, because every type of molecule occurs exactlyonce in each simulation. We thus do not have to consider missingmolecules, or multiple molecules of the same type that take theirindividual position by chance. !(S, S') has the nice property that itis invariant under translation and rotation of S and S'. An exampleof this technique is shown in Fig. 7.

4.3.3. PhylogramWe use a phylogram as a qualitative analysis tool of structural

similarity and diversity. It displays the relationship of a set of struc-tures with a distance measure ! (cf. Eq. (5)): Each structure S is aleaf at the bottom of the three. Internal nodes (branches) repre-sent sets of structures, containing the branches connected to them.The height of a node (distance to the bottom) is proportional to theaverage distance between the structures within the representedset. Two dissimilar structures only share nodes close to the root.Similar structures share nodes close to the leafs. The phylogram iscomputed using agglomerate clustering and the UPGMA distance

Fig. 7. Distance distribution of the wild-type with itself, the Cenp-Q-Cenp-U and the Cenp-S-Cenp-X mutant. Histograms of !(S, S') computed according to Eq. (5) of theinter-specific distance distribution of the wild-type (red), the Cenp-Q-Cenp-U (blue) and the Cenp-S-Cenp-X (green) mutant. The plot in the upper right corner comparesthe wild-type distribution (red) with a simulation without any reactions; all molecules move freely in space (black). Considering the histogram as vector, each bin is onedimension, enables computing an Euclidean distance between distance distributions. The distance between the wild-type histogram and the displayed structures are:1.10 & 10"2 for the CenpQ-CenpU mutant, 5.39 & 10"2 for the Cenp-S-Cenp-X mutant, and 27.3 & 10"2 for the interaction-free simulation. Wild-type and Cenp-Q-Cenp-Umutations have distance distributions with multiple modes of the same position and magnitude. This hints at intra-specific micro-structural diversity that is identicallyexpressed in both variants. In contrast, deleting Cenp-S-Cenp-X results in a slightly shifted curve with a similar shape, but different modes. This can be interpreted as analteration of the structure with an amount of intra-specific structural diversity comparable to the wild-type. The model stripped of all bonds (black) has a much higherdistance to the wild-type than any other tested mutation. Nor is the mode structure of distribution replicated. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of the article.)

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Fig. 8. Phylograms of wild-type and mutants Cenp-S-Cenp-X and Cenp-Q-Cenp-U. (A) Phylogram displaying the distance ! (cf. Eq. (5)) between simulated structures of thewild-type. The phylogram of the wild-type is composed of three parts: two families of different structures and a single outlier. The families differ in the orientation of theCenp-C chain. (B) The same phylogram for the Cenp-S-Cenp-X mutant. It still has the two families of structures, but the segmentation within the families is different. The splitinto the subfamilies occurs further away from the root (short branches). This is evidence that the structures within this family are more similar, compared with the wild-typefamily. (C) Comparison phylograms of wild-type and Cenp-S-Cenp-X mutant. Each leaf is colored according to its genotype: wild-type (blue) or mutant (gray). Although thebasic phylograms look quite similar (panel (A) and (B)), the mixed tree with both of them included, is nearly complete separated. The similarity within the wild-type andmutation (Cenp-S-Cenp-X) structure is greater than the similarity between them, based on !. According to this, the removal of Cenp-S-Cenp-X effects a significant alterationof the structure. (D) In contrast the comparison phylogram of wild-type and Cenp-Q-Cenp-U mutation. Internal nodes are nearly perfectly mixed. Thus the mutant exhibitsthe same phenotype as the wild-type and does not change inner-kinetochore structure at all. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of the article.)

measure (Prager and Wilson, 1978). The slightly different versionWPGMA is not appropriate here, due to all simulations are treatedequally and thus contribute equally to the phylogram (unweightedinstead of weighted arithmetic mean).

When visualizing a set of structures of identical genotype, forexample the wild-type, the phylogram allows one to discover if asingle genotypes yields a single phenotype, or multiple clusters ofstructures. In the later case one expects a few branches close to

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the root separated by a gap from all other branches. To discoverwhether a mutation significantly alters the structure of the wild-type, we compute the phylogram of the set union of N wild-typesand N mutants of identical genotype. Each structure is labeled withits genotype. Two examples are displayed in Fig. 8. We can deducepartial and total structural differences from the mixture of geno-types in the branches of the phylogram: When the structures of aphylogram are drawn from two identical distributions, they are ofthe same genotype or a mutation that has no structural effect. Thenin each branch both genotypes are equiprobable. If a branch con-taining many structures that are only composed of structures fromone genotype, but not the other, then both genotypes yield signifi-cantly different structures. If all branches are divergent in this way,both phenotypes are always different. If only some branches of thetree are divergent, this hints at either reduced fidelity or the abilityof the mutant to reproduce some, but not all conformations.

4.3.4. Statistical momentsIn the third method to test structure equality, we considered

mean and variance of individual particle positions. But their posi-tion in the Euclidean space is subject to random rotation. Insteadwe consider transformed coordinates:

(x, y, z) ( (x, r). (6)

The kinetochore assembles between two cube shaped nucleo-somes, with only one protein being able to bind. This makes therotation axis of the complex span between both of nucleosomes.During simulation they are situated on fixed points on the x-ordinate. Thus the coordinates (x, r) refers to the position on therotation axis (x) and the distance to it (r):

(x, r) = (x,p

y2 + z2). (7)

For each mutation m and for each molecule i we compute meanM1 and variance M2 of its transformed coordinates over all sim-ulations and compare those values for wild-type and mutant inabsolute (cn,m,i) in relative (c!

n,m,i) fashion.

M1,m,i = E(xm,i), (8)

M2,m,i = E(x2m,i) " E(xm,i)

2, (9)

cn,m,i = Mn,m,i " Mn,W,i, (10)

c!n,m,i = ln

✓Mn,m,i

Mn,W,i

◆. (11)

The values for the wild-type (W) where computed. The variance ofa molecule signifies its mobility across different phenotypes (notin time). If neither mean nor variance differ more than 30% (thatis 0.7 < c!

n,m,i < 1.3) from the wild-type, we consider a mutant asidentical in structure. A negative value implies the stabilization ofthe molecule in the complex, whereas a positive one shows therelaxation of it’s position. Fig. 4 displays an example.

4.3.5. Mutual informationOur last analysis technique mentions the information flow

through a system. As method for measuring this flow in biolog-ical system proved Shannons information-theory’s understandingof the entropy oneself. Calculating the mutual information betweena pair of bonds determines the correlation of those two bonds. If thisvalue alters significantly under a mutation or a bond deletion, thisis related to a structural change. The entropy of a finite set Z can becalculated as follows:

H = "X

z)Z

pzlog2pz. (12)

Considering this set as the probability that a certain bond was real-ized (Z = {0, 1}), the entropy for every bond X is determined by

H(X) = "{0,1}X

x

P(X = x)log2P(X = x). (13)

Calculating the joint entropy for every pair of bonds (X, Y) considerswhether regarding bonds occur in the same simulation run or onlyone of them (none, respectively):

H(X, Y) = "X

x,y

P(X = x, Y = y)log2P(X = x, Y = y). (14)

The mutual information between two bonds is given as

MI(X, Y) = H(X) + H(Y) " H(X, Y). (15)

MI was calculated for every mutation between every pair of bondsand gives insight into the information of the current structure. Amutual information of more than 0.75 bits is significantly high forus (cf. Fig. 6). Considering this as measurement of nonlinear correla-tion between the two sources, this method reveals how related eachpair of bond is. While a low mutual information means the indepen-dence of two interactions, a high value of MI depicts the coherenceof two bonds. An alteration of the mutual information, affected byany kind of mutation, reveals a shift of the relation between thebonds. This implies that the structure formed in a different way,changing its information. Concluding, the variation of the mutualinformation may indicate another structure than the wild-type. Butit makes no statement about the kind of variation.

Acknowledgments

This work was supported by the German Research Founda-tion priority program InKoMBio (SPP 1395, Grant DI 852/10-1),the European Commission NeuNeu Project (248992), the EuropeanCommission HIERATIC Project (316705), and the COBRA project(62098).

Appendix A. Supplementary Data

Supplementary data associated with this article can befound, in the online version, at http://dx.doi.org/10.1016/j.biosystems.2014.11.004.

References

Amano, M., Suzuki, A., Hori, T., Backer, C., Okawa, K., Cheeseman, I.M., Fukagawa, T.,2009. The CENP-S complex is essential for the stable assembly of outer kineto-chore structure. J. Cell Biol. 186, 173–182.

Ando, S., Yang, H., Nozaki, N., Okazaki, T., Yoda, K., 2002. CENP-A, -B, and -C chro-matin complex that contains the I-type alpha-satellite array constitutes theprekinetochore in HeLa cells. Mol. Cell. Biol. 22, 2229–2241.

Bergmann, J.H., Rodriguez, M.G., Martins, N.M., Kimura, H., Kelly, D.A., Masumoto, H.,Larionov, V., Jansen, L.E., Earnshaw, W.C., 2011. Epigenetic engineering showsH3K4me2 is required for HJURP targeting and CENP-A assembly on a synthetichuman kinetochore. EMBO J. 30, 328–340.

Black, B.E., Cleveland, D.W., 2011. Epigenetic centromere propagation and the natureof CENP-A nucleosomes. Cell 144, 471–479.

