3 Signal Transduction in Biological Systems and its …...cellular signal transduction: (1) How to formally represent kernel information of cellular signal transduction; (2) How to

57LIU Jian-Qin and PEPER Ferdinand

1 Introduction

Signal transduction plays a crucial role inthe complex dynamics of living cells to theextent that it is considered a fundamentalinformation processing mechanism in livingsystems. The recent availability of data onsignal transduction has the potential for thecreation of artificial systems conducting com-putation and communication using its inherentmechanisms. It also promises to give inspira-tion on building computation and communica-tion systems in our world that are based onnovel principles only used to date in biologicalorganisms.

It is yet unknown what advantages can begained from using biologically-inspired mech-anisms in the application to information pro-

cessing systems, but given the high efficiencyby which biological organisms function, itmakes sense to study them, especially in theframework of an information processing para-digm. The boundary conditions of theprocesses in biological systems tend to bequite different from what we are used to in ourdaily lives. Noise, for example, is large com-pared to signal levels. Mechanisms to copewith it in traditional systems include error cor-recting codes, and it is an interesting issue toinvestigate whether and to what extent suchtechniques are applicable in biological sys-tems. To investigate issues like these it isimportant to have a formal model describingbiological signal transduction. The most com-monly used model is the network, which hastopological features such as hubs in a scale-

3 Signal Transduction in Biological Systems and its Possible Uses in Computation and Communication Systems

LIU Jian-Qin and PEPER Ferdinand

Cellular signal transduction is crucial for cell communication and is described by signalingpathway networks that sustain the biological functions of living cells. The robustness of the mol-ecular mechanism of cellular signal transduction forms an important inspiration in the design offuture communication networks based on the information processing mechanisms of cellular sig-nal transduction. This paper discusses some important aspects of the computational issues oncellular signal transduction: (1) How to formally represent kernel information of cellular signaltransduction; (2) How to get a fixed point from a pathway network with feedbacks; and (3) Howto encode information in signal transduction pathways by error-correcting codes, such as toincrease the fault-tolerance of the system, while at the same time conform to the unstructurednature of such pathways. The above results provide a basis for innovative future communica-tions networks, with biological signaling pathway networks acting as references for systems withimproved performance in factors like robustness.

KeywordsCell communication, Signal transduction, Signaling pathway networks, Error Correcting codes

58 Journal of the National Institute of Information and Communications Technology Vol.55 No.4 2008

free network［1］. This suggest the explorationof efficient ways to systematically understandthe robustness of networks in terms of graphs,where the building block of the signal trans-duction networks that are treated as complexsystems called network motif in［2］are defined.Based on the technology of networks, we canmodel the dynamics of signal transductionnetworks and find a quantitative description ofits signaling mechanism that sustains therobustness of the corresponding cellular sig-naling processes, which have been widelyreported in the signal transduction networksfor chemotaxis, heat shock response, ultrasen-sitivity, and cell cycle control［3］.

In this paper, we formulate a model of sig-nal transduction in terms of graph theory inorder to increase our understanding of itsinformation processing role in biological sys-tems. A graph is a set of nodes, the so-calledvertexes, with relations between them calledthe edges. In the framework of a communica-tion network, for example, the edges representthe communication channels between thenodes between which communication takesplace. Applied to biological systems, weobtain a formulation of the structure of signal-ing pathways of signal transduction. We illus-trate our concepts through a particular proteincalled MAPK (Mitogen-Activating ProteinKinase), which plays an important role inintra-cellular communication processes. Wethen add concepts related to error correctingcodes to study how robustness of the aboveprocesses can be improved, as well as a newconcept of fixed points in biological systems,which serve to restore signals through nonlin-ear dynamics with feedback.

This paper is focused on computationalaspects of signal transduction in cells. In Sec-tion 2, based on the biochemical features ofsignal transduction processes, the data struc-ture of a graph is presented to formulate thesignaling pathway of signal transduction, inwhich MAPK is discussed as an instance ofpathways for describing the dynamicalprocesses of signal transduction networks. InSection 3, a fixed point phenomenon is stud-

ied as well as the robustness factor for devel-oping molecular communication systems. InSection 4, molecular codes for error correctionare described from the network level of signaltransduction.

2 Cellular signal transduction networks and their formalmodel

Signal transduction in cells is a biochemi-cal process that is of fundamental importancefor their functioning. In living cells signaltransduction is carried out by series of bio-chemical reactions that are regulated by genet-ic factors. The signals used in signal transduc-tion are usually quantified by concentrationsof the corresponding chemicals.

