Modeling In Systems Biology Modeling In Systems Biology

By: Payman Gifani

AmirKabir University of Technology Biomedical Engineering Department

Advanced System Modeling28/2/87

System biology RootsSystem biology RootsSystem biology RootsSystem biology RootsWhereas the foundations of systems biology-at-large are generally recognizedWhereas the foundations of systems biology at large are generally recognized as being as far apart as 19th century systems biology has origin in the expansion of molecular biology to genome-

id lwide analyses.the emergence of this ‘new’ field constitutes a ‘paradigm shift’ for molecular biology, which ironically has often focused on reductionist thinking.biology, which ironically has often focused on reductionist thinking. Systems thinking in molecular biology will likely be dominated by formal integrative analysis going forward rather than solely being driven by high-h h h l ithroughput technologies.

It is, however, incorrect to state that integrative thinking is new to molecular biology.biology.High-throughput technologies have made the scale of such inquiries much larger, enabling us to view the genome as the ‘system’ to study. Thus, the

l i f bi l b i hpopular contemporary view of systems biology may be synonymous with ‘genomic’ biology.

System Biology RootsSystem Biology Roots

first root—which stemmed from fundamental discoveries abo t the nat re of genetic material str ct ralabout the nature of genetic material, structural characterization of macromolecules and later developmentsin recombinant and high throughput technologiesin recombinant and high-throughput technologiesSecond root, which sprung from nonequilibrium thermodynamics theory in the 1940s the elucidation ofthermodynamics theory in the 1940s, the elucidation of biochemical pathways and feedback controls in unicellular organisms and the emerging recognition of networks inorganisms and the emerging recognition of networks in biology.

Metabolic control analysisMetabolic control analysisMetabolic control analysisMetabolic control analysisDistributed Control Mechanism instead Distributed Control Mechanism instead

of Centralized Controlof Centralized Controlof Centralized Controlof Centralized Controlcontemporary systems biology has an historical root outside mainstream molecular biology, ranging from basic principles of self-organization in nonequilibrium thermodynamics, through large-scale flux and kinetic models to ‘genetic g g gcircuit’ thinking in molecular biology.‘Systems thinking’ differs from ‘component thinking’Systems thinking differs from component thinking and requires the development of new conceptual frameworksframeworks.

C t l t t t b tC t l t t t b tControl was not a component property but a Control was not a component property but a network propertynetwork propertyp p yp p y

M t b li t l l i (MCA) d l d i th l ti t d k lMetabolic control analysis (MCA), developed in the early seventies, presented a key example of approaches to characterize properties of networks of interacting chemical reactions.At this time, thinking in biochemistry was dominated by the concept that there had to be a i l ‘ t li iti ’ t t th b i i f ll t b li thsingle ‘rate-limiting’ step at the beginning of all metabolic pathways.

However, the application of these criteria to some metabolic pathways suggested that they contained more than a single rate-limiting step. Network thinking through MCA helped to

l thi dresolve this paradox.Instead, they found a theorem stating that all the flux-control coefficients must sum to unity. This result then suggested that there need not be a single rate-limiting step to a pathway and h i d ib i l l h l f h k Ththat instead many enzymes can contribute simultaneously to the control of the network. Thus,

control was not a component property but a network property.

CintrolCintrol & Network & Network An important aspect of systems biology is to relate the system properties to the molecular properties of components that comprise a network.The kinetics based sensitivity analysis by MCA showed that by focusingThe kinetics-based sensitivity analysis by MCA, showed that by focusing on the properties of an individual component, one cannot properly decipher its role in the context of a whole network.The connectivity laws proven by MCA pinpointed how that distribution ofThe connectivity laws proven by MCA, pinpointed how that distribution of control relates to network structure and the kinetic properties of all network components simultaneously. Similarly the topological analyses of network structure have revealed theSimilarly, the topological analyses of network structure have revealed the existence of network-based definitions of pathways that can be used mathematically to represent all possible functional states of reconstructed networksnetworks. Thus, a growing number of methods now exist to analyze the properties mathematically of the large-scale networks that we are now able to reconstruct based on high throughput datareconstruct based on high-throughput data.