Blinov, M.L., Faeder, J.R., Goldstein, B., Hlavacek, W.S., 2004. Bionetgen: softwarefor rule-based modeling of signal transduction based on the interactions ofmolecular domains. Bioinformatics 20, 3289–3291.

Carroll, C.W., Milks, K.J., Straight, A.F., 2010. Dual recognition of CENP-A nucleosomesis required for centromere assembly. J. Cell Biol. 189, 1143–1155.

Carroll, C.W., Silva, M.C., Godek, K.M., Jansen, L.E., Straight, A.F., 2009. Centromereassembly requires the direct recognition of CENP-A nucleosomes by CENP-N.Nat. Cell Biol. 11, 896–902.

Caydasi, A.K., Lohel, M., Grunert, G., Dittrich, P., Pereira, G., Ibrahim, B., 2012. Adynamical model of the spindle position checkpoint. Mol. Syst. Biol. 8, 582.

Chan, G.K., Liu, S.T., Yen, T.J., 2005. Kinetochore structure and function. Trends CellBiol. 15, 589–598.

Cheeseman, I.M., Desai, A., 2008. Molecular architecture of the kineto-chore–microtubule interface. Nat. Rev. Mol. Cell Biol. 9, 33–46.

584

585

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609

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611

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620

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680

681

682

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685

686

687

688

689

690

691

692

693

694

695

696




Cheeseman, I.M., Hori, T., Fukagawa, T., Desai, A., 2008. KNL1 and the CENP-H/I/Kcomplex coordinately direct kinetochore assembly in vertebrates. Mol. Biol. Cell19, 587–594.

Cimini, D., Degrassi, F., 2005. Aneuploidy: a matter of bad connections. Trends CellBiol. 15, 442–451.

Dalal, Y., Bui, M., 2010. Down the rabbit hole of centromere assembly and dynamics.Curr. Opin. Cell Biol. 22, 392–402.

Doncic, A., Ben-Jacob, E., Barkai, N., 2005. Evaluating putative mechanismsof the mitotic spindle checkpoint. Proc. Natl. Acad. Sci. U. S. A. 102,6332–6337.

Eskat, A., Deng, W., Hofmeister, A., Rudolphi1, S., Emmerth, S., Hellwig, D., Ulbricht, T.,Döring, V., Bancroft, J.M., McAinsh, A.D., Cardoso, M.C., Meraldi, P., Hoischen, C.,Leonhardt, H., Diekmann, S., 2012. Step-wise assembly, maturation and dynamicbehavior of the human CENP-P/O/R/Q/U kinetochore sub-complex. PLoS ONE 7,e44717.

Faeder, J.R., Blinov, M.L., Hlavacek, W.S.,2005. Graphical rule-based representationof signal-transduction networks. In: Proceedings of the ACM Symposium onApplied Computing. ACM, New York, USA, pp. 133–140.

Foltz, D.R., Jansen, L.E., Black, B.E., Bailey, A.O., Yates, J.R., Cleveland, D.W., 2006. Thehuman CENP-A centromeric nucleosome-associated complex. Nat. Cell Biol. 8,458–469.

Fukagawa, T., Brown, W.R., 1997. Efficient conditional mutation of the vertebrateCENP-C gene. Hum. Mol. Genet. 6, 2301–2308.

Görlich, D., Escuela, G., Gruenert, G., Dittrich, P., Ibrahim, B., 2013. Molecular codes inQ3the human inner-kinetochore model: relating CENPS to function. J. Biosemiotics,submitted for publication.

Gruenert, G., Ibrahim, B., Lenser, T., Lohel, M., Hinze, T., Dittrich, P., 2010. Rule-based spatial modeling with diffusing, geometrically constrained molecules.BMC Bioinform. 11.

Hanissian, S.H., Akbar, U., Teng, B., Janjetovic, Z., Hoffmann, A., Hitzler, J.K., Iscove, N.,Hamre, K., Du, X., Tong, Y., Mukatira, S., Robertson, J.H., Morris, S.W., 2004. cDNAcloning and characterization of a novel gene encoding the MLF1-interactingprotein MLF1IP. Oncogene 23, 3700–3707.

Hellwig, D., Emmerth, S., Ulbricht, T., Döring, V., Hoischen, C., Martin, R., Samora,C.P., McAinsh, A.D., Carroll, C.W., Straight, A.F., Meraldi, P., Diekmann, S., 2011.Dynamics of CENP-N kinetochore binding during the cell cycle. J. Cell Sci. 124,3871–3883.

Hellwig, D., Hoischen, C., Ulbricht, T., Diekmann, S., 2009. Acceptor-photobleachingfret analysis of core kinetochore and NAC proteins in living human cells. Eur.Biophys. J. 38, 781–791.

Hori, T., Amano, M., Suzuki, A., Backer, C.B., Welburn, J.P., Dong, Y., McEwen, B.F.,Shang, W.H., Suzuki, E., Okawa, K., Cheeseman, I.M., Fukagawa, T., 2008a. CCANmakes multiple contacts with centromeric DNA to provide distinct pathways tothe outer kinetochore. Cell 135, 1039–1052.

Hori, T., Okada, M., Maenaka, K., Fukagawa, T., 2008b. CENP-O class proteins form astable complex and are required for proper kinetochore function. Mol. Biol. Cell19, 843–854.

Howman, E.V., Fowler, K.J., Newson, A.J., Redward, S., MacDonald, A.C., Kalitsis,P., Choo, K.H.A., 2000. Early disruption of centromeric chromatin organizationin centromere protein A (Cenpa) null mice. Proc. Natl. Acad. Sci. U. S. A. 97,1148–1153.

Hua, S., Wang, Z., Jiang, K., Huang, Y., Ward, T., Zhao, L., Dou, Z., Yao, X., 2011. CENP-U cooperates with Hec1 to orchestrate kinetochore-microtubule attachment. J.Biol. Chem. 286, 1627–1638.

Ibrahim, B., Diekmann, S., Schmitt, E., Dittrich, P., 2008a. In-silico modeling of themitotic spindle assembly checkpoint. PLoS ONE 3.

Ibrahim, B., Dittrich, P., Diekmann, S., Schmitt, E., 2007. Stochastic effects in acompartmental model for mitotic checkpoint regulation. J. Integr. Bioinform.4, 66.

Ibrahim, B., Dittrich, P., Diekmann, S., Schmitt, E., 2008b. Mad2 binding is not suffi-cient for complete cdc20 sequestering in mitotic transition control (an in silicostudy). Biophys. Chem. 134, 93–100.

Ibrahim, B., Henze, R., Gruenert, G., Egbert, M.M., Huwald, J., Dittrich, P., 2013. Rule-based modeling in space for linking heterogeneous interaction data to large-scale dynamical molecular complexes. Cells 2, 506–544.

Ibrahim, B., Schmitt, E., Dittrich, P., Diekmann, S., 2009. In silico study of kinetochorecontrol, amplification, and inhibition effects in MCC assembly. BioSystems 95,35–50.

Izuta, H., Ikeno, M., Suzuki, N., Tomonaga, T., Nozaki, N., Obuse, C., Kisu, Y., Goshima,N., Nomura, F., Nomura, N., Yoda, K., 2006. Comprehensive analysis of the ICEN(interphase centromere complex) components enriched in the CENP-A chro-matin of human cells. Genes Cells 11, 673–684.

Kalitsis, P., Fowler, K.J., Earle, E., Hill, J., Choo, K.H., 1998. Targeted disruption ofmouse centromere protein C gene leads to mitotic disarray and early embryodeath. Proc. Natl. Acad. Sci. U. S. A. 95, 1136–1141.

Kato, H., Jiang, J., Zhou, B.R., Rozendaal, M., Feng, H., Ghirlando, R., Xiao, T.S., Straight,A.F., Bai, Y., 2013. A conserved mechanism for centromeric nucleosome recog-nition by centromere protein CENP-C. Science 340, 1110–1113.

Kreyssig, P., Escuela, G., Reynaert, B., Veloz, T., Ibrahim, B., Dittrich, P., 2012. Cyclesand the qualitative evolution of chemical systems. PLoS ONE 7, e45772.

Kreyssig, P., Wozar, C., Veloz, T., Ibrahim, B., Dittrich, P., 2014. Effects of small particlenumbers on long-term behaviour in discrete biochemical systems. Bioinformat-ics, in press.

Lohel, M., Ibrahim, B., Diekmann, S., Dittrich, P., 2009. The role of localization in theoperation of the mitotic spindle assembly checkpoint. Cell Cycle 8, 2650–2660.

Maiato, H., DeLuca, J., Salmon, E.D., Earnshaw, W.C., 2004. The dynamic kineto-chore–microtubule interface. J. Cell Sci. 117, 5461–5477.

McClelland, S.E., Borusu, S., Amaro, A.C., Winter, J.R., Belwal, M., McAinsh, A.D.,Meraldi, P., 2007. The CENP-A NAC/CAD kinetochore complex controls chro-mosome congression and spindle bipolarity. EMBO J. 26, 5033–5047.

Muro, Y., Masumoto, H., Yoda, K., Nozaki, N., Ohashi, M., Okazaki, T., 1992. Cen-tromere protein B assembles human centromeric alpha-satellite DNA at the17-bp sequence, CENP-B box. J. Cell Biol. 116, 585–596.