2.1 Some preliminaries of the biochemistry of signal transduction

Cellular signal transduction is defined as aphenomenon, process, or mechanism that real-izes a series of biochemical reactions in cellsin response to stimuli of chemical signals out-side the cells; this function includes so-calledcell communication.

Cell communication is the term used forthe communication processes in cells that takethe form of chemical signals and that is real-ized by the biochemical reactions in cellsthrough cellular signal transduction. Cellcommunication can be distinguished intointer-cellular and intra-cellular communica-tions.

Inter-cell communication describes howcells interact with each other. An importantmechanism in inter-cell communication isformed by signaling molecules, which are alsoknown as first messengers. Inter-cell commu-nication has four types［4］: (1) contact-depen-dent signaling, in which cells have directmembrane-to-membrane contact to exchangesignals, (2) paracrine signaling, in which cellsrelease signals into the extracellular space toact locally on neighboring cells, (3) synapticsignaling, in which neuronal cells transmit sig-nals electrically along their axons and release


neurotransmitters at synapses, and (4)endocrine signaling, in which hormonal sig-nals are secreted into the blood stream to bedistributed on a wide scale throughout anorganism’s body.

Intra-cell communication concerns thecommunication within cells. When an incom-ing first messenger molecule reaches a cell, itcannot directly pass the cell membrane, but isbound by specific receptors that effectuate theactivation of certain signaling molecule pro-teins within the cell. Referred to as secondarymessengers, these signals are relayed by achemical reaction process through a signalingcascade, which relays the signals to the nucle-us of the cell. In this paper we will mainlywrite about intra-cell communication. Impor-tant signaling molecules in cells are proteins.We will be especially interested in proteinsthat can bind to a phosphate molecule. Suchproteins are called phospho-proteins. When aphosphate is attached to a protein it becomesphosphorylized through a phosphorylationprocess; when it becomes detached it becomesdephosphorylized through a dephosphoryla-tion process. To switch between the twostates, special enzymes are required. Theenzyme that realizes phosphorylation is calledkinase, the enzyme that realizes dephosphory-lation is called phosphatase.

The phosphorylation / dephosphorylationstate of a protein will be used in the followingto encode the binary state of a variable. Thisstate can be detected by immunofluorescenceanalysis, which provides us with a possibletool to read out such a variable.

The signaling pathway in cells is a seriesof biochemical reactions, which have specificbiological functions.

2.2 Graph representation for signal transduction

The reactions in a signaling pathway willbe described by a directed graph with inputand output. By the graph representation, wecan get the information form of the signaltransduction network, from which we caninvestigate the structure, encoding, and net-

works of signal transduction in cells.To study the structural relations concern-

ing the transduction of signals, we definetransduction in a spatial form as a graph. Leta molecule be represented by a vertex (node)in a graph and let a biochemical reaction berepresented by an edge (link), then we obtain agraph

G = <V, E>,

where the vertex set is defined as the setV = {V1, V2, .., Vn} and the edge set is definedas E = E ( Vi,Vj) ( Vi,Vj∈ V ). Any parame-ters of a biochemical reaction represented byan edge are depicted as labels to that edge.

The direction of a biochemical reaction isrepresented in this formalism by a directededge in the graph, which is graphically depict-ed as an arrow from one vertex to another ver-tex. In case a biochemical reaction is bidirec-tional, the corresponding edge is also bidirec-tional, and it will be depicted as merely a linebetween its vertices. Figure 1 shows a graphrepresenting a pathway from a substrate vertexto a product vertex.

The actual phosphorylation process followsthe so-called Michaelis-Menten equation［5］,which appears in the graph as a label on theedge between the substrate-vertex and theproduct-vertex. The Michaelis-Menten equa-tion is described as follows. Let the reactantdenote the input to the pathway, then the prod-uct is calculated by the Michaelis-Mentenequation as

d/dt (product (t)) =k3*enzyme*substrate (t) / (substrate (t) + km)

where product (t) is the product concentra-tion at time t, substrate (t) is the substrate con-centration, enzyme is the enzyme concentra-tion, and k1, k2, and k3 are the coefficients ofthe biochemical reaction, whereas km = k2 / k1

and it is assumed that k3 ≪ k2.The above formalism is used for describ-

ing an individual pathway (Cf. Fig. 2). If apathway can not be divided into any other


pathways, the pathway is called indivisible oratomic. Atomic pathways are the buildingblocks from which more complex pathwaynetworks are constructed. Such complex path-ways are called interacted pathways. Theentire pathway network will be constructed bythese building blocks. In case of phosphoryla-tion and dephosphorylation states, for exam-ple, it is possible to switch between thesestates via the interacted pathways that are reg-ulated by kinases and phosphotases.

