mathematical modelling for systems biology - eth zürich · mathematical modelling for systems...

Mathematical Modelling for Systems Biology

March 7, 2016

Contents

1 Biochemical Reaction Modelling 11.1 An Introduction to Modelling . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic Reaction Types . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 0th Order Reactions - Constant Reaction Rates . . . . . . . . 31.2.2 1st Order Reactions - Monomolecular Reactions . . . . . . . 41.2.3 2nd Order Reactions - Bimolecular Reactions . . . . . . . . . 6

1.3 Rule-based modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 Rule-based modeling concepts . . . . . . . . . . . . . . . . . 81.3.2 Contact maps . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Simplifying Approximations . . . . . . . . . . . . . . . . . . . . . . 101.4.1 Michaelis-Menten Kinetics . . . . . . . . . . . . . . . . . . . 101.4.2 Hill Kinetics - Cooperativity . . . . . . . . . . . . . . . . . . 141.4.3 Inhibitory interactions . . . . . . . . . . . . . . . . . . . . . 151.4.4 Goldbeter-Koshland Kinetics . . . . . . . . . . . . . . . . . . 16

2 Model Development and Analysis 192.1 Development of simplified models . . . . . . . . . . . . . . . . . . . 192.2 Non-dimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Phase Plane Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Linearization around the steady state . . . . . . . . . . . . . . 232.4.2 Solution to the linearised ODE . . . . . . . . . . . . . . . . . 242.4.3 Eigenvalues and Eigenvectors of the Jacobian . . . . . . . . . 262.4.4 Stability of the steady-states . . . . . . . . . . . . . . . . . . 27

2.5 A worked example: A model for TGF-β signaling . . . . . . . . . . . 342.5.1 Model Development . . . . . . . . . . . . . . . . . . . . . . 342.5.2 Non-dimensionalisation . . . . . . . . . . . . . . . . . . . . 352.5.3 Equilibrium concentrations . . . . . . . . . . . . . . . . . . . 362.5.4 Phase Plane Analysis . . . . . . . . . . . . . . . . . . . . . . 362.5.5 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . 372.5.6 Linearization around the steady state . . . . . . . . . . . . . . 38

2.6 Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.7 Delay Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3

4 CONTENTS

2.8 Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 472.8.1 Transcritical Bifurcation . . . . . . . . . . . . . . . . . . . . 482.8.2 Saddle-Node Bifurcation . . . . . . . . . . . . . . . . . . . . 482.8.3 Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 49

Chapter 1

Biochemical Reaction Modelling

This chapter provides an introduction to the formulation and analysis of differential-equation-based models for biological regulatory networks. We will discuss basicreaction types and the use of mass action kinetics and of simplifying approxima-tions in the development of models for biological signaling.

1.1 An Introduction to Modelling

The cell is a large dynamic system with thousands of interacting components. To pre-dict how a dynamical system evolves over time and what equilibrium it assumes wecan formulate differential equations that describe the state of the system. Each systemcomponent is represented by a state variable, x. Typically x is a concentration, but xcould also represent a density or the number of molecules. The change of x, ∆x, pertime interval ∆t depends on the rate v+ at which x is generated, the ”gain” rate, and onthe loss rate v− at which x is removed. Here the rate of generation reflects all processesthat lead to an increase in x, i.e. synthesis, change of chemical modification and manymore while the loss rate includes all processes that lead to a decrease in the value of x.More formally we can write

∆x∆t

= gain rate - loss rate = v+− v−. (1.1)

Instead of considering finite time intervals ∆t we will consider the change in an infisiti-mal small time interval dt, such that

dxdt

= gain rate - loss rate = v+− v−. (1.2)

In many cases there are multiple components and multiple compartments (i.e. cyto-plasm and nucleus) with distinct pools of x so that instead of a single x we have manystate variables xi ∈ {x1(t),x2(t), . . .}. The value of each xi then also changes due to theformation of complexes and due to shuttling between compartments. We then write for

1

2 CHAPTER 1. BIOCHEMICAL REACTION MODELLING

a change of each state variable xi(t) in this time interval dt

dxi

dt= synthesis - degradation±shuttling±complex formation±chemical modification±·· ·