Nishino, T., Takeuchi, K., Gascoigne, K.E., Suzuki, A., Hori, T., Oyama, T., Morikawa, K.,Cheeseman, I.M., Fukagawa, T., 2012. CENP-T-W-S-X forms a unique centromericchromatin structure with a histone-like fold. Cell 148, 487–501.

Obuse, C., Yang, H., Nozaki, N., Goto, S., Okazaki, T., Yoda, K., 2004. Proteomics anal-ysis of the centromere complex from HeLa interphase cells: UV-damaged DNAbinding protein 1 (DDB-1) is a component of the CEN-complex, while BMI-1 istransiently co-localized with the centromeric region in interphase. Genes Cells9, 105–120.

Okada, M., Cheeseman, I.M., Hori, T., Okawa, K., McLeod, I.X., Yates, J.R., Desai, A., Fuk-agawa, T., 2006. The CENP-H-I complex is required for the efficient incorporationof newly synthesized CENP-A into centromeres. Nat. Cell Biol. 8, 446–457.

Orthaus, S., Biskup, C., Hoffmann, B., Hoischen, C., Ohndorf, S., Benndorf, K., Diek-mann, S., 2008. Assembly of the inner kinetochore proteins CENP-A and CENP-Bin living human cells. ChemBioChem 9, 77–92.

Perpelescu, M., Fukagawa, T., 2011. The ABCs of CENPs. Chromosoma 120, 425–446.Plimpton, S., 1995. Fast parallel algorithms for short-range molecular dynamics. J.

Comput. Phys. 117, 1–19.Prager, E.M., Wilson, A.C., 1978. Construction of phylogenetic trees for proteins and

nucleic acids: empirical evaluation of alternative matrix methods. J. Mol. Evol.11, 129–142.

Przewloka, M.R., Venkei, Z., Bolanos-Garcia, V.M., Debski, J., Dadlez, M., Glover, D.M.,2011. CENP-C is a structural platform for kinetochore assembly. Curr. Biol. 21,399–405.

Quenet, D., Dalal, Y., 2012. The CENP-A nucleosome: a dynamic structure and roleat the centromere. Chromosome Res. 20, 465–479.

Rohn, H., Ibrahim, B., Lenser, T., Hinze, T., Dittrich, P., 2008. Enhancing parameterestimation of biochemical networks by exponentially scaled search steps. Lec-ture Notes in Computer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics), LNCS, vol. 4973., pp. 177–187.

Stoler, S., Keith, K.C., Curnick, K.E., Fitzgerald-Hayes, M., 1995. A mutation in CSE4, anessential gene encoding a novel chromatin-associated protein in yeast, causeschromosome nondisjunction and cell cycle arrest at mitosis. Genes Dev. 9,573–586.

Suijkerbuijk, S.J., Kops, G.J., 2008. Preventing aneuploidy: the contribution of mitoticcheckpoint proteins. Biochim. Biophys. Acta 1786, 24–31.

Sullivan, K.F., Hechenberger, M., Masri, K., 1994. Human CENP-A contains a histoneH3 related histone fold domain that is required for targeting to the centromere.J. Cell Biol. 127, 581–592.

Suzuki, N., Nakano, M., Nozaki, N., Egashira, S.I., Okazaki, T., Masumoto, H., 2004.CENP-B interacts with CENP-C domains containing mif2 regions responsible forcentromere localization. J. Biol. Chem. 279, 5934–5946.

Takeuchi, K., Nishino, T., Mayanagi, K., Horikoshi, N., Osakabe, A., Tachiwana, H.,Hori, T., Kurumizaka, H., Fukagawa, T., 2013. The centromeric nucleosome-likeCENP-T-W-S-X complex induces positive supercoils into DNA. Nucl. Acids Res.

Topp, C.N., Zhong, C.X., Dawe, R.K., 2004. Centromere-encoded RNAs are inte-gral components of the maize kinetochore. Proc. Natl. Acad. Sci. U. S. A. 101,15986–15991.

Tschernyschkow, S., Herda, S., Gruenert, G., Doring, V., Gorlich, D., Hofmeister, A.,Hoischen, C., Dittrich, P., Diekmann, S., Ibrahim, B., 2013. Rule-based modelingand simulations of the inner kinetochore structure. Prog. Biophys. Mol. Biol 113,33–45.

Varma, D., Wan, X., Cheerambathur, D., Gassmann, R., Suzuki, A., Lawrimore, J., Desai,A., Salmon, E.D., 2013. Spindle assembly checkpoint proteins are positioned closeto core microtubule attachment sites at kinetochores. J. Cell Biol. 202, 735–746.

Yang, C.H., Tomkiel, J., Saitoh, H., Johnson, D.H., Earnshaw, W.C., 1996. Identificationof overlapping DNA-binding and centromere-targeting domains in the humankinetochore protein CENP-C. Mol. Cell. Biol. 16, 3576–3586.

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Int. J. Mol. Sci. 2014, 15, 19074-19091; doi:10.3390/ijms151019074OPEN ACCESS

International Journal ofMolecular Sciences

ISSN 1422-0067www.mdpi.com/journal/ijms

Article

Active Transport Can Greatly Enhance Cdc20:Mad2 FormationBashar Ibrahim 1,2,3,†,* and Richard Henze 3,†

1 Al-Qunfudah University College, Umm Al-Qura University, 1109 Makkah Al-Mukarramah,Saudi Arabia

2 Al-Qunfudah Center for Scientific Research (QCSR), 21912 Al-Qunfudah, Saudi Arabia3 Bio Systems Analysis Group, Institute of Computer Science, Jena Center for Bioinformatics and

Friedrich Schiller University, 07743 Jena, Germany; E-Mail: [email protected]

† These authors contributed equally to this work.

* Author to whom correspondence should be addressed; E-Mail: [email protected];Tel.: +49-3641-9-46460; Fax +49-3641-9-46302.

External Editor: William Cho

Received: 28 August 2014; in revised form: 30 September 2014 / Accepted: 11 October 2014 /Published: 21 October 2014

Abstract: To guarantee genomic integrity and viability, the cell must ensure properdistribution of the replicated chromosomes among the two daughter cells in mitosis. Themitotic spindle assembly checkpoint (SAC) is a central regulatory mechanism to achieve thisgoal. A dysfunction of this checkpoint may lead to aneuploidy and likely contributes to thedevelopment of cancer. Kinetochores of unattached or misaligned chromosomes are thoughtto generate a diffusible “wait-anaphase” signal, which is the basis for downstream eventsto inhibit the anaphase promoting complex/cyclosome (APC/C). The rate of Cdc20:C-Mad2complex formation at the kinetochore is a key regulatory factor in the context of APC/Cinhibition. Computer simulations of a quantitative SAC model show that the formationof Cdc20:C-Mad2 is too slow for checkpoint maintenance when cytosolic O-Mad2 has toencounter kinetochores by diffusion alone. Here, we show that an active transport of O-Mad2towards the spindle mid-zone increases the efficiency of Mad2-activation. Our in-silicodata indicate that this mechanism can greatly enhance the formation of Cdc20:Mad2 andfurthermore gives an explanation on how the “wait-anaphase” signal can dissolve abruptlywithin a short time. Our results help to understand parts of the SAC mechanism thatremain unclear.

Int. J. Mol. Sci. 2014, 15 19075

Keywords: spindle assembly checkpoint; chromosome segregation; systems biology ofmitosis; simulation; modeling

1. Introduction

Correct distribution of the replicated genome during mitosis is essential in all proliferating cells.In order to accomplish this, the cell must guarantee that each chromosome has established a tightbipolar attachment to the spindle apparatus before sister-chromatid separation is initiated in anaphase.The mitotic Spindle Assembly Checkpoint (SAC; [1]) is a surveillance mechanism that delays theonset of anaphase until all chromosomes have made their attachments to the mitotic spindle. Evenone misaligned chromosome is sufficient to keep the checkpoint active, yet the mechanism by whichthis is achieved is still elusive. It is thought that unattached or misaligned kinetochores catalyze theformation of a “wait-anaphase” signal. This signal eventually diffuses to counter the activation ofthe ubiquitin ligase anaphase promoting complex/cyclosome (APC/C) by its co-activator Cdc20 (CellDivision Cycle 20 homolog [2]). The activation of APC by Cdc20 triggers chromosome segregation byubiquitination of securin and cyclin B [3–6] (for review see [7]). A dysfunction of the SAC may lead toaneuploidy [8,9] and furthermore its reliable function is important for tumor suppression [10–12].

The core proteins involved in the SAC, conserved in all eukaryotes, are MAD (“Mitotic ArrestDeficient”; Mad1, Mad2, and Mad3 (in humans BubR1) [13]) and BUB (“Budding Uninhibited byBenzimidazole”; Bub1, and Bub3 [14]). These proteins work to regulate APC activity and its co-activatorCdc20. In addition to these two core proteins, the SAC also involves several other components thatparticipate in essential aspects of this mechanism. Among these components are Aurora-B [15] andthe “Multipolar spindle-1” protein (Mps1) [16]. These two proteins are required for SAC signalamplification and the formation of the Mitotic Checkpoint Complex (MCC). Moreover, several othercomponents, involved in carrying out essential aspects of the SAC mechanism, have been identified inhigher eukaryotes. Those are for example the RZZ complex [17,18] which is composed of “Rough Deal”(Rod) [19,20], Zeste White 10 [20–23] and Zwint-1 [24].