The interaction of different states needs tobe investigated considering their influence onbiological functions of cells. The MAPK cas-cade is one of the important pathways withsuch features.

2.3 Example: the MAPK cascadeA MAPK cascade is an important pathway

that is at the base of many biological functions

in cells, such as in a phosphorylation process.Involved in a MAPK cascade is a number ofkinases with the following names:

MAPK: mitogen activating protein kinase,MAPKK: mitogen activating protein

kinase kinase,MAPKKK: mitogen activating protein

kinase kinase kinase,MAPKKKK: mitogen activating protein

kinase kinase kinase kinase,

The resulting pathway is called k-layered,with k being an integer denoting the numberof stages in the cascade. The structures of twoMAPK cascades are is illustrated in Figs. 3and 4. From top to down in the order goingfrom upstream to down stream, MAPKKKKphosphorylates MAPKKK, MAPKKK phos-phorylates MAPKK, MAPKK phosphorylates

Fig.1 A pathway: a graph vs. signals

Fig.2 The concept of pathway as a system


MAPK, which effectuates a phosphorylatedprotein as output to the process.

When building information models of sig-nal transduction, it is important to structurallyanalyze the functionality of a signaling path-way network. As an example of a structuralmodel, we show the MAPK cascade of bud-ding yeast Schizosaccharomyces pombe inFig. 4, which involves the kinases Ste 20,

Ste 11, Ste 7, Kss 1 / Fus 3 and Far 1 / Ste 12.In this cascade, MAPKKKK is Ste 20, MAP-KKK is Ste 11, MAPKK is Ste 7, and there aretwo MAPK factors, Kss 1 and Fus 3. Further-more, there are proteins at the bottom of theMAPK cascade, called Far 1 and Ste 12; theseproteins, which form the output of the cascade,play an active role in the reproduction of cells.In technical terms, Far 1 is a CDK (Cyclin-

Fig.4 An example of a MAPK cascade

Fig.3 The hierarchical structure of MAPK cascade


Dependent Kinase) inhibitor, and Ste 2 is a TF(Transcription Factor) Ste 12. Other examplesof signal pathways can be found in KEGG［6］.

Up to now we have witnessed that theobjects — nodes and links of pathway net-works — can be represented by the vertexesand edges of graphs. Accordingly, the dynam-ics features of networks can be investigatedbased on the computational formulation givenabove.

3 Dynamical analysis of signaltransduction networks

3.1 Temporal dynamics of signaltransduction networks

Based on the graph structure we formulat-ed in previous sections, we will discuss thedynamical features of signal transduction net-works in order to systematically understandtheir information processing mechanism. Basi-cally, the dynamics of signal transduction canbe investigated by two major aspects: spatialdynamics and temporal dynamics of signaltransduction. In signal transduction, the spatialdynamics is mainly reflected in diffusionprocesses. Kholodenko’s review paper［7］onspatial dynamics uses the diffusion equationwith polar coordinates to formulate the con-centration values of kinases in the MAPK cas-cade constrained by the distance of diffusion.

The current paper focuses mostly on theinformation processing aspect of signal trans-duction networks, which clearly have a tempo-ral character, so we limit our discussion to thetemporal dynamics of signal transduction.Such dynamics is usually formulated by dif-ferential equations, among which theMichaelis-Menten equation mentioned in Sec-tion 2 is the most fundamental. The Michaelis-Menten equation describes the biochemicalreaction among molecules used for signaling incells. In the following, we will denote the con-centration of any chemical X as [X], describ-ing the number of molecules per unit volume.

A substrate X is the input to the pathwayunder the regulation of an enzyme E, and aproduct Y is the output of the pathway. The

enzyme plays the role of catalyzer, which trig-gers the biochemical reaction and transformsthe initial substrate into the resulting product.

k1 k3

S + E→ S • E→ P + Ek2

S • E→ S + E

The expression S•E means that S is bound to E.The parameters k1, k2, and k3 describe the con-version rates and

km = (k2 + k3) / k1

which is called the Michaelis constant.We can obtain a simple differential equa-

tion system for the kinetic dynamics of theproduct P, where E0 is the total concentrationof the enzyme.

d/dt [P] = k3 [E0] * [X] / (km + [X]).