(1.3)The values of all state variables at a given time point t constitute the state of the systemat time t. When gain and loss rates balance the variable no longer changes with time,i.e. dxi

dt = 0. When the gain and loss rates of all variables balance then the systemreaches an equilibrium point, also referred to as steady state or fixed point. In general,the gain and loss rates change as the values of the state variables change. This is thebasis of all feedback regulation. Mathematically, we say that the system of ODEs iscoupled, i.e. the differential equations for the different variables depend on each other

dxi

dt= f (x1,x2, ...,xn). (1.4)

This means that we need to consider the entire set of equations simultaneously andcannot solve the different equations separately. Software packages (such as Matlabor Mathematica) are available that provide algorithms to solve these sets of equationsnumerically. In the following we will discuss how the rates of some typical biologi-cal reaction types are affected by changes in the values (i.e. concentrations) of statevariables.

1.2 Basic Reaction TypesThe most accurate model can be obtained when the law of mass action is used to for-mulate kinetic laws for all elementary reactions in Eq. 1.3.

Mass Action Kinetics According to the law of mass action, the rate of areaction is proportional to the probability of a collision of reactants. Thisprobability, in turn, is proportional to the concentrations of the participatingmolecules to the power of the molecularity, i.e. the number in which theyenter the specific reaction.

Thus, if the molecules participating in the gain reaction have concentrations ci andmolecularites mi, then the gain rate according to the general mass action rate law isgiven by

v+ = k+∏i

cmii . (1.5)

Similarly, if the molecules participating in the loss reaction have concentrations c j andmolecularites m j, then the loss rate according to the general mass action rate law isgiven by

v− = k−∏j

cm jj . (1.6)

1.2. BASIC REACTION TYPES 3

The equilibrium constant Keq is then given by

Keq =k+k−

=∏ j c

m jj

∏i cmii

(1.7)

where the concentrations are those in equilibrium. Here, it should be noted that thisonly holds as long as there are no parallel inputs to the gain or loss rates. Parallel inputswould have to be included as sums, i.e.

v+ = ∑j

v+ j = ∑j

k+ j ∏i

cm jii . (1.8)

We will continue with the simple version. Here, we can distinguish between differentorder reactions and relate these to the basic reaction types that are frequently found inbiological settings. Thus, for

dci

dt= kcmi

i (1.9)

the order of the reaction depends on mi. For ∑i mi = 0, we speak of a zero-orderreaction, for ∑i mi = 1 of a first-order reaction, and for ∑i mi = 2 of a second orderreaction.

1.2.1 0th Order Reactions - Constant Reaction Rates0th order reactions are the simplest of all reactions because the rate of the reaction doesnot depend on the state variables. This kinetic law is used frequently to describe thesynthesis of a molecular component.

Constant Synthesis Assuming that the species X is produced at a constant rate kprodwe write for the concentration of X , [X ],

d[X ]

dt= kprod . (1.10)

This equation can be solved as [X(t)] = [X(t0)]+kprod(t− t0) and we note that the con-centration of X at time t depends only on the initial value of X at time t0 and on thetime interval t− t0 that has passed. Accordingly, the rate at which X is produced doesnot change when the concentration of X is changed (Fig. 1.1A, a).

MATLAB Exercise

Simulate equation 1.10 with kprod = 1 and [X(t0)]=0 for t ∈ [0,10] and compare thesolution to the analytical solution:

function ODE_model1_prod()x0 = 0; % initial concentration


k_prod = 1;tspan = [0, 10];

[t,x] = ode15s(@rhs_prod, tspan, x0, [], k_prod);plot(t, x, ’r.’, t, x0+k_prod*t, ’k-’)xlabel(’Time t’)ylabel(’Concentration x’)legend(’numerical solution’, ’analytical solution’)end

function dxdt = rhs_prod(t,x,p0)dxdt = [p0];

end

1.2.2 1st Order Reactions - Monomolecular Reactions

Most biological reactions are catalyzed or affected by components whose concentra-tions vary with time. Reactions that only depend on one such component are referredto as monomolecular reactions. Important examples include the decay of a molecularspecies or its transport between compartments, i.e. cytoplasm and nucleus. Mathemat-ically the dynamics of the state variable linearly depends on the state variable in 1storder reactions.