The Cdc20-binding protein Mad2 was suggested as a candidate for the “wait-anaphase” signal, as itis stabilized in a conformation with increased affinity to Cdc20 specifically at unattached kinetochores.The resulting Cdc20:C-Mad2 complex is diffusible and can bind to a complex of Bub3 and BubR1 toform the possibly transient mitotic checkpoint complex (MCC), which is a potent inhibitor of the APC.Additional, Cdc20:C-Mad2 can bind directly to the APC and form an inactive complex [25].

Several mathematical models have been developed during the last decade to evaluatepossible mechanisms for signal generation and propagation. Doncic et al. [26] as well as Sear andHoward [27] analyzed simple spatial models of potential checkpoint mechanisms with focus on buddingyeast or metazoan, respectively. They observed theoretically that a diffusible signal can generallyaccount for checkpoint operation. A more detailed model of the human SAC, including many ofthe confirmed interactions, has been proposed by Ibrahim et al. [28–30]. It is based on the “Mad2template” model by De Antoni et al. [31] and has been supported with in-vitro experiments by

Int. J. Mol. Sci. 2014, 15 19076

Simonetta et al. [32]. Lohel et al. [33] achieved an improvement of this model by taking into accountspecies localization and binding sites at kinetochores.

Here, we discuss a spatial model of the SAC, assuming the rate of Cdc20:C-Mad2 complex formationas the key regulatory factor for SAC activation and maintenance. Therefore, we develop a 3-dimensionalmodel of one mitotic cell and its last unattached kinetochore, based on the “Mad2 template”model [31]. All core proteins and their reactions are enabled with in-vitro measured concentrations andinteraction rates. Thus, our simulations give a reliable insight into the behavior of the real SAC system.To mimic the majority of experimental findings we mainly focus on human data but also use buddingyeast results for a comparison. Taking into account the fact that animals undergo an “open” mitosis,where the nuclear envelope breaks down before the chromosomes separate, while Saccharomycescerevisiae (yeast) undergoes a “closed” mitosis, where chromosomes divide within an intact cell nucleus.To overcome the issue of a “closed” mitosis for the budding yeast Saccharomyces cerevisiae model,we define the nucleus as a compartment model which is exactly the meaning of a “closed” mitosis.However, from a theoretical point of view if the diffusion is sufficient for the whole cell this implies thatthe diffusion must be sufficient for a smaller sub-volume too, like the nucleus.

First, we implemented the full “Mad2 template” model, including amplification, to check whether thishas an effect on the formation of Cdc20:C-Mad2. As the diffusion rate of O-Mad2 is not certainly knownyet, we estimated a suitable value. Therefore, we assumed that a high concentration of Cdc20:C-Mad2supports the SAC, varied the diffusion of free Mad2 in a range from 0.0–50.0 µm2s�1and used theamount of formed Cdc20:C-Mad2 as score. Furthermore, the idea of an active O-Mad2 transport cameup to support the formation of Cdc20:C-Mad2 at the kinetochore [27]. We show that the effectiveCdc20:C-Mad2 formation rate does not only depend on its chemical kinetics, but in addition requiresa high flux of free Mad2 towards the spindle mid-zone in human cells.

2. The Model

2.1. Mad2 Template Model

DeAntoni et al. suggested a simple SAC model [31], which describes the mechanism of Mad2recruitment to the kinetochore and the transport of Mad2 to Cdc20. This model got known as the“Mad2 template” model as Mad1 and Mad2 form a compound in the vicinity of a kinetochore toactivate Cdc20. The dynamic of the six species O-Mad2, Mad1:C-Mad2, Mad1:C-Mad2:O-Mad2*,Cdc20, Cdc20:C-Mad2, and Cdc20:C-Mad2:Mad2* is described by the reaction Equations (1)–(5). TheEquations (1)–(3) represent the basic “template model”, while the additional Equations (4) and (5) referto the effect of amplification (autocatalytic loop).

Int. J. Mol. Sci. 2014, 15 19077

Cdc20 +O-Mad2k1��*

)��k�1

Cdc20:C-Mad2 (1)

Mad1:C-Mad2 +O-Mad2k2��*

)��k�2

Mad1:C-Mad2:Mad2⇤ (2)

Mad1:C-Mad2:Mad2⇤ + Cdc20k3��*

)��k�3

Mad1:C-Mad2 + Cdc20:C-Mad2 (3)

Cdc20:C-Mad2 +O-Mad2k4��*

)��k�4

Cdc20:C-Mad2:Mad2⇤ (4)

Cdc20:C-Mad2:Mad2⇤ + Cdc20k5��*

)��k�5

2Cdc20:C-Mad2 (5)

The biochemical reactions of SAC activation and maintenance mechanism can be divided into akinetochore and a cytosolic part. The former serves to communicate the attachment status of eachkinetochore to the remainder of the cell while the latter one accounts for the actual inhibition ofthe APC.

2.1.1. Mad2-Activation and Its Function in Sequestering Cdc20

Mad2-activation at the kinetochore is commonly seen as the central part of the SAC mechanism.According to the “Mad2 template” model, Mad2 in its open conformation (O-Mad2) is recruited tounattached kinetochores by Mad1-bound Mad2 in its close conformation (C-Mad2) to form the ternarycomplex Mad1:C-Mad2:O-Mad2* (cf. Reaction (2)). In this complex O-Mad2* is the “activated”Mad2, i.e., it is stabilized in a conformation which can interact with Cdc20 to form Cdc20:C-Mad2(cf. Reaction (3)). The kinetic rate coefficients for this interactions have been determined in-vitroby [32] (cf. Table 1). In addition, O-Mad2 can likewise be activated by Cdc20 to increase Cdc20:C-Mad2autocatalytically [32] (cf. Reaction (1)).

2.1.2. Autocatalytic Amplification of Cdc20:C-Mad2 Formation

The addition of reactions (4) and (5) results in reaction (1). For that reason the amplification is alsoknown as autocatalytic loop. The contribution of this pathway is minor with the kinetic data givenin [32] (cf. Table 1). The presence of a highly contributing autocatalytic loop would also counteractthe checkpoint deactivation and is therefore not desirable for living cells from a theoretical point ofview [28]. While Mad1:C-Mad2 only exists at the kinetochore, Cdc20:C-Mad2 can be seen as structuralequivalent of it in the cytosol. Thus, Cdc20:C-Mad2 converts more O-Mad2 into Cdc20 bound C-Mad2(cf. Reaction (4)) and subsequently binds Cdc20 (cf. Reaction (5)). This implies that the amplificationhas just a minor contribution to the formation of Cdc20:C-Mad2, as also observed by [28].

Int. J. Mol. Sci. 2014, 15 19078

Table 1. Parameters for the mitotic spindle assembly model.

Parameter Human Budding Yeast RemarksRate constants: k1 1.00⇥ 10�3 µM�1s�1 4.83⇥ 10�5 µM�1s�1 [34]/[32]

k2 2.00⇥ 10�1 µM�1s�1 3.00⇥ 10�1 µM�1s�1 [35]/[32]k3 1.00⇥ 102 µM�1s�1 3.00⇥ 10�3 µM�1s�1 [28]/[32]k4 1.00⇥ 10�2 µM�1s�1 NA [28]k5 1.00⇥ 102 µM�1s�1 NA [28]

k�1 1.00⇥ 10�2 s�1 4.83⇥ 10�6 s�1 [28]/[32]k�2 2.00⇥ 10�1 s�1 4.50⇥ 10�1 s�1 [35]/[32]k�3 0.00 s�1 2.00⇥ 10�4 µM�1s�1 [28]/[32]k�4 3.00⇥ 10�2 s�1 NA [28]k�5 0.00 s�1 NA [28]

Initial amount:Cdc20 0.22 µM 0.1 µM [36,37]/[32]

O-Mad2 0.15 µM 0.2 µM [35]/[32]Cdc20:C-Mad2 0 µM 0 µM [29]/[33]Mad1:C-Mad2 0.05 µM 0.00616 µM [31]/[32]

Mad1:C-Mad2:Mad2* 0 µM 0 µM [31]/[32]Cdc20:C-Mad2:Mad2* 0 µM NA [31]

Diffusion constants:Cdc20 19.5 µm2s�1 19.5 µm2s�1 [38]

O-Mad2 0.0� 50.0 µm2s�1 0.0� 50.0 µm2s�1

Cdc20:C-Mad2 0.0� 14.0 µm2s�1 0.0� 14.0 µm2s�1

Mad1:C-Mad2 0 0Mad1:C-Mad2:Mad2* 0 0Cdc20:C-Mad2:Mad2* 0.0� 11.0 µm2s�1 NA

Environment:radius of the kinetochore 0.1 µm 0.015 µm [39]/[40,41]

radius of the cell 10 µm 2 µm [42]/[43]