The above formulation, however, is onlyvalid under quasi steady-state conditions, i.e.,conditions in which the concentration of thesubstrate-bound enzyme changes at a muchslower rate than those of the product and sub-strate. This allows the enzyme to be treated asa constant, which is E0 in the formula.

A question arising naturally from here ishow to use the above form to explain theMAPK cascade we already met before. So, wereformulate it as a set of coupled differentialequations in which each stage in the cascadecorresponds to one equation:

d/dt [MAPKKK] =k3 (MAPKKK) [MAPKKKK (0)] *[MAPKKK] / (km (MAPKKK) + [MAPKKK]),

d/dt [MAPKK] = k3 (MAPKK) [MAPKKK (0)] * [MAPKK] / (km (MAPKK) + [MAPKK]),

d/dt [MAPK] = k3 (MAPK) [MAPKK (0)] *[MAPK] / (km (MAPK) + [MAPK]),


d/dt [output-of-MAPKcascade] =k3 (output-of-MAPKcascade) [MAPK (0)] *[output-of-MAPKcascade] / (km (output-of-MAPKcascade) + [output-of-MAP-Kcascade]),

With the appropriate parameters, usuallyobtained from empirical observations, the sys-tem MAPK cascade can be numerically calcu-lated. Basically, the general behavior can bedescribed as an amplification of the initialsubstrate concentration S(0), resulting in anenhanced signal with higher concentration ofthe resulting product P (final moment).

One of the well-known functions of theMAPK cascade in cellular signal transductionnetworks is to act as an amplifier for intracel-lular signaling processes. An unexpected phe-nomenon — a fixed-point that occurs at afour-layered MAPK cascade where a feedbackis embedded — is observed by simulation［7］,which shows that in theory the second mes-sengers’ signals can be kept at a constantvalue during their relay processes within cells.

3.2 Fixed point of pathways with feedbacks

Nonlinear dynamical features, such asbifurcation［3］, have been reported in signaltransduction networks. In order to study the

computational and communication capacity ofsignal transduction networks, it is necessary tomake sure how the cellular signals are con-trolled so that the information flow can bequantitatively measured. This frameworkforms the basis of the architectural design andperformance analysis of engineered ICT sys-tems inspired by signal transduction networksin cells. In this section, we take the fixed-pointphenomena as an instance to demonstrate thenonlinear phenomena of signal transductionnetworks, as reported in［8］.3.2.1 The model for the simulation

As shown in Fig. 5 and Fig. 6, the struc-ture of a MAPK cascade is layered, with feed-back being embedded into each layer.

The set of coupled differential equations inthe case feedback is present then becomes:

d/dt [phosphor-protein in MAPKKK layer] =－ k3 [MAPKKK] [MAPKKKK (0)] *[MAPKKK] / (km (MAPKKK) + [MAPKKK]) － ∫ [MAPKK] dt,

d/dt [phosphor-protein in MAPKK layer] =－ k3 (MAPKK) [MAPKKK (0)] *[MAPKK] / (km (MAPKK) + [MAPKK]) －∫ [MAPK] dt,

Fig.5 Three-layered MAPK cascade with feedback


d/dt [phosphor-protein in MAPK layer] =－ k3 (MAPK) [MAPKK (0)] * [MAPK] / (km (MAPK) + [MAPK])－ ∫ [output-of-MAPK-cascade] dt,

d/dt [MAPKKK] = k3 (MAPKKK) [MAPKKKK (0)] * [MAPKKK] / (km (MAPKKK) + [MAPKKK]),

d/dt [MAPKK] = k3 (MAPKK) [MAPKKK (0)] * [MAPKK] / (km (MAPKK) + [MAPKK]),

d/dt [MAPK] = k3 (MAPK) [MAPKK (0)] *[MAPK] / (km (MAPK) + [MAPK]),

d/dt [output-of-MAPKcascade] =k3 (output-of-MAPKcascade) [MAPKK (0)] *[output-of-MAPKcascade] / (km (output-of-MAPKcascade) + [output-of-MAPKcascade]),

where the integral part is calculated frominitial time 0 to the current time t.3.2.2 Simulation

Owing to the fact that the essential processof intracellular communication exhibits non-linear dynamical behaviors in a biochemicalframework in the model above, it is possibleto figure out what kind of parameters ofMichaelis-Menten kinetics behind the fixed-point phenomena is used by the molecularmechanism of signal transduction networksthrough the MAPK cascade.

The conditions are set as follows:

• The initial concentration of the enzyme =0.45.