Linear Degradation The rate at which a protein, mRNA or similar is removed orinactivated is often proportional to its own abundance, i.e. it changes linearly with itsown concentration (Fig. 1.1A, b). We write for the concentration of such a componentX

d[X ]

dt=−kdeg[X ]. (1.11)

This equation can be solved as [X(t)] = [X(t0)]exp(−kdeg(t− t0)) and we note thatthe concentration of X decays exponentially over time. An important measure is thecharacteristic time t1/2 =

ln(2)kdeg

by which the initial concentration [X(t0)] has decreasedby half.

MATLAB Exercise

Simulate equation 1.11 with kdeg = 1 and [X(t0)]=1 for t ∈ [0,10] and compare thesolution to the analytical solution:


function ODE_model2_degradation()x0 = 1; % initial concentrationk_deg = 1;tspan = [0, 10];

[t,x] = ode15s(@rhs_prod, tspan, x0, [], k_prod);plot(t, x, ’r.’, t, x0 * exp(-k_deg*t), ’k-’)xlabel(’Time t’)ylabel(’Concentration x’)legend(’numerical solution’, ’analytical solution’)end

function dxdt = rhs_prod(t,x,p0)dxdt = [-p0*x];

end

Shuttling between Compartments Similarly the shuttling between two compart-ments (i.e. nucleus and cytoplasm) can be described by two coupled differential equa-tions for the concentrations of X in the nucleus, [Xn], and in the cytoplasm [Xc]. Im-portantly, we need to take the volume difference between the two compartments intoaccount. Let us denote the different volumes of cytoplasm and nucleus as Vc and Vn. Ifa concentration [Xc] shuttles from the cytoplasm to the nucleus then Vc[Xc] moleculesof Xc leave the cytoplasm and enter the nucleus. In the nucleus, these Vc[Xc] moleculescorrespond to a concentration Vc/Vn · [Xc], where Vn is the volume of the nucleus. Ac-cordingly, if X is exported from the nucleus at rate kout and is imported from the cyto-plasm at rate kin then we have

d[Xn]

dt= kin

Vc

Vn[Xc]− kout [Xn]

d[Xc]

dt=−d[Xn]

dt= −kin[Xc]+ kout [Xn]

Vn

Vc. (1.12)

Since the total amount (NOT concentration) of X is conserved inside the cell, i.e.[Xc]Vc + [Xn]Vn = T = const, we can decouple these two ODEs. We can then write[Xc] =

T−[Xn]VnVc

and obtain a differential equation that is similar to Eq. 1.11 except foran additional constant term, kin

TVn

,

d[Xn]

dt= kin

Vc

Vn

T − [Xn]Vn

Vc− kout [Xn] = kin

TVn− (kin + kout)[Xn]. (1.13)


MATLAB Exercise

Combine the previous MATLAB examples (production and liner decay) to simulateequation 1.13.

1.2.3 2nd Order Reactions - Bimolecular ReactionsMost reactions in biology involve some form of complex formation and therefore de-pend on the interaction of more than one time-varying component. Here it is importantto distinguish between homo- and heterodimerization.

Complex formation - heterodimers The formation of heterodimers, XY , is the resultof the interaction of two components X and Y (Fig. 1.1A, c). The reaction rate dependslinearly on both the concentrations of X and of Y . Assuming that the reaction proceedsat rate kon and that the total concentrations of both components is constant we have

d[XY ]dt

= kon[X ][Y ] = kon(XT − [XY ])(YT − [XY ]) (1.14)

where XT = [X ]+ [XY ] and YT = [Y ]+ [XY ] are the total concentrations of X and Y re-spectively. This equation can be solved to give [X ](t)=XT−[XY (t)]= YT−XT

YTXT

exp((YT−XT )kt)−1.

MATLAB Exercise

Simulate equation 1.14 with kon = 1, [XT ] = [YT ]=1 and [XY (0)]=0 for t ∈ [0,10] andcompare the solution to the analytical solution.