2.1.3. APC Inhibition

Although Cdc20:C-Mad2 can bind to and inhibit APC directly, its inhibitory potency increasesgreatly in synergy with the Bub3:BubR1 complex [34]. It was shown that Cdc20:C-Mad2 together withBub3:BubR1 forms the tetrameric mitotic checkpoint complex MCC which is a powerful inhibitor ofAPC [44]. Also the ternary complex Bub3:BubR1:Cdc20 alone is an effective inhibitor ofAPC [45]. However, the rate of its uncatalyzed formation in the cytosol is slow [44]. The formationof Bub3:BubR1:Cdc20 is accelerated in the presence of unattached chromosomes [46] and it might bethat MCC forms as an intermediate complex from which O-Mad2 rapidly dissociates [45–47]. Thedemanding mechanisms for binding the inhibitory complexes to the APC are subject to current research.Despite its possibly transient nature in Bub3:BubR1:Cdc20 formation, MCC has been found stablybound to APC in mitosis [48]. The MCC might form more stably at unattached kinetochores and

Int. J. Mol. Sci. 2014, 15 19079

recruit APC from the cytoplasm. The MCC sub-complexes Bub3:BubR1:Cdc20 and Cdc20:C-Mad2can bind to the APC independently from unattached kinetochores although their binding might befacilitated in a kinetochore-dependent manner [45–47]. Also, the MCC might bind to APC in akinetochore-independent manner to eventually inhibit APC by releasing O-Mad2, forming the stableBub3:BubR1:Cdc20:APC complex. The mitotic checkpoint factor 2 protein (MCF2) is a highly potentAPC:inhibitor, yet the mechanism of binding to the APC and its regulation is still unknown [49,50]. Withthe exception of MCF2, all complexes inhibiting APC rely on the presence of Cdc20:C-Mad2, whichrequires unattached kinetochores for adequately fast formation. We can categorize the Cdc20:C-Mad2complex as “interface” which links signaling from unattached kinetochores to APC inhibition. Hence,in this article we focus our analysis on the dynamics of the Cdc20 and Mad2 concentrations. We didnot consider APC explicitly due to the fact that there are other direct inhibitory pathways for APC, likethe MCC or MCF2 complexes. Including solely an APC reaction to Cdc20 would provide no additionalinformation from the simulation. Additionally, there is less kinetic data available on such level of detail.However, we think future work of detailed spatial APC models is required.

2.2. Mathematical Treatment and Simulation

2.2.1. Reaction-Diffusion-Convection System

The models reactions are translated to mathematical language of coupled ordinary differentialequations (ODEs). Adding a second spatial-derivative as diffusion term transforms the system in coupledpartial differential equations (PDEs) known as Reaction-Diffusion system:

@[Ci

]

@t

= Di

r2[Ci

]| {z }Diffusion

+ R

j

({[Ci

]}, P)| {z }Reaction

. (6)

where [Ci

] refers to the concentration of species i = {1, ..., 6}. The first term on the right hand siderepresents the diffusion and the second one represents the biochemical reactions R

j

= {R1, ..., R5}where species i is involved. The constant D

i

denotes its species diffusion. The operator r refers to thespatial gradient ( ~r = ~e

x

@

@x

+ ~e

y

@

@y

+ ~e

z

@

@z

), t adverts to the time and P symbolizes phenomenologicalparameters. The full reaction-diffusion system, including six species and five reactions, can be writtenwith PDEs as following:

Int. J. Mol. Sci. 2014, 15 19080

@[Cdc20]@t

= D1r2[Cdc20]� r1 + r�1 � r3 + r�3 � r5 + r�5 (7)

@[O-Mad2]@t

= D2r2[O-Mad2]� r1 + r�1 � r2 + r�2 � r4 + r�4 (8)

@[Cdc20:C-Mad2]@t

= D3r2[Cdc20:C-Mad2] + r1 � r�1 + r3 � r�3 � r4 + r�4 + r

25 � r

2�5

(9)@[Mad1:C-Mad2]

@t

= D4r2[Mad1:C-Mad2]� r2 + r�2 + r3 � r�3 (10)

@[Mad1:C-Mad2:Mad2*]@t

= D5r2[Mad1:C-Mad2:Mad2*] + r2 � r�2 � r3 + r�3 (11)

@[Cdc20:C-Mad2:Mad2*]@t

= D6r2[Cdc20:C-Mad2:Mad2*] + r4 � r�4 � r5 + r�5. (12)

where by r

j

denotes the reaction rate of reaction j and r�j

the rate of the inverse reaction. By adding aconvectional term, the system can be extended to a Reaction-Diffusion-Convection system.

@[Ci

]

@t

= Di

r2[Ci

]| {z }Diffusion

+ Qi

r[Ci

]| {z }Convection

+ R

j

({[Ci

]}, P)| {z }Reaction

(13)

The additional symbol Qi

denotes the constant convection coefficient of species i. With the definitionof the r operator and its square, the full system looks like the following in Cartesian coordinates:

@[Ci

]

@t

= Di

✓@

2[Ci

]

@x

2+

@

2[Ci

]

@y

2+

@

2[Ci

]

@z

2

◆+ Q

i

✓@[C

i

]

@x

+@[C

i

]

@y

+@[C

i

]

@z

◆+R

j

({[Ci

]}, P) . (14)

2.2.2. Model Assumptions

The mitotic cell is considered as 3-ball with radius R. The last unattached kinetochore is a 2-spherewith radius r in the center of the cell (see Table 1 for the species-dependent values of the radius).

We use a lattice based model, which implies that the reaction volume of the mitotic cell is segmentedinto equal compartments. The initial concentrations of Cdc20 and O-Mad2 are distributed randomly overall compartments of the mitotic cell. As Mad1:C-Mad2 and Mad1:C-Mad2:Mad2* are only present at thekinetochore, their initial amount is located on the surface of the modeled 2-sphere. While reaction (1) cantake place in any compartment, the reactions containing either Mad1:C-Mad2 or Mad1:C-Mad2:Mad2*(reactions (2) and (3)) only occur at those compartments, attached to the kinetochore. In order to observea more accurate spatial behavior of the “Mad2 template” model we do not consider any symmetricalrestrictions. All boundary conditions are reflective in order that the amount of particles is conserved. Weassume mass-action-kinetics for all reactions. As the diffusion constant of O-Mad2 is not known yet,we used parameter estimation to determine a suitable value (see Section 3). Active transport of O-Mad2has been suggested to increase the inhibition of Cdc20:C-Mad2. Thus, we applied a convectional forceto the species O-Mad2 which is direct proportional to the distance to the kinetochore with an unknown

Int. J. Mol. Sci. 2014, 15 19081

factor. We also used parameter estimation to determine the most realistic value for the convectionalforce term (see Section 3). All initial concentrations, reaction rates and environment parameter valuesare taken from literature (cf. Table 1). We did not model the Mad1:Mad2 complex formation andconsider it as a preformed species in our model. We should point out here that this complex is atetrameric 2:2 Mad1:Mad2 and not a monomer complex. However, considering single species frommathematical point of view would not make any difference in this case as long as we have one form in ourmodel. All previous mathematical models have considered the same assumption to the template model(e.g., [28,32,33]).

2.2.3. Numerical Simulation

We run our simulations using the Virtual Cell software [51]. We create the reaction volume accordingto the model geometry (cf. Section 2). Each dimension is divided into 51 parts, which results in132.651 compartments in total. All parameters are set up consistent with the model assumptions. Thesystem of PDEs with boundary and initial conditions is solved using the“Fully implicit finite volumewith variable time-ste” method. This method employs Sundials stiff solver CVODE for time stepping(method of lines) [51]. The derivations, necessary for diffusion and convection, are computednumerically. We simulate the human system for 300 s which is sufficient to reach steady state (buddingyeast took 5000 seconds), with a maximum time-step of 0.1 s and an absolute and relative toleranceof 1.0⇥ 10�7.

One simulation run takes between 2 and 20 h, dependent on the parameter-set. The time dependentconcentration plots add up the amount of every species over all compartments and are generated withMatLab [52].

3. Results and Discussion

3.1. Quantitative Analysis of the SAC Model

The spindle checkpoint has to be fully active only abruptly after the cell enters mitosis and itsactivity must be maintained even with decreasing numbers of unattached kinetochores. After the lastchromosome has established its connection to the mitotic spindle, the SAC must be rapidly deactivated.We showed in previous studies [28–30] that for fast checkpoint deactivation the destabilization ofAPC:inhibitor complexes might be required. Here, we focus on the formation of the Cdc20:C-Mad2complex which is most certainly an APC:inhibitor.

Recently, it has been discussed in detail that the checkpoint activation is too slow with the measuredin-vitro rate coefficients for the “Mad2 template” model in budding yeast. In agreement with this,non-spatial simulations of the model show an insufficient APC inhibition when using the measuredyeast rate coefficients [32]. We could verify this hypothesis in our study. The inhibition of Cdc20 byO-Mad2 takes about 5000 min (data in supplement) which is way to long for an effect that has to occurnearly instantaneous. However, with a significant increase of all interaction rates the simulation of themodel reaches steady state considerably faster. This supports that fast Cdc20:C-Mad2 formation indeed

Int. J. Mol. Sci. 2014, 15 19082

requires faster rates than the measured ones in budding yeast. With this result we will focus on thehuman model and use the budding yeast data only for comparison.