The evolution of molecular biology The evolution of molecular biology The evolution of molecular biology The evolution of molecular biology into systems into systems biologybiology

foundational discoveries of the structure and information coding of DNA and protein, molecular biology◦ The description of restriction enzymes and cloning were major breakthroughs in the 1970s, p y g j g ,

ushering in the era of genetic engineering and biotechnology.◦ In the 1980s, we began to see the scale-up of some of the fundamental experimental approaches

of molecular biology.◦ Automated DNA sequencers began to appear and reached genomescale sequencing in the mid-◦ Automated DNA sequencers began to appear and reached genomescale sequencing in the mid-

1990s.◦ Automation, miniaturization and multiplexing of various assays led to the generation of

additional ‘omics’ data types.Th l l f d t t d b th h l d t id th i thThe large volumes of data generated by these approaches led to rapid growth in the field of bioinformatics, again largely emanating from the reductionist perspective. Although this effort was mostly focused on statistical models and object classification approaches in the late 1990sclassification approaches in the late 1990s, it was recognized that a more formal and mechanistic framework was needed to analyze multiple high-throughput data types systematically. This need led to efforts toward genome-scale model building to analyze theThis need led to efforts toward genome scale model building to analyze the systems properties of cellular function.

Molecular selfMolecular self--organizationorganization

integration of multiple molecular processes is fundamental to the living cell. Biochemical processes necessitate the production of entropy (chaos in the thermodynamic sense) as driving forcedriving force.The paradox felt by many, but expressed by Schrödinger in his war-time lectures, was how one could explain the progressive ordering that occurs in developmental biology(that is, the ‘self-organization,’ decrease in chaos) when entropy (‘chaos’) must be increased.

The answer was that one process could produce order (negative entropy or negentropy) provided it was coupled to a second process that produced more chaos (entropy): coupling, another word for integration of processes, is therefore essential for life.a law for such systems of coupled processes: close to equilibrium the dependence of the one process rate on the driving force of the other process should equal the dependence of the other process rate on the former driving force.These approaches were called nonequilibrium thermodynamics and constituted a prelude to systems pp q y p ybiology at the cell and molecular levels.

NNonequilibriumonequilibrium thermodynamicsthermodynamicsThe problem of biological self-organization was to understand how structures, oscillations or waves arise in a steady and homogenous environment, a phenomenon called symmetry breaking.Turing led the way, but the Prigogine school and others developed the topic from the perspective of nonequilibrium thermodynamics in molecular contexts.q y

They demonstrated how having a sufficient number of nonlinearly interacting chemical processes in a single system , could lead to symmetry-breaking as a result of self-amplification of random fluctuationsfluctuations.Only relatively recently has systems-based analysis been used to reveal that the oscillations are simultaneously controlled by many steps in the intracellular network and how the oscillations in the individual cells synchronize actively.

with the more recent experimental capability to inspect single cells dynamically, more and more cells are seen to exhibit asynchronous oscillations of all sorts and some of these cases are up for systems biology analysis.

a network of reactions was shown to be essential for systems biology reaching one step beyond cell biology, again by combining mathematical modeling with experimental molecular information.

the need for theory model the need for theory model the need for theory, model the need for theory, model building and simulationbuilding and simulation

Systems modeling and simulation in molecular biology was once seen y g gyas purely theoretical and not particularly relevant to understanding ‘real’ biology.

However, now that molecular biology has become such a data-richfield, the need for theory, model building and simulation has emerged.

The systems-directed root always had the ambition of discovering fundamental principles and laws, such as those of nonequilibriumthermodynamics and MCAthermodynamics and MCA.This ambition should now extend to Systems biology.