Complex formation - homodimers Similarly, the kinetics of homodimer formationbetween two X components can be described by the following quadratic rate law

d[X2]

dt= kon[X ]2 = kon(XT −2[X2])

2 (1.15)

where XT is the total amount of X which we again assume to be constant. Here therate of homodimer X2 formation depends non-linearly on the concentration of themonomers X (Fig. 1.1A, d). The dynamics of X can be described by

d[X ]

dt=−2

d[X2]

dt=−2kon[X ]2 (1.16)


Figure 1.1: Basic Reaction Types (A): (a) Constant Synthesis. (b) MonomolecularReactions: Linear Degradation. (c) Bimolecular Reactions: Heterodimer Formation.(d) Bimolecular Reactions: Homodimer Formation. Simplifying approximations (B):(a) Michaelis-Menten Kinetics. (b) Hill Kinetics. (c) Hill Kinetics with allosteric orcompetitive inhibition. (d) Goldbeter-Koshland Kinetics. The reaction scheme and aplot of the representative reaction rate versus the concentration of the reactant X (A) orthe enzyme E (B) are depicted. In (B, d) the steady state concentrations of Xp and Xare plotted versus the signal strength S.


The equation can be solved to give [X ](t) = XT2XT kt+1 .

We note that there are many cases in which the total concentrations are not constant.The above simplification would then not apply and a set of coupled ODEs for themonomers and the dimers would then need to be solved. In case of higher order com-plexes the formation can, in general, be modeled as a sequential step of bimolecularreactions.

1.3 Rule-based modelingBased on mass action, accurate models can be formulated even for large networks, aslong as both the components (i.e. the proteins, compartments, complexes etc) and therules and kinetics of their interactions are known. Typically such models are based onnetwork cartoons of the form shown in Fig 1.2A. To translate such cartoons into math-ematical models we assign a single state variable xi(t) to each icon. One state variablewould be the unbound ligand x1(t). Another state variable would be the ligand-receptorcomplex, x2(t), and so on. The set of values of all state variables {x1(t),x2(t), ...} at agiven time point t constitutes the state of the system at time t.

In most biological networks, models based on mass action will lead to huge dy-namic systems that are based on hundreds of ODEs to describe the interactions be-tween less than 10 components, a problem referred to as ”combinatorical complexity”.The problem arises because proteins typically have multiple binding sites such thateven a simple network as shown in Figure 1.2A translates into a much larger networkas shown in Figure 1.2B when these are taken into account. The technical problemof generating such large system of ODEs can be overcome by rule-based modelling.The algorithm generates the system of ODEs from the formulated rules of interactions.Using only the set of sensible biochemical rules for what is known about our system itis easy to generate a comprehensive system and avoid making any errors due to miss-ing interactions and/or unjustified assumptions. There are different software that havebeen designed to enable rule-based modeling. Among other softwares, most popularare BioNetGen (book find citation) and Kappa (Vincent Danos, missing citation...).The syntax and example below are based on the BioNetGen language (BNGL).

1.3.1 Rule-based modeling conceptsWe can summarize the possible rules in rule-based modeling in the following fivebasic transformations. (1) Complex formation: a bond can be formed to link twomolecules through their available binding sites. (2) Complex dissociation: an ex-isting bond between two molecules can be removed (3) Change the state-label of acomponent: a molecule undergoes a certain post-translational state modification (e.g.become phosphorylated), or alters the state-label of its functional shape or conforma-tion (e.g. open/closed conformation of integrins). (4) Add a molecule: production ofa species. (5) Delete a molecule: degradation of a species. The above stated rules canbe expressed as unidirectional transformations, but some of them can be also bidirec-tional (e.g. binding/unbinding, phosphorylation/de-phosphorylation, opening/closing).

1.3. RULE-BASED MODELING 9

Figure 1.2: Examples of network representation. (Up-left) Cartoon of the regulatoryinteractions in the network that controls σF during sporulation in Bacillus subtilis. Theimage was reproduced from Figure 1 in [?]. (Down-left) Manually created network ofinteractions (Right) Contact map

Based on these concepts, already with a very small set of rules and molecule types wecan generate enormous systems.