To establish checkpoint signaling within few minutes after entry into mitosis, the combined actionof all kinetochores is required. However, when reducing the number of contributing kinetochores toone, i.e., the situation right before the last attachment, checkpoint maintenance is no longer possiblefor the human and the yeast model (formation of Cdc20:C-Mad2 takes too long). When we lookedclosely to the species amounts at individual unattached kinetochores in-silico, we found the amount ofO-Mad2 and activated O-Mad2* molecules is very close to zero. When removing reaction (3) from themodel (O-Mad2 binding to Mad1:C-Mad2), the formed amount of Cdc20:C-Mad2 was nearly zero. Thisindicates that the catalytic rate of Cdc20:C-Mad2 formation (reaction (1)) is not limiting. The majority ofCdc20:C-Mad2 is formed in vicinity of the kinetochore which is done by the catalyzing reaction (3). Thisimplies that diffusive O-Mad2 alone cannot compensate the consumption of O-Mad2 at the kinetochores.Recent work [53] showed that the human protein Tpr binds Mad2 in the region of the mitotic spindleeven in the absence of microtubules. Mad2-localization to the spindle region was formerly attributedonly to dynein binding. Hence, we speculate that O-Mad2 might also be actively transported towards thespindle mid-zone to increase restoration of O-Mad2 at unattached kinetochores.

Note that the combined contribution of all kinetochores may be sufficient to activate the checkpointeven without such a mechanism, especially in combination with other mechanisms like Cdc20sequestration by Emi1 (only meiotic, we do not need it here) [54,55] or preformed MCC [34].Taken together, with our current given knowledge, SAC control alone is insufficient to guardchromosome attachment.

3.2. Reaction-Diffusion System of the “Mad2 Template” Model

Ibrahim et al. [28] found that in the “Mad2 template” model the amplification by the autocatalyticloop (reactions (4) and (5)) is vanishing if the reaction rate constants are low. To make sure thatthese results are still valid with our spatial reaction-diffusion system, we execute a simulation byintegrating the autocatalytic amplification of Mad2. As the diffusion rate of Mad2 is unknown, wevary it between 0–50 µm2s�1. However, the changes when including the reactions of the autocatalyticloop are not perceptible when compared to the basic “template model” (cf. Figure 1A,B). This is becausedirect binding (reaction (1)) dominates Cdc20 sequestering [33], while kinetochore dependent catalysis(reactions (2) and (3)) becomes increasingly important during the inhibition mechanism. Thus, weremove the reactions (4) and (5) from the network (cf. Figure 2 black graph) as they have no effecton the outcome of the simulation in steady-state.

Int. J. Mol. Sci. 2014, 15 19083

Figure 1. Dynamical behavior of the spatial human SAC model. The figures show thetotal concentrations over time for every species with different parameter sets. (A) Outcomeof the simulated “Mad2 template” model (experimental interactions, diffusion rates and noconvection, cf. Table 1). It takes about 5 min to reach steady state. O-Mad2 and Cdc20are reduced to 70% and 80% of their initial amount, respectively. Cdc20:C-Mad2 has aconcentration of 4.3⇥ 10�2

µM (20% of Cdc20) in the steady state. According to literature,this amount is not enough and its formation too slow to dissipate the “wait anaphase” signal.Mad1:C-Mad2 and Mad1:C-Mad2:Mad2* do not change their concentration over time andthus do not contribute to the formation of Cdc20:C-Mad2 significantly; (B) The results of thereduced “template model” (slow diffusion constant and deleted reactions (4) and (5)) do notdeviate significantly from the full model, cf. panel A. Neither the amount of Cdc20:C-Mad2is increased nor the time it takes to reach steady state decreased. Thus, the autocatalyticloop, especially the formation of the Cdc20:C-Mad2:Mad2* complex, has no influence onthe model and can be omitted; (C) The simulation of the model with a 4-fold higher diffusionconstant of O-Mad2 (D

i

= 20µm2s�1). The species’ dynamical behavior is qualitative thesame as in the standard simulation (panel B). However, with a higher diffusion constantthe final amount of Cdc20:C-Mad2 increased to 6.3 ⇥ 10�2

µM (29% of Cdc20), wherebyO-Mad2 and Cdc20 are decreased to 57% and 71% of their initial concentration, respectively;(D) The simulation of the model with an active transport of O-Mad2 (cf. Equation (13)). Ittakes 1�2 min to reach steady state. The active transport of O-Mad2 towards the kinetochorepromotes the rates of reactions (2) and (3). Thus, the inhibition level of Cdc20 raises to 60%

of its total amount. Mad1:C-Mad2 and Mad1:C-Mad2:Mad2* catalyze the formation ofCdc20:C-Mad2 and reduce free Mad2 to 11% of its initial concentration.

0 50 100 150 200 250 3000 50 100 150 200 250 300

0

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0 20 40 60 80 100 120

Cdc20O-Mad2

Cdc20:C-Mad2Mad1:C-Mad2Mad1:C-Mad2:Mad2*Cdc20:C-Mad2:Mad2*

Time in seconds

Conc

entr

atio

n in

A B

C D

Int. J. Mol. Sci. 2014, 15 19084

Figure 2. Schematic diagram of the “Mad2 template” model. Depicted is a projectionof our mitotic cell. The gray disk corresponds to the cell while the green disk presentsthe last unattached kinetochore. Inscribed is the reaction network of the “Mad2 template”model with species localization. The core model consists of five species (yellow boxes) andtheir three reactions (black boxes R1, R2 and R3). The full model furthermore containsthe amplification pathway, which is symbolized by the blue boxes (Cdc20:C-Mad2:Mad2*,R4 and R5). The arrows’ directions indicate whether a species is a reactant or a product.Mad1:C-Mad2 and Mad1:C-Mad2:Mad2* are localized predominantly in the vicinity of thekinetochore and the later catalyzes reactions R2 and R3. The autocatalytic amplificationreactions of Cdc20:C-Mad2 occur in the cytosol (see Section 2 for details).

Using only the reduced SAC model we figure out the most suitable value for the diffusion constantof free Mad2. Therefore, we vary the unknown diffusion constant of O-Mad2 for wide range between0–50 µm2s�1. The amount of formed Cdc20:C-Mad2 and time until the simulation reaches steady stateare our measurements for the quality of the SAC model. We found no major effects if the diffusionexceeds 20 µm2s�1(see Figure 3A). Since the highest diffusion rate in the model is 19.5 µm2s�1

(from Cdc20), we mainly focus our comparison on high diffusion rate of Mad2 (20 µm2s�1) versusslow diffusion rate (5 µm2s�1). The result of this comparison shows only minor effects for the formation

Int. J. Mol. Sci. 2014, 15 19085

of the Cdc20:Mad2 complex (see Figure 1B,C). This outcome is consistent for both human and buddingyeast models (yeast data in supplement, due to its effect is even less).

Figure 3. Diffusion and convection parameter estimation of the species O-Mad2. (A)Depicted is the time dependent concentration of Cdc20:C-Mad2 with different diffusionconstants of O-Mad2. The variation of this rate is shown next to the plot with the colorgradient. The black curve denotes a slow diffusion (0 µm2s�1) while red curves presentfast diffusion values (up to 50 µm2s�1). At the end of the simulations it is a maximum of0.07µM of the Cdc20:C-Mad2 complex formed. If the diffusion constant D

i

is greater than20 µm2s�1no significant alteration can be observed in terms of produced Cdc20:C-Mad2.Thus, we used this value for all other simulations (dashed line); (B) presented is againthe time dependent concentration of Cdc20:C-Mad2. This time the convection rate ofO-Mad2 is varied according to the color gradient. The blue curve has no convection andconsequently is the same like the dashed plot in panel A. Red curves have a convection upto 10 µms�1. Exceeds the convection 4 µm2s�1, no alteration of the formed Cdc20:C-Mad2can be observed in steady state. As a result we used this value for the force, transportingO-Mad2 towards the spindle mid-zone.

0 50 100 150 200 250 300

0

0.01

0.02

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0.07

Time in seconds

Co

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tio

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nµ

M

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50 10

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M

Time in seconds

3.3. Mad2 Active Transport towards Spindle Mid-Zone

We propose an active transport of O-Mad2 towards the spindle mid-zone using convection(see Section 2). This results in a so called reaction-diffusion-convection system. As the magnitudeof the convection is completely unknown (symbol Q

i

in Formula (13)), we estimate the best valueregarding the amount of formed Cdc20:C-Mad2. We run simulations with a convection rate constantbetween 0–10 µms�1. Our results show that any convection constant larger than 4 µms�1has nosignificant influence to the formation of the Mad2:Cdc20 complex (cf. Figure 3B). Thus, we use aconvection rate of ( ~Q

i

= �4.0 ⇥ ~e

i

µMs�1) for O-Mad2 to achieve an active transport towards themidpoint of the cell.

With the human model settings (cf. Table 1), this transport of O-Mad2 towards the spindle mid-zonegreatly enhances the formation of Cdc20:C-Mad2 (see Figure 1D). Interestingly, the level of Cdc20inhibition is even higher with the active transport model (about three times the amount of the standard

Int. J. Mol. Sci. 2014, 15 19086

model). Moreover, the time required to reach steady state in the active transport model was very fastand near to reality (about 1–2 min, cf. Figure 1D). In comparison it takes more than twice as long inthe model without an active transport (about 5 min, cf. Figure 1A). As expected, with the budding yeastmodel settings (see Table 1) no significant improvements are recorded (data shown in supplement). Thisis due to the fact that budding yeast cells are relatively small in size and diffusion alone can provideenough O-Mad2 at the kinetochore.