• In the MAPKKKK, MAPKKK, MAPKK,MAPK layer, the product acts as thekinase for the succeeding MAPKKK,MAPKK, MAPK layers and output of theentire MAPK cascade at its bottom.

• In each layer of MAPK cascade km = 0.1and k3 = 0.01,

Fig.6 Four-layered MAPK cascade with feedback


• The initial concentration of substrates(phosphor-proteins) in all the pathway-like units is set as 0.45.

• The step/sample number is 10• The initial concentration of product is set

as 0.001.

Now follows a simulation of a 4-layeredMAPK cascade

Let y = f (x) denote the signaling processfrom MAPK denoted as x to protein of theentire MAPK cascade y, we observed that

f (0.0001030) = 0.0001030.

where x = y = 0.0001030.

This is a fixed-point-like phenomenon.The value 0.0001030 determines the crucialpoint of the phase transition between themonotonic decreasing mode and the fixed-point-like mode.

3.3 RobustnessAt the system level of a pathway network,

dynamical features of pathway networks arethe key to understand the cellular signalingmechanism. One of the most importantdynamical features of pathway networks isrobustness. The robustness of a pathway net-work can be investigated through differentmeans, for example, stability analysis is anefficient one in the case of the Mos-p MAPKcascade pathway, where Mos-p is a kinase/pro-tein that is in the phosphorylation state, where-as Mos is the same protein in the dephospho-rylation state.

The cell has high robustness against exter-nal disturbances. As Kitano［9］points out, can-cer is an example of robustness in a cell. Incontrast with the robustness of pathway net-works in the cell, conventional communica-tion networks are very fragile, for example,the Achelles’ heel phenomenon in internetnetworks occurs when failures occur. Thiscontrast motivates us to quantitativelydescribe the biological robustness［10］of path-way networks in order to investigate the possi-

bility of applying the knowledge of the biolog-ical robustness in pathway networks to thedesign of information networks in the future.

In the previous sections, we discussed thegraph structure that is usual in computer sci-ence. It is obvious that the robustness of cellu-lar signaling processes described in terms ofnonlinear dynamics is tightly connected withthe “dynamical” graph structure of pathwaynetworks. In general, the parameters of thedifferential equations that describe the bio-chemical reactions of pathway networks canbe defined as labels that correspond to thegraph, where the vertices and edges of thegraph are defined for the pathway network inthe previous section. These topics belong tothe field called Dynamical Networks, whichrefers to the integration of nonlinear dynamicsand graph theory. In this section we definenonlinear dynamical systems in a matrix for-mulation that corresponds to the graph of thepathway network.

Based on these schemes, we can formulatea robustness mechanism of cellular signalingpathway networks.3.3.1 Basic concepts

The concept of robustness is defined asfollows.

Definition of Robustness:Robustness refers to a mechanism that can

guarantee and realize the state transition of a(usually dynamic) system from vulnerable andunstable states to sustainable (stable) stateswhen the system suffers from disturbance thatis outside the environment and that is unex-pected in most cases［11］.

Let us define a nonlinear system W todescribe the cellular signaling mechanism incells:

W (X, S, U,Y, Q)

where X — input to the system Y — output of the systemS — the states of the systemU — feedback


Q — parameter vector

The system description is given as fol-lows:

d/dt (X) = A X + B S + C UY = D S

where A, B, C, D are constant matrixes,described by Q.

All the above-mentioned variables are vec-tors.

We focus on the definition of robustness interms of the system state. Then we define therobustness by the function G (.) satisfying thecondition that

S = G (S, E)

when the system suffered disturbance EThe robustness feature of a pathway is

expressed in several different aspects of path-way networks. Stability is among them. Theabove-mentioned formulation makes it possi-ble to use the states of the system for describ-ing the robustness where the robustness mech-anism is interpreted as the mechanism for pro-viding the steady states. The Mos-p MAPKpathway［12］is an example explaining/analyz-ing the pathway stability for the robustness ofthe corresponding pathway network. In thispathway, the term “stable steady states” denot-ed as SS is used to describe the state transitionof the pathway network within a certaindomain［12］. The Mos-p MAPK pathwayincludes a MAPK cascade. As Huang et al［13］reported, the nonlinear dynamical features ofdifferent phosphorylation processes in aMAPK cascade vary, i.e., the phosphorylationconcentration versus time curve is differentfor each layer of a MAPK cascade. The signif-icance of this phenomenon is obvious if wereveal the fixed-point of the MAPK cascadepresented in the previous section.3.3.2 Stability analysis: From an exam-

ple of the Mos-p MAPK pathwayfor explanation of stability

The Mos-p MAPK pathway that demon-

strates the transition between the stable/unsta-ble steady states in cells is reported in［12］. Inorder to make the formulation of the corre-sponding model, we need the Hill coefficientand the Hill equation.

The Hill equation is given as

δ= ([L] n ) / (Kd + [L] n )= [L] n / (KA n + [L] n ),

where

δ is the concentration of the phosphory-lated protein,[L] is the ligand concentration, Kd is the equilibrium dissociation constant, KA is the ligand concentration occupying halfof the binding sites,and n is the Hill coefficient denoting the coop-erativity of binding. It describes the nonlineardegree of the product’s response to the ligand.

The cooperativity indicates the degree ofthe biochemical reactions for binding the sub-strate and enzyme. Here ligand means anenzyme protein that can bind with anothermolecule. In a Mos-p MAPK pathway, Mos-pis the ligand.

The coefficient n mentioned above is nor-mally denoted as nHill.

nHill = 1 refers to the case of Michaelis-Menten kinetics, which corresponds to thecase of completely independent binding,regardless of how many additional ligands arealready bound.

nHill > 1 shows the case of positive cooper-ativity,

nHill < 1 shows the case of negative cooper-ativity.

By the above, the SS state can be quantita-tively described. When the SS is achieved, theactivation degree (concentration) of Mos-pcan be formulated as a fixed-point of the fol-lowing form:

Mos-p = f (Mos-p),


where f (.) refers to the Mos-p MAPK cas-cade pathway.

This shows the steady state of Mos-p,which is consistent with the phenomenareported by Ferrell et al.［12］.

Based on the ultrasencitivity quantization,we have a closer look at the Mos-p MAPKpathway from two viewpoints:3.3.2.1 A brief look at the system

We consider the input of the system as theproteins Mos-P and malE-Mos, and the output asthe protein MAPK. Here two signals concerningMAPK are involved — activated MAPK andphosphorylated MAPK (phos • MAPK forshort).

Based on the results from the experimentreported in［12］, we can use the Michaelis-Menten equation to obtain that the response ofMek is monotonic to Mos-P, to malE-Mos, orto Mos-P and malE-Mos. Additionally, the

response of activate • MAPK or phos • MAPKis monotonic to the systems’ inputs Mos-P orto malE-Mos or to Mos-P and male-Mos,under the condition that there is no feedbackin the pathway networks, which are branchedat the routes from Mos-P and malE-Mos toMek. The phosphorylation effect can be wit-nessed at Mek, MAPK and Mos-P.

A graph description for this pathway net-work is given in Fig. 6

Considering the robustness again, we real-ize that we need the feedback (Cf. Fig. 7) andrelated nonlinear cellular signaling mechanismto help us understand the robustness withinthis pathway network.3.3.2.2 Modeling the temporal

dynamics of the systemLet us use the Hill-coefficient-based for-

mula to describe the temporal behavior of cel-lular signaling. Assuming the time series of

Fig.7 Cascade without feedback and with feedback


malE-Mos as

d/dt [moleE-Mos] = f1 ([molE-Mos], coefficients)

where f1 () takes the form of a monotic func-tion with a decreasing order, we obtain

[moleE-Mos] (n+1) － [molE-Mos] (n) =coefficient [molE-Mos] (t)

This is the time difference equation used fornumerical calculation.

Let m denote the step, m is an integer and the[mole-Mos] is set as 0. Then we obtain

[active MAPK tot] (m+1) =([Mos-P] (m) + 1000) H

/ (EC50 H + ([Mos-P] (m) + 1000) H )

whereH refers to the Hill coefficient nH = 5EC50 = 20 (nM)[molE-Mos] (t = 0) is set as 1000

Then

[Mos-P] (m+2) = 0.5 * [active MAPK](m+1)= 0.5 * [MAPK (tot)] ([Mos-P] (m) +

1000) H / (EC50 H + ([Mos-P] (m) + 1000) H )

Now assume that [Mos-P] (m+3) = [Mos-P] (m+2) + [Mos-P] (m). Let the moment n+1 correspond to m+3, n correspond to mwhen we set the time reference of Mos-Paccording to the initial time of going throughthe pathway and the final time. The momentsof m+1 and m+2 refer to the internal states ofthe pathway network as a system. Consequent-ly we have

[Mos-P] (n+1) = 0.5 * ([Mos-P] (n) + 1000) H

/ (EC50 H + ([Mos-P] (n) + 1000) H ) + [Mos-P] (n)

The Mos-P MAPK pathway derived com-puting process then becomes

[x (n+1)] － [x (n)] = 0.5 * ([x (n)] + 1000) 5

/ (20 5 + ([x (n)] + 1000) 5 ) － [x (n)]

where

x is an integer > = 0,[x (0)] is set as 5.

Let f ((x (t)) = 0.5 * (x (t) + 1000) 5

/ (20 5 + (x (t) + 1000) 5 ) － x (t)

d/dt (x (t)) = x (t+1) － x (t) = f ((x (t))

We define a function in general:

d/dt (x (t)) = f ((x (t))

when [molE-Mos] is treated as a constant.

It is obvious that the above system is stillnonlinear even though the control input[molE-Mos] is a constant.

Considering the dynamical feature of con-trol input [molE-Mos], we have that

d/dt (x (t)) = 0.5 * (x (t) + u (t)) 5

/ (20 5 + (x (t) + u (t)) 5 ) － x (t)

where x (t) = Mos-p refers to the state of thesystem and u (t) = [molE-Mos] refers to thecontrol input.

Since this is a coupled system, it is neces-sary to decouple the different signals in orderto efficiently control the system.

The description of the pathway network asa system is normally established by a differen-tial equation. Denote the system by the fol-lowing equations

d [X (t)] / dt = f (t, X (t), U (t))Y (t) = g (t, X (t))

where t is time, U (t) is the input, X (t) is thestate, and Y (t) is the output. This is a general


form that includes the case of nonlinear sys-tems.

Based on the instance of Mos-p MAPKpathway we discussed before, the cellularpathway network is modeled as a controller-centered system where feedback is embedded.Different constraints can be used to formulatespecific objects in applications, e.g., a matrix-based representation could be

dX (t) / dt = A (t) X (t) + B (t) U (t)Y (t) = C (t) X (t)

where A (t), B (t) and C (t) are matrixes.

In this instance, the variable is one dimen-sional and a nonlinear relation exists betweenstate transitions. So we have that

d/dt (x (t)) = f (x (t), u (t))

wherex (t) = [Mos-p], this is the state; u (t) = [molE-Mos], this is the control

input; this equation is established under thesteady states of the Mos-p MAPK pathway.

The output for detecting the signals ofphosphorylation is given as follow:

y (t) = phos • [MAPK] = g ((x (t), u (t))

where

g ((x (t), u (t)) = 0.5 * (x (t) + u (t)) 3

/ (20 5 + (x (t) + u (t)) 3 )

under the condition that the Mos-p MAPK path-way is in the steady state.

From the above discussion of pathwaysystems, we have presented a dynamics-basedrepresentation for formulating the robustnessof a dynamical system, which is motivated bythe biological robustness in cellular pathwaynetworks, but which can be modified andextended to any abstract dynamical systemwhere feedback is embedded.

4 Error correcting codes for cellular signaling pathways

How can we carry out reliable informationprocessing by pathway networks with dynami-cal features? An important element underlyingcellular signaling is the robustness of molecu-lar pathways. Mechanisms such as thoseresembling error correcting codes may play anessential role in this framework.

4.1 Molecular coding for molecularcommunication

A reversible molecular switch is the basisof information representation in cellular infor-matics as shown in Fig. 7. Two kinds ofreversible molecular switches exist in cells.

The molecular switch of phosphorylationand dephosphorylation: The phosphorylationstate of a signaling protein is defined as 1,whereas the dephosphorylation state of a sig-naling protein is defined as 0. The phosphory-lation process is regulated by kinase, thedephosphorylation process is regulated byphosphatase.

The switch of GTP-bound and GDP-bound :As shown in Fig. 8, the GTP-bound state ofGTPase set by the so-called GEF and GEFpathway is defined as 1 and the GDP-boundstate of GTPase set by the so-called GAP andGAP pathway is defined as 0.

The MAPK cascade consists of severalphosphorylation processes. So multiple binarycodes can be generated, e.g. four bits generatedby a four-layered MAPK cascade (see Fig. 9).

The above molecular switches allow us toformulate a mathematical model for informa-tion processing based on an abstraction of thedata structure corresponding to the signalingpathways.

The above framework only involves switch-es, and it does not include the redundancy typi-cally associated with error correcting codes. Ifwe intend to include such codes, then the result-ing mechanisms should be compatible with themechanisms encountered in cell communication.


4.2 LDPC coding for pathways Error correcting codes are mathematical

constructs, and their design involves a differ-ent philosophy from the way in which biologi-cal mechanisms have evolved, which includesan element of randomness. Fortunately, thereis an error correcting code, of which thedesign---though mathematically well-found-

ed---involves a random element as well.Called Low Density Parity Code (LDPC), thiscode is, surprisingly, very efficient, in thesense that it allows transmission of informa-tion at rates very close to the theoretical maxi-mum. These codes were developed in 1960 byGallager, and they had been long forgottendue to the perception of them being impracti-

Fig.8 Molecular Switch for binary codes (bits)

Fig.9 Four bits generated from a MAPK cascade


cal, only to be rediscovered by MacKay in1996. Ironically, the randomness in LDPCcodes is very attractive in the framework ofbiological systems, and this is the reason whywe describe them in more detail.

The LDPC code is defined by a partitegraph BG = <V, E>, where the vertex setV = V1 ∪ V2. V1 is an ordered set of nodes,each of which is labeled by a binary number.The set V1 thus denotes a binary code word.V2 is a set of which each node is labeled by abinary number equaling the parity value of thesum of the labels of the nodes in V1 that areconnected to the node in V2 by an edge in E.

As shown in Fig. 10, at first let V12 be 1and V21 be 0 (parity summation), then V11

should accordingly be assigned the value 1.This is the encoding scheme. The decoding iscarried out in a reverse way, as shown inFig. 11.

Assume that V11 is lost during informationtransmission, then in order to restore the value

of V11 we need to use the information of V12

and V21. Because V12 is 1 and the parity sum-mation is 0, we can infer that V11 should be 1.

Figure 12 gives the encoding process of 5bits in V1 (labeled as L1), where the knownbits are put into the condition part of an “IFTHEN” rule and the unknown bits are put intothe conclusion part of this rule.

The nodes in V1 are called L1-units and theones in V2 are called L2-units, owing to thefact that they are described by two separatedvertex sets in a bipartite graph.

Within the signal transduction model fol-lowed in this paper, the two bipartite partsassociated with the generation of an LDPCcan be designed as shown in Fig. 13, wherethe information processing units correspond-ing to the phosphorylation/dephosphorylationpathway and GEF/GAP pathway are denotedas L1-unit and L2-unit, respectively.

Fig.10 A simple example of an LDPC encoding follows below

Fig.11 A simple example of an error correcting LDPC code


As shown in the above figure, the bipartitegraph to encode an LDPC code is mappedfrom a pathway whose input is V12 when inthe phosphorylation state and V21 when in theGTP-bound state and whose output is V11

when in the phosphorylation state. This path-way corresponds to the graph in Figs. 10 and 11for encoding and decoding an LDPC code.

The above model allows us to encode phos-phorylation pathways and the dephosphoryla-tion pathways in terms of an LDPC code, whichis capable of approaching the Shannon limit.The derived encoding/decoding model is bidi-rectional, symmetrical and “implicitly-binary”(i.e., its binary form can be formulated in termsof n bits of the L1-units of the model), whichdiffers from the previous model given in［14］that is one-directional, asymmetrical (from

phosphorylation/dephosphorylation to GTP-bound/GDP-bound states) and “explicitly-bina-ry” (i.e., the phosphorylation/dephosphorylationmechanism directly encodes the code z).

5 Conclusions

In this paper we have described pathwaysin biological organisms that behave de-factolike communication systems on cellularscales. The robustness of biological communi-cation systems offers an important lesson forthe design of man-made communication sys-tems: key concepts in this context are paral-lelism, adaptability, and structural stability.We have also sketched LDPC codes, whichhave much in common---due to their random-ness---with the unstructured character of bio-

Fig.12 The rules for generating words

Fig.13 An example of a pathway that is mapped to the model in Fig. 11


logical systems. This may suggest that LDPCcodes form an important avenue of research inthe realization of communication systems thatinclude the above key concepts. Cellular sig-nal transduction networks form a rich sourceof inspiration in the design of next-generation

communication systems, which will havegreater robustness, greater capacity, andgreater adaptability, but which will also beless visible to its users, forming a transparentever-present environment, such as the overlay-network in［15］.

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Jian-Qin Liu, Ph.D. / Dr.Informatics

Expert Researcher, Kobe AdvancedICT Research Center, Kobe ResearchLaboratories

Bioinformatics, Network Informatics,Biomolecular Computation

Ferdinand Peper, Ph.D.

Senior Researcher, Kobe AdvancedICT Research Center, Kobe ResearchLaboratories

Nanocomputing, Cellular Automata,Noise

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