1.3.2 Contact mapsLarge regulatory networks are difficult to visualise. On the one hand, the conceptualcartoons that are usually found in text books are too abstract to reflect the full systemthat needs to be modeled mathematically. On the other hand, detailed network mapsare typically to dense to be readable. In rule-based modeling, contact maps are used.Contact maps depict the set of all possible interactions among the basic elements ofthe system; the actual transitions from reactants to products are not given explicitly. InFigure 1.2, an example of a signaling pathway is illustrated both in all three ways.

1.3.3 Simple exampleThe following figure illustrates some basic elements with a small example. Assumethere are two interacting molecules, a ligand L and a receptor R. Let the ligand have onebinding site for the receptor, and the receptor to have two binding sites, one for ligand-binding and one that enables homodimerization with another receptor. Also assumethat the receptor has a tyrosine site Y with two state-labels, phosphorylated P andunphosphorylated U (Figure 1.3(a)). Having defined the molecules and their propertieswe can start defining patterns, e.g. free receptors irrespective of their state-label (Figure1.3(b)). Now we can write down a set of simple rules that will summarize our set ofinteractions: the ligand and the receptor can form a heterodimer complex, this complexcan further homodimerize, and this allows the receptors to further trans-phosphorylateeach other (Figure 1.3(c)). This scenario of interactions could be further continuedby just binding of adapter molecules on the phosphorylated site, or by enabling themodulation of other interactions upon phosphorylation.


Figure 1.3: Rule-based modeling concepts using BioNetGen language (BNGL). (a)Molecules (b) Patterns (c) Simple example of interactions

1.4 Simplifying ApproximationsIf we formulate the kinetics of large networks based on first principles then the de-scription becomes very complex and will be accurate only if we are able to determinea large number of parameters with high accuracy. In particular, in case of coopera-tive enzymes it can be very difficult to obtain accurate data on the reaction rates of allintermediate complexes. Most of the times we do not know all elementary/molecularinteractions that regulate a particular reaction. Therefore there are many situations inwhich simplifications are sufficient and in fact preferable. Even from a computationalpoint of view, it can make calculations more efficient.

1.4.1 Michaelis-Menten Kinetics

One frequently used approximation is quasi-stationarity of a reaction. Here the differ-ent time scales are exploited on which reactions proceed. If some reactions proceedmuch faster than others then certain concentrations are constant early on while otherconcentrations barely change at a later time. This is used in the derivation of Michaelis-Menten kinetics for the enzymatic turn-over of a substrate (Fig. 1.1B (a)). In a basicenzymatic reaction a substrate X binds to an enzyme E to form a substrate-enzymecomplex C. Complex formation is a reversible reaction while the formation of theproduct P is irreversible,

X +Ek1

GGGGGBFGGGGG

k−1C

k2−→ E +P. (1.17)

1.4. SIMPLIFYING APPROXIMATIONS 11

Figure 1.4: The Kinetics of the Michaelis Menten Reaction (a, c)The kinetics ofsubstrate X and product P on linear and log scale. (b,d) The kinetics of enzyme E andsubstrate-enzyme complex C on linear and log scale.

The elementary reaction rates for the enzymatic turn-over of a substrate are:

d[X ]

dt= k−1[C]− k1[X ][E] (1.18)

d[E]dt

= (k−1 + k2)[C]− k1[E][X ] (1.19)

d[C]

dt= −(k−1 + k2)[C]+ k1[E][X ] (1.20)

d[P]dt

= k2[C] (1.21)

with initial conditions [X(0)] = XT , [E(0)] = ET , [C(0)] = [P(0)] = 0. We notice thatd[E]dt + d[C]

dt = 0, and thus [E]+ [C] = ET , i.e. the total amount of enzyme is conserved(Fig. 1.4B,D). Moreover, the differential equation for the product P is uncoupled fromthe other differential equations since P does not impact on X , C, or E. We can thereforereduce the set of 4 differential equations to a set of 2 coupled differential equations:

d[X ]

dt= k−1[C]− k1[X ](ET − [C]) (1.22)

d[C]

dt= −(k−1 + k2)[C]+ k1(ET − [C])[X ] (1.23)