Taking together, our in-silico data suggest that an active transport of O-Mad2 towards the spindlemid-zone may increase the efficiency of the “Mad2 template” model in human cells. Activated Mad2inhibits significantly more Cdc20 via the Cdc20:C-Mad2 complex then in other models, without anyconvectional forces.

4. Conclusions

In eukaryotic cells, the mitotic spindle assembly checkpoint is an important regulatory mechanismfor accurate chromosome segregation. The checkpoint guarantees that each chromosome has establishedits attachment to the spindle apparatus before sister-chromatid separation is initiated in anaphase.Consequently, its failure has been implicated in tumorigenesis [10,56]. Quantitative analysis andcomputational modeling are very important tools to elucidate how such elaborate systems work. Sofar, mathematical models have helped to enlighten kinetochores structure and with that the mitoticcheckpoint mechanism [26–30,32,33,57–62]. These models helped to pinpoint advantages and problemsof putative regulatory mechanisms. However, the conclusion of these models is that SAC mechanismalone is insufficient to guard chromosome attachment.

In this paper, we have analyzed a spatial SAC model from two different organisms, human andbudding yeast Saccharomyces cerevisiae, with different parameter sets. We first verified the hypothesisthat the autocatalytic loop is not contributing to the formation of Cdc20:C-Mad2. Based on this result, wereduce the model and determine the diffusion constant of free Mad2 as 20 µm2s�1. Even with a higherdiffusion the required Mad2 at the kinetochore could not be covered. Therefore, we propose an additionalmechanism for SAC functioning which is an active transport of O-Mad2 towards the spindle mid-zone inhuman cells. The simulation of our model indicates that this mechanism greatly enhances Cdc20:Mad2formation. The experimental validity of this prediction is pending and needs to be investigated in thefuture. The outcome of the yeast simulations are consistent with the in-vitro experiments, if we assume100-fold higher interaction rates.

This reveals that the model for the budding yeast is incomplete or in a lack of experimental findings.In contrast to the human setup, the needed Mad2 at the kinetochore is covered alone by diffusion, asbudding yeast cells are way smaller than human cells. Future model validation in an integrative approach,theoretical and experimental, will help to reveal the SAC mechanism on the molecular level. An examplefor an analogous mechanism is the spindle position checkpoint [63,64]. We anticipate that a systemsbiological approach of the SAC mechanism will serve as a basis to integrate future findings and evaluatenovel hypothesis related to checkpoint architectures and regulation.

Int. J. Mol. Sci. 2014, 15 19087

Supplementary Materials

Supplemenatry figure can be found at http://www.mdpi.com/1422-0067/15/10/19074/s1.

Acknowledgments

We thank Jakob Fischer for critical reading the manuscript. This work was supported by the EuropeanCommissions HIERATIC Grant 062098/14.

Author Contributions

Bashar Ibrahim conceived and designed the model. Richard Henze performed the simulations.Bashar Ibrahim and Richard Henze analyzed the data. Bashar Ibrahim wrote the manuscript with helpof Richard Henze.

Conflicts of Interest

The authors declare no conflict of interest.

References

1. Minshull, J.; Sun, H.; Tonks, N.K.; Murray, A.W. A MAP kinase-dependent spindle assemblycheckpoint in Xenopus egg extracts. Cell 1994, 79, 475–486.

2. Hartwell, L.H.; Culotti, J.; Reid, B. Genetic control of the cell-division cycle in yeast. I. Detectionof mutants. Proc. Natl. Acad. Sci. USA 1970, 66, 352–359.

3. King, R.W.; Peters, J.M.; Tugendreich, S.; Rolfe, M.; Hieter, P.; Kirschner, M.W. A 20S complexcontaining CDC27 and CDC16 catalyzes the mitosis-specific conjugation of ubiquitin to cyclin B.Cell 1995, 81, 279–288.

4. Sethi, N.; Monteagudo, M.C.; Koshland, D.; Hogan, E.; Burke, D.J. The CDC20 geneproduct of Saccharomyces cerevisiae, a beta-transducin homolog, is required for a subset ofmicrotubule-dependent cellular processes. Mol. Cell. Biol. 1991, 11, 5592–5602.

5. Shirayama, M.; Zachariae, W.; Ciosk, R.; Nasmyth, K. The Polo-like kinase Cdc5p and theWD-repeat protein Cdc20p/fizzy are regulators and substrates of the anaphase promoting complexin Saccharomyces cerevisiae. EMBO J. 1998, 17, 1336–1349.

6. Sudakin, V.; Ganoth, D.; Dahan, A.; Heller, H.; Hershko, J.; Luca, F.C.; Ruderman, J.V.;Hershko, A. The cyclosome, a large complex containing cyclin-selective ubiquitin ligase activity,targets cyclins for destruction at the end of mitosis. Mol. Biol. Cell 1995, 6, 185–197.

7. Robbins, J.A.; Cross, F.R. Regulated degradation of the APC coactivator Cdc20. Cell Div. 2010,5, 23.

8. Cimini, D.; Degrassi, F. Aneuploidy: A matter of bad connections. Trends Cell Biol. 2005, 15,442–451.

9. Suijkerbuijk, S.J.; Kops, G.J. Preventing aneuploidy: The contribution of mitotic checkpointproteins. Biochim. Biophys. Acta 2008, 1786, 24–31.

Int. J. Mol. Sci. 2014, 15 19088

10. Morais da Silva, S.; Moutinho-Santos, T.; Sunkel, C.E. A tumor suppressor role of the Bub3 spindlecheckpoint protein after apoptosis inhibition. J. Cell Biol. 2013, 201, 385–393.

11. Holland, A.J.; Cleveland, D.W. Boveri revisited: Chromosomal instability, aneuploidy andtumorigenesis. Nat. Rev. Mol. Cell Biol. 2009, 10, 478–487.

12. Li, M.; Fang, X.; Wei, Z.; York, J.P.; Zhang, P. Loss of spindle assembly checkpoint-mediatedinhibition of CDC20 promotes tumorigenesis in mice. J. Cell Biol. 2009, 185, 983–994.

13. Li, R.; Murray, A.W. Feedback control of mitosis in budding yeast. Cell 1991, 66, 519–531.14. Hoyt, M.A.; Totis, L.; Roberts, B.T. S. cerevisiae genes required for cell cycle arrest in response to

loss of microtubule function. Cell 1991, 66, 507–517.15. Vagnarelli, P.; Earnshaw, W.C. Chromosomal passengers: The four-dimensional regulation of

mitotic events. Chromosoma 2004, 113, 211–222.16. Fisk, H.A.; Mattison, C.P.; Winey, M. A field guide to the Mps1 family of protein kinases.

Cell Cycle 2004, 3, 439–442.17. Karess, R. Rod-Zw10-Zwilch: A key player in the spindle checkpoint. Trends Cell Biol. 2005,

15, 386–392.18. Lu, Y.; Wang, Z.; Ge, L.; Chen, N.; Liu, H. The RZZ complex and the spindle assembly checkpoint.

Cell Struct. Funct. 2009, 34, 31–45.19. Buffin, E.; Emre, D.; Karess, R.E. Flies without a spindle checkpoint. Nat. Cell Biol. 2007,

9, 565–572.20. Raff, J.W.; Jeffers, K.; Huang, J.Y. The roles of Fzy/Cdc20 and Fzr/Cdh1 in regulating the

destruction of cyclin B in space and time. J. Cell Biol. 2002, 157, 1139–1149.21. Saffery, R.; Irvine, D.V.; Griffiths, B.; Kalitsis, P.; Choo, K.H. Components of the human

spindle checkpoint control mechanism localize specifically to the active centromere on dicentricchromosomes. Hum. Genet. 2000, 107, 376–384.

22. Saffery, R.; Irvine, D.V.; Griffiths, B.; Kalitsis, P.; Wordeman, L.; Choo, K.H. Humancentromeres and neocentromeres show identical distribution patterns of >20 functionally importantkinetochore-associated proteins. Hum. Mol. Genet. 2000, 9, 175–185.

23. Williams, B.C.; Li, Z.; Liu, S.; Williams, E.V.; Leung, G.; Yen, T.J.; Goldberg, M.L. Zwilch, a newcomponent of the ZW10/ROD complex required for kinetochore functions. Mol. Biol. Cell 2003,14, 1379–1391.

24. Kops, G.J.P.L.; Weaver, B.A.A.; Cleveland, D.W. On the road to cancer: Aneuploidy and themitotic checkpoint. Nat. Rev. Cancer 2005, 5, 773–785.

25. Fang, G.; Yu, H.; Kirschner, M.W. The checkpoint protein MAD2 and the mitotic regulator CDC20form a ternary complex with the anaphase-promoting complex to control anaphase initiation.Genes Dev. 1998, 12, 1871–1883.

26. Doncic, A.; Ben-Jacob, E.; Barkai, N. Evaluating putative mechanisms of the mitotic spindlecheckpoint. Proc. Natl. Acad. Sci. USA 2005, 102, 6332–6337.

27. Sear, R.P.; Howard, M. Modeling dual pathways for the metazoan spindle assembly checkpoint.Proc. Natl. Acad. Sci. USA 2006, 103, 16758–16763.

Int. J. Mol. Sci. 2014, 15 19089

28. Ibrahim, B.; Dittrich, P.; Diekmann, S.; Schmitt, E. Mad2 binding is not sufficient for completeCdc20 sequestering in mitotic transition control (an in silico study). Biophys. Chem. 2008,134, 93–100.