I f tI f tIn futureIn future

Systems biology is a thorough science with its own quest for scientific principles at the interface of physics, chemistry and biology, with its

k bl i t f f ti lit h t i ti i ti dremarkable mixture of functionality, hysteresis, optimization and physical chemical limitations.

Al h h h h i l d li f h l b d hAlthough the mathematical modeling of whole-body human systems cannot yet be linked to genome-wide data and models, data analysis and modeling are likely to contribute to the success of realizing the goal

f i di id li d di iof individualized medicine.

Even if we have to rely on less precise models than the currently available genome-scale models of microorganisms, systems biology may soon lead to better diagnosis and dynamic therapies of human disease than the qualitative methodology presently in use.

Computational Molecular BiologyComputational Molecular Biology

Complex assemblies of interacting proteins carry out most of the interesting jobs in a cell, such as metabolism, DNA synthesis, movement and i f ti iinformation processing.

These physiological properties play out as a subtle molecular dance, choreographed by underlying regulatory networks.

To understand this dance, a new breed of theoretical molecular biologists reproduces thesetheoretical molecular biologists reproduces these networks in computers and in the mathematical language of dynamical systems.

The ultimate goal of molecular cell biology is toThe ultimate goal of molecular cell biology is to understand the physiology of living cells in terms of the information that is encoded in the genome of the cell.

Modeling ScaleModeling ScaleModeling ScaleModeling Scale

sophisticated computational methods will be needed to manage, interpret and understand the complexity of biological information.The analysis of nucleotide strings — to assemble genomic sequences and to identify protein-coding regions — is already big businessregions is already big business. Database services, such as multiple sequence alignments, are now familiar computational tools that are easily used by experimentalists. Conversely, the prediction of protein structure from primary sequence, despite its urgency, is not so easyeasy.

Many scientific and computational problems remain to be solved before protein-structure prediction becomes a reliable and readily available tool.

The next level of organization — predicting protein function from structure — involves new theoretical challenges. Useful tools exist to predict protein–ligand binding, based on molecular dynamics simulations. Sophisticated methods, based on quantum mechanics, statistical mechanics y p , q ,and electrostatics, have been used to understand diverse protein-mediated processes

Even if we could predict the physicochemical properties (binding constants, rate constants, transport coefficients and so on) of a protein from its three-dimensional structure we would still be in thecoefficients and so on) of a protein from its three dimensional structure, we would still be in the dark about the properties of the networks in which individual proteins operate.

D i I t ti D i I t ti Dynamic Interaction Dynamic Interaction

The dynamical properties of a cell are implicit in the ptopology of the protein networks that underlie cell physiology.

The wiring diagram implies a set of dynamical relationships among itsrelationships among its component and, therefore, demands to be converted into a set of mathematical equations that describe the temporal and spatial evolution of the system

BBrains of the cellrains of the cellBBrains of the cellrains of the cell

Of special significance are the regulatory pathways that control cell behaviour, because these pathways are the ‘brains’ of the cell.

They receive information from outside and inside the cell by signal-transduction pathways, process the information to make ‘decisions’, then trigger responses that are appropriate to the survival and reproduction of the cell.

The molecular signals that control the proliferation of mammalian cells are extremely complex.

What tools do we have to help us understand how the precise spatio-temporal organization of a cell arises from the molecular interactions of the protein machinery inside? Are there ways to compute physiology from network topology?machinery inside? Are there ways to compute physiology from network topology? And is there a solid theoretical foundation for such computations?

Modeling ApproachesModeling Approaches

The type of equations to be used depends on the bi l i l ti d id ti F i tbiological questions under consideration. For instance, genetic regulatory circuits might be modelled by differential equations or by Boolean networksdifferential equations or by Boolean networks.

Spatial signalling might be modelled by partialSpatial signalling might be modelled by partial differential equations or by cellular automata. Functional or integro-differential equations are called for when g qsignificant time delays enter the process, or when spatial averaging is involved. When small numbers of

l l i l d t h ti d l t b dmoleculesare involved, stochastic models must be used.

Dynamical System AnalysisDynamical System AnalysisDynamical System AnalysisDynamical System Analysis

In any case we need a more intuitive, but still reliable, approach to the solution of differential equations.

We need insight into why a control system behaves the way it does, and how this behavior depends on parameter values., p p

With such insight we have a better chance of developing well t i d d l th t hi b th l ‘ tit tiparameterized models that achieve both goals : ‘quantitative

predictions’ and ‘general principles’.

Are there general principles and useful tools to sort out the complex behavior of realistic models of molecular regulatory

t k ?networks?

BIFURCATION theoryBIFURCATION theoryBIFURCATION theoryBIFURCATION theory

these transmutations can be visualized by choosing a single ariable as being representati e of all thesingle variable as being representative of all the interacting proteins in the network, and a single

t t ti f ll th t t tparameter as representative of all the rate constants that are involved in these reactions.

By plotting the proxy variable against the proxy parameter (a bifurcation diagram), we get a visual representation of the behaviour of the dynamical system in dependence on its parameters.

M t t h t fi th M t t h t fi th Mutant phenotypes confirm the Mutant phenotypes confirm the modelmodel

In model, to move from S/G2 to M, a cell must grow large enough to get past theIn model, to move from S/G2 to M, a cell must grow large enough to get past the ‘nose’ of the bifurcation diagram (size = 1.75 ). Cell size at the end of the nose depends sensitively on the expression levels of wee1 and cdc25 in the model, exactly as in experiments. In particular, in wee1− cells (non-functional wee1 protein), the nose of the S/G2 state disappears; the cell enters mitosis and divides at an abnormally small size, and is then captured by the G1/S module. Th t t ll i t ll d t th G1/S t iti ( i 0 8) dThese mutant cells are now size controlled at the G1/S transition (size = 0.8) and divide at about half the size of wild-type cells, exactly as observed.Indeed, the ‘wee’phenotype of these mutant cells attracted Paul Nurse’s attention to the significance of a trio of genes (wee1 cdc2 and cdc25) that provided the firstto the significance of a trio of genes (wee1, cdc2 and cdc25) that provided the first glimpse of the cell-cycle regulatory network in eukaryotes.

For this discovery he received the Nobel Prize for Physiology or Medicine inFor this discovery, he received the Nobel Prize for Physiology or Medicine in 2001, together with Leland Hartwell and Tim Hunt.

The central themes The central themes Cell-cycle control is a particularly compelling example of the dynamical-systems approach to cell physiology, because cell-cycle transitions are so evidently connected to bifurcations of vector fieldsevidently connected to bifurcations of vector fields.

The same approach has been used to advantage in numerous studies of other molecular regulatory systemsother molecular regulatory systems.

Although much of this work uses nonlinear differential equations and bif ti l i i it i i t t t h tibifurcation analysis, in some cases it is more appropriate to use stochasticequations, Boolean networks, or cellular automata, and different tools from the theory of dynamical systems.

But in all cases, the central themes are the same: a dynamical system describes how the network evolves in state space, and these changes d i h h i l i l b h i f h lldetermine the physiological behaviour of the cell.

CCommon languageommon languageSpontaneous spatial and temporal organization of dynamical systems is the unifying principle of the ‘last step’ of computational molecular biology.

Analysis and simulation of mathematical models provides a rigorous way to correlate cell physiology with underlying molecular networks.

Experimentalists need to be familiar with the basic ideas that make this correlation possible (vector fields, stability and bifurcations),

just as theoreticians, if they are to make a useful contribution to the life sciences, need to know the basic principles of molecular genetics, protein interactions and biochemical regulation. g

Only when the two communities speak a common language can they collaborate fruitfully on problems of network dynamics and cell physiology.

Bifurcation Definition Bifurcation Definition Consider an autonomous system of ordinary differential equations (ODEs)

where f is smooth. A bifurcation occurs at parameter if there are parameter values arbitraily close to with dynamics topologically inequivalent fromdynamics topologically inequivalent from those at . For example, the number or stability of equilibria or periodic orbits of may change

i h b i f fwith perturbations of from . One goal of bifurcation theory is to produce parameter space maps or bifurcation diagrams that divide the parameter space g p pinto regions of topologically equivalentsystems. Bifurcations occur at points that do not lie in the interior of one of these regions.

An example of a bifurcation (Saddle nodeAn example of a bifurcation. (Saddle-node bifurcation on the plane in the system: and ).

Bifurcation theory Bifurcation theory Bifurcation theory provides a strategy for investigating the bifurcations that occur within a family. It does so by identifying ubiquitous patterns of bifurcations Each identifying ubiquitous patterns of bifurcations. Each bifurcation type or singularity is given a name.

Cl ifi ti f bif ti Classification of bifurcations One can view bifurcations as a failure of structural stabilitywithin a family. A starting point for classifying bifurcation

i h K k S l h h li h i types is the Kupka-Smale theorem that lists three generic properties of vector fields: ◦ hyperbolic equilibrium points

h b l d b◦ hyperbolic periodic orbits◦ transversal intersections of stable and unstable manifolds of

equilibrium points and periodic orbits.

Bif ti i iBif ti i iBifurcation in neuroscienceBifurcation in neuroscience

Bifurcation TypeBifurcation Type• Equilibria ◦ Saddle-Node Bifurcation ◦ Andronov-Hopf Bifurcation • Periodic Orbits• Periodic Orbits ◦ Fold Limit Cycle Bifurcation ◦ Flip Bifurcation (aka Period Doubling bifurcation) ◦ Neimark-Sacker Bifurcation (aka Torus bifurcation) • Global Bifurcations ◦ Homoclinic Bifurcation of equilibria ◦ Homoclinic tangencies of stable and unstable manifolds of periodic orbits ◦ Heteroclinic Bifurcation of equilibria and periodic orbitsHeteroclinic Bifurcation of equilibria and periodic orbits

• The classification of bifurcation types becomes more complex as their codimension increases. There are five types of "local" codimension two bif ti f ilib ibifurcations of equilibria:

◦ Bautin Bifurcation ◦ Bogdanov-Takens Bifurcation ◦ Cusp Bifurcation p◦ Fold-Hopf Bifurcation ◦ Hopf-Hopf Bifurcation

Bifurcation in Bifurcation in Bifurcation in Bifurcation in neuroscienceneuroscience

Codimension 1 bifurcations corresponding to a transition fromcorresponding to a transition from equilibrium to oscillatory dynamics.Fold bifurcation occurs when the Jacobian matrix at the equilibriumhas a zero eigenvalue. We refer to it as the saddle-node

i i t i l bif tion invariant circle bifurcation when the center manifold makes a loop.Andronov Hopf bifurcationAndronov-Hopf bifurcation occurs when the matrix has a pair of complex-conjugate eigenvalues with zero real partp

State Space SeparationState Space SeparationEquivalent Equivalent EquribriaEquribriaEquivalent Equivalent EquribriaEquribria

Chaos & BifurcationChaos & BifurcationBifurcation theory has intensively investigated varied topics that bear on chaotic and quasiperiodic dynamics. Much of this theory has been developed in the context of discrete timeMuch of this theory has been developed in the context of discrete time dynamical systems defined by iteration of mappings. In some areas, bifurcation theory of discrete systems goes farther than that for continuous time systems In particular an extensive deep theoryfor continuous time systems. In particular, an extensive, deep theory describing the properties of iterations of one dimensional mappings was developed over the last quarter of the twentieth century. This theory characterizes universal sequences of bifurcations and theThis theory characterizes universal sequences of bifurcations and the existence of chaotic attractors.

S f hi h i h i f i ibl i iSome of this theory carries over to the setting of invertible mappings in higher dimensions and to continuous time dynamical systems via Poincar'emaps.