Non-dimensionalization To simplify all subsequent analyses, we first non-dimensionalizethe equations. As such, each variable and each parameter needs to be transformed intoa dimensionless counterpart. There is no general rule as to how to non-dimensionalize.However, there are some guidelines: (1) If there is a maximal value that a variable canattain then it is sensible to normalize the variable with respect to this maximal value.We therefore write s = [X ]

XT, c = [C]

ET. (2) Parameters should be grouped so as to reduce

the total number of parameters. We write τ = k1ET t, κ1 =k−1+k2

k1XT, and κ2 =

k−1k1XT

. (3)If possible, very small and very large parameters should be generated so as to enablethe use of perturbation methods. Here we exploit that the substrate concentration, XT ,is much larger than the total enzyme concentration, [ET ] and thus ε = ET

XT� 1. We then

obtain

dsdτ

= −s+ c(s+κ2)

εdcdτ

= s− c(s+κ1) (1.24)

with initial conditions s(0) = 1 and c(0) = 0.

Quasi-steady-state approximation A quasi-steady-state approximation can be usedwhen processes occur on very different timescales such that within a certain time inter-val a variable barely changes in value. In case of the Michaelis-Menten model ε

dcdτ≈ 0

once dcdτ≤ O(1). In that case ε

dcdτ

= s− c(s+ κ1) ≈ 0 and the quasi-state approxi-mation thus yields as quasi-steady state c = s

s+κ1. In dimensional form we then have

[C] = ETk1[X ]

k1[X ]+k−1+k2, and thus for the rate at which the product (P) is formed the well-

known Michaelis-Menten kinetics

d[P]dt

= k2[C] = k2ETk1[X ]

k1[X ]+ k−1 + k2= vmax

[X ]

[X ]+Km. (1.25)

k2ET is the maximal rate, vmax, at which this reaction can proceed when the substrateconcentration is large ([X ]� Km). Km =

k−1+k2k1

is the Michaelis-Menten constant andspecifies the substrate concentration at which the reaction proceeds at half-maximalrate. Importantly, the rate at which product is formed versus the substrate concentra-tion yields a hyperbolic graph (Fig. 1.1B (a) RHS). While the conditions for Michaelis-Menten kinetics do not always strictly apply, such dependency of the reaction rate onthe substrate concentration is observed more generally. In such cases the reaction rateν can be approximated by ν = νmax

[X ][X ]+Km

.

Note that this approximation is only valid at sufficiently long times. If we checkthe initial conditions we realize that we obtain a contradiction, i.e.

[C(t)] = [ET ][X(t)]

[X(t)]+Km⇒ C(0) = 0 6= ET

XT

XT +Km. (1.26)

There is thus an initial time interval in which C changes rapidly before assuming arelatively stable quasi-steady state value (Fig. 1.4D). We can estimate the length of


the relevant time scales. The first time scale Tc on which C changes rapidly while Xremains about constant(Fig. 1.4D) can be estimated by setting [X ] = XT in Eq. 1.23,i.e.

d[C]

dt= −(k−1 + k2)[C]+ k1(ET − [C])XT

⇔ d[C]

k1(XT +Km)dt= −[C]+ET

XT

XT +Km. (1.27)

With [C](0) = 0 this equation can be solved as

⇔ [C](t) = ETXT

XT +Km(1− exp(−k1(XT +Km)t)) . (1.28)

The quasi-steady-state concentration of C, [C]qstst = ET[X ]

[X ]+Km, is thus reached expo-

nentially fast and

Tc =1

k1(XT +Km)(1.29)

represents the time within which the concentration of C reached its quasi-steady statevalue up to 1− exp(−1) ∼ 63%. This characteristic time can thus be used as the firsttime scale Tc over which [C] is changing rapidly. The subsequent timescale on whichX changes significantly can be estimated as

Ts =XT

| dXdt |max

∼ XT +Km

k2ET. (1.30)

Here we used d[X ]/dt =−d[P]/dt =−k2ET[X ]

[X ]+Kmwhich applies once [C] has reaches

a quasi-steady state; the reaction speed is maximal initially while [X ]∼ XT .


Quasi-stationarity requires

1. a separation of the timescales on which C and X change rapidly, i.e.Tc� Ts and thus

k2ET

k1(XT +Km)2 � 1 (1.31)

2. that the substrate concentration is almost constant in the time intervalTc, i.e.

∣∣∣∆[X ]XT

∣∣∣� 1 with |∆[X ]| ∼ k1ET XT Tc since in this time interval[X ] is changing mainly due to binding of ET . Therefore∣∣∣∣∆[X ]

XT

∣∣∣∣∼ k1ET XT

XT

1k1(XT +Km)

=ET

XT +Km� 1. (1.32)

The second condition is more restrictive and quasi-stationarity thus appliesif XT � ET , as characteristic for most metabolic reactions, but typicallynot valid for protein signaling networks where both substrate and enzymeare typically proteins and XT ∼ ET . Alternatively, if XT ≤ ET we can haveET ,XT � KM such that the enzymatic reaction is limited by the formationof the complex relative to the processing rate (and C is therefore low, butalmost constant) and the reaction then proceeds at speed v� vmax.

While the quasi-steady state approximation yields the very useful Michaelis-Mentenequation we may still want a full solution. It is not possible to get analytically a closedform solution for Eq. 1.24, but singular perturbation methods can be used to obtainapproximate solutions.

1.4.2 Hill Kinetics - CooperativityMany proteins have more than one binding site for their interaction partners (Fig. 1.1B(b) LHS). Binding of the first ligand can trigger a conformational change that alters thebinding characteristics at all binding sites (Fig. 1.1B (b) 2nd column). The detailedmodeling of all interactions and transitions is tedious. It can be shown [?] that if thefirst ligand binds with very low affinity (i.e. large K1 =

k−1+k2k1

), and all subsequentligands i = 2...n binds with an increasing affinity (i.e. smaller Ki), then

d[P]dt

= vmax[X ]n

Kn +[X ]n. (1.33)

Strictly speaking this formula is obtained in the limit K1→∞ and Kn→ 0 while keepingK1Kn finite. [X ]n

Kn+[X ]n is referred to as Hill function with Hill constant K = (∏ni=1 Ki)

1n

and Hill coefficient n. If we plot the rate at which product is formed versus the substrateconcentration we obtain a sigmoid graph (Fig. 1.1B (b) RHS). The Hill constant Kcorresponds to the concentration at which the reaction proceeds at half-maximal speed.The Hill factor n determines the steepness of the response. Typically n is smaller than


the total number of binding sites because the idealized limits from above do not apply.For a more detailed discussion see standard text books in Mathematical Biology [?].

1.4.3 Inhibitory interactionsInhibitors of a chemical reaction either fully prevent a reaction or reduce the reactionrate. When the effect of an inhibitor is reversible, the steady state of the inhibitedspecies is reduced, whereas in the case of irreversible inhibition the steady state is zero.Here we will only focus on reversible inhibitions. An important regulatory paradigmis the use of inhibitors and activators to modulate the speed of reactions. Inhibitorscan either compete with the substrate for the catalytic cleft (competitive inhibition) oralternatively inhibitors can induce a conformational change that alters the activity ofthe enzyme (allosteric inhibition).

Competitive Inhibition Inhibitors that bind to the active site of an enzyme and com-pete with substrate for access are termed competitive inhibitors (Fig.1.1B (c)). Theset of differential equations which describes the system is (with C2 referring to the CIcomplex):

d[X ]

dt= k−1[C1]− k1[E][X ] (1.34)

d[E]dt

= (k−1 + k2)[C1]− k1[E][X ] (1.35)

d[I]dt

= −k3[E][I]+ k−3[C2] (1.36)

d[C1]

dt= −(k−1 + k2)[C1]+ k1[E][X ]+ k−3[C2]− k3[E][I] (1.37)

d[C2]

dt= −k−3[C2]+ k3[E][I] (1.38)

d[P]dt

= k2[C1] (1.39)

We have d[E]dt + d[C1]

dt + d[C2]dt = 0

∫⇒ [E]+ [C1]+ [C2] = ET . Using again a quasi steady-

state approximation for C1 and C2 we have

C1 =ET [X ]KI

[X ]KI +KIKm +[I]Km(1.40)

C2 =ET [I]Km

[X ]KI +KIKm +[I]Km(1.41)

(1.42)

where Km =k−1+k2

k1and KI =

k−3k3

. The product is then produced according to

d[P]dt

= k2[C1] =k2ET [X ]KI

[X ]KI +KIKm +[I]Km=Vmax

[X ]

[X ]+Km(1+[I]KI)

(1.43)


A higher amount of substrate is therefore required to achieve a particular reaction rate:the half-saturation constant increases from Km to Km(1+

[I]KI), where KI is the dissoci-

ation constant for the enzyme-inhibitor interaction. Similarly, in case of Hill kineticscompetitive inhibition is modelled by an increase in the Hill constant K by a factor of1+ [I]

KI.

Allosteric Inhibition Allosteric inhibitors do not bind to the substrate binding sitebut affect the reaction rate by binding to a different site where they may induce a con-formational change (Fig.1.1B (c)). While this conformational change can, in principle,also affect the binding affinities in the active site, allosteric inhibitors, in general, re-duce the maximal velocity of the reaction vmax (i.e. k′2� k2 in Fig. 1.1B (c)), and wehave

v =vmax

1+ [I]KI

[X ]

Km +[X ]. (1.44)

1.4.4 Goldbeter-Koshland KineticsThe biological activity of signaling proteins is often controlled by a reversible chemicaltransformation, i.e. phosphorylation, methylation etc. If we were to model all stepsexplicitly the models would again be complex (Fig. 1.1B (d)), and experimental datamay lack to parameterize the model. These enzymatic reactions are therefore oftenapproximated with Michaelis-Menten reactions, i.e.

X +E1k1

GGGGGBFGGGGG

k−1C1

k2−→ E1 +Xp

Xp +E2p1

GGGGGGBFGGGGGG

p−1C2

p2−→ E2 +X .

We then have for the kinetics of the phosphorylated and unphosphorylated forms, Xpand X respectively,

d[Xp]

dt=−d[X ]

dt= kphosS

XT − [Xp]

KM1 +XT − [Xp]− kdephos

[Xp]

KM2 +[Xp](1.45)

Here kphos and kdephos are the vmax of the enzymatic reactions. S refers to an externalsignal that is assumed to only affect the kinase and thus the phosphorylation reaction.In equilibrium

d[Xp]

dt=

d[X ]

dt= 0 (1.46)

and we obtain the Goldbeter-Koshland formula

X∗p =[Xp]

XT= G(u1,u2,J1,J2) =

2u1J2

B+√

B2−4(u2−u1)u1J2. (1.47)

where u1 = kphosS,u2 = kdephos,J1 =KM1XT

,J2 =KM2XT

, and B = u2−u1 + J1u2 + J2u1. XT

refers to the total concentration of the signal protein X, i.e XT = [X ] + [Xp]. In the


context of larger regulatory networks with such regulatory motif (Fig. 1.1B (d)), theGoldbeter-Koshland formula can be used to approximate the fraction of active enzymedependent on the input signal S as long as quasi-stationarity for the reaction that regu-lates the enzyme relative to the rest of the network is a reasonable assumption.

Chapter 2

Model Development andAnalysis

In this chapter we will discuss tools to identify the long-term attractors of dynamicsystems. This will allow us to predict the time evolution of ODE systems and toevaluate the sensitivity and robustness of dynamic systems to perturbations. Wewill start by describing the development and non-dimensionalization of mathe-matical models based on biological network cartoons, and then introduce phaseplane, linear stability and bifurcation analysis.

2.1 Development of simplified models

The development of a good model is an art as it requires a good judgement of whatdetails are essential to obtain a model given the goal(s) of the analysis and whichdetails only increase the problem size and thus render the analysis difficult withoutcontributing to the studied effects. Conceptually it is easiest to define the componentsto be included in the model (i.e. the proteins, compartments, complexes etc) as wellas the dynamical options of each component, and to then translate these into a set ofODEs based on the laws of mass action as discussed in the previous Chapter. How-ever, even if we can generate such systems of ODEs efficiently, simulations may not besufficiently informative to grasp the regulatory complexity and possibilities in such anetwork. Therefore simple, meaningful approximations to such networks are typicallystudied first to better understand and predict the qualitative behaviour of such complexnetworks. In developing simple models it is important to identify the key variables inthe problem