29. Ibrahim, B.; Diekmann, S.; Schmitt, E.; Dittrich, P. In-silico modeling of the mitotic spindleassembly checkpoint. PLoS One 2008, 3, e1555.

30. Ibrahim, B.; Schmitt, E.; Dittrich, P.; Diekmann, S. In silico study of kinetochore control,amplification, and inhibition effects in MCC assembly. BioSystems 2009, 95, 35–50.

31. De Antoni, A.; Pearson, C.G.; Cimini, D.; Canman, J.C.; Sala, V.; Nezi, L.; Mapelli, M.;Sironi, L.; Faretta, M.; Salmon, E.D.; Musacchio, A. The Mad1/Mad2 complex as a templatefor Mad2 activation in the spindle assembly checkpoint. Curr. Biol. 2005, 15, 214–225.

32. Simonetta, M.; Manzoni, R.; Mosca, R.; Mapelli, M.; Massimiliano, L.; Vink, M.; Novak, B.;Musacchio, A.; Ciliberto, A. The influence of catalysis on mad2 activation dynamics. PLoS Biol.2009, 7, e10.

33. Lohel, M.; Ibrahim, B.; Diekmann, S.; Dittrich, P. The role of localization in the operation of themitotic spindle assembly checkpoint. Cell Cycle 2009, 8, 2650–2660.

34. Musacchio, A.; Salmon, E.D. The spindle-assembly checkpoint in space and time. Nat. Rev. Mol.Cell Biol. 2007, 8, 379–393.

35. Howell, B.J.; Hoffman, D.B.; Fang, G.; Murray, A.W.; Salmon, E.D. Visualization of Mad2dynamics at kinetochores, along spindle fibers, and at spindle poles in living cells. J. Cell Biol.2000, 150, 1233–1250.

36. Fang, G. Checkpoint protein BubR1 acts synergistically with Mad2 to inhibit anaphase-promotingcomplex. Mol. Biol. Cell 2002, 13, 755–766.

37. Stegmeier, F.; Rape, M.; Draviam, V.M.; Nalepa, G.; Sowa, M.E.; Ang, X.L.; McDonald, E.R.;Li, M.Z.; Hannon, G.J.; Sorger, P.K.; et al. Anaphase initiation is regulated by antagonisticubiquitination and deubiquitination activities. Nature 2007, 446, 876–881.

38. Wang, Z.; Shah, J.V.; Berns, M.W.; Cleveland, D.W. In vivo quantitative studies of dynamicintracellular processes using fluorescence correlation spectroscopy. Biophys. J. 2006, 91, 343–351.

39. Cherry, L.M.; Faulkner, A.J.; Grossberg, L.A.; Balczon, R. Kinetochore size variation inmammalian chromosomes: An image analysis study with evolutionary implications. J. Cell Sci.1989, 92, 281–289.

40. Joglekar, A.P.; Bloom, K.; Salmon, E.D. In vivo protein architecture of the eukaryotic kinetochorewith nanometer scale accuracy. Curr. Biol. 2009, 19, 694–699.

41. Haase, J.; Mishra, P.K.; Stephens, A.; Haggerty, R.; Quammen, C.; Taylor, R.M.;Yeh, E.; Basrai, M.A.; Bloom, K. A 3D map of the yeast kinetochore reveals the presence ofcore and accessory centromere-specific histone. Curr. Biol. 2013, 23, 1939–1944.

42. Morgan, D. The Cell Cycle: Principles of Control; Sinauer Associates, Inc.: Sunderland, MA,USA, 2007.

43. Yamaguchi, M.; Namiki, Y.; Okada, H.; Mori, Y.; Furukawa, H.; Wang, J.; Ohkusu, M.;Kawamoto, S. Structome of Saccharomyces cerevisiae determined by freeze-substitution and serialultrathin-sectioning electron microscopy. J. Electron. Microsc. 2011, 60, 321–335.

Int. J. Mol. Sci. 2014, 15 19090

44. Sudakin, V.; Chan, G.K.; Yen, T.J. Checkpoint inhibition of the APC/C in HeLa cells is mediatedby a complex of BUBR1, BUB3, CDC20, and MAD2. J. Cell Biol. 2001, 154, 925–936.

45. Malureanu, L.A.; Jeganathan, K.B.; Hamada, M.; Wasilewski, L.; Davenport, J.;van Deursen, J.M. BubR1 N terminus acts as a soluble inhibitor of cyclin B degradation byAPC/C(Cdc20) in interphase. Dev. Cell 2009, 16, 118–131.

46. Kulukian, A.; Han, J.S.; Cleveland, D.W. Unattached kinetochores catalyze production of ananaphase inhibitor that requires a Mad2 template to prime Cdc20 for BubR1 binding. Dev. Cell2009, 16, 105–117.

47. Medema, R.H. Relaying the checkpoint signal from kinetochore to APC/C. Dev. Cell 2009,16, 6–8.

48. Herzog, F.; Primorac, I.; Dube, P.; Lenart, P.; Sander, B.; Mechtler, K.; Stark, H.; Peters, J.M.Structure of the anaphase-promoting complex/cyclosome interacting with a mitotic checkpointcomplex. Science 2009, 323, 1477–1481.

49. Braunstein, I.; Miniowitz, S.; Moshe, Y.; Hershko, A. Inhibitory factors associated withanaphase-promoting complex/cylosome in mitotic checkpoint. Proc. Natl. Acad. Sci. USA2007, 104, 4870–4875.

50. Eytan, E.; Braunstein, I.; Ganoth, D.; Teichner, A.; Hittle, J.C.; Yen, T.J.; Hershko, A. Twodifferent mitotic checkpoint inhibitors of the anaphase-promoting complex/cyclosome antagonizethe action of the activator Cdc20. Proc. Natl. Acad. Sci. USA 2008, 105, 9181–9185.

51. Loew, L.M.; Schaff, J.C. The virtual cell: A software environment for computational cell biology.Trends Biotechnol. 2001, 19, 401–406.

52. MATLAB. version 8.3.0.532 (R2014a); The MathWorks Inc.: Natick, MA, USA, 2014.53. Lince-Faria, M.; Maffini, S.; Orr, B.; Ding, Y.; Florindo, C.; Sunkel, C.; Tavares, A.;

Johansen, J.; Johansen, K.; Maiato, H. Spatiotemporal control of mitosis by the conserved spindlematrix protein megator. J. Cell Biol. 2009, 184, 647 – 657.

54. Reimann, J.D.; Freed, E.; Hsu, J.Y.; Kramer, E.R.; Peters, J.M.; Jackson, P.K. Emi1 is a mitoticregulator that interacts with Cdc20 and inhibits the anaphase promoting complex. Cell 2001,105, 645–655.

55. Wang, W.; Kirschner, M.W. Emi1 preferentially inhibits ubiquitin chain elongation by theanaphase-promoting complex. Nat. Cell Biol. 2013, 15, 797–806.

56. Marzo, I.; Naval, J. Antimitotic drugs in cancer chemotherapy: Promises and pitfalls.Biochem. Pharmacol. 2013, 86, 703–710.

57. Ibrahim, B.; Dittrich, P.; Diekmann, S.; Schmitt, E. Stochastic effects in a compartmental modelfor mitotic checkpoint regulation. J. Integr. Bioinform. 2007, 4, 66.

58. Mistry, H.B.; MacCallum, D.E.; Jackson, R.C.; Chaplain, M.A.; Davidson, F.A. Modeling thetemporal evolution of the spindle assembly checkpoint and role of Aurora B kinase. Proc. Natl.Acad. Sci. USA 2008, 105, 20215–20220.

59. Kreyssig, P.; Escuela, G.; Reynaert, B.; Veloz, T.; Ibrahim, B.; Dittrich, P. Cycles and theQualitative Evolution of Chemical Systems. PLoS One 2012, 7, e45772.

Int. J. Mol. Sci. 2014, 15 19091

60. Ibrahim, B.; Henze, R.; Gruenert, G.; Egbert, M.M.; Huwald, J.; Dittrich, P. Rule-based modelingin space for linking heterogeneous interaction data to large-scale dynamical molecular complexes.Cells 2013, 2, 506–544.

61. Tschernyschkow, S.; Herda, S.; Gruenert, G.; Doring, V.; Gorlich, D.;Hofmeister, A.; Hoischen, C.; Dittrich, P.; Diekmann, S.; Ibrahim, B. Rule-based modelingand simulations of the inner kinetochore structure. Prog. Biophys. Mol. Biol. 2013, 113, 33–45.

62. Kreyssig, P.; Wozar, C.; Peter, S.; Veloz, T.; Ibrahim, B.; Dittrich, P. Effects of smallparticle numbers on long-term behaviour in discrete biochemical systems. Bioinformatics 2014,30, i475–i481.

63. Caydasi, A.K.; Lohel, M.; Grunert, G.; Dittrich, P.; Pereira, G.; Ibrahim, B. A dynamical model ofthe spindle position checkpoint. Mol. Syst. Biol. 2012, 8, 582.

64. Caydasi, A.; Ibrahim, B.; Pereira, G. Monitoring spindle orientation: Spindle position checkpointin charge. Cell Div. 2010, 5, 28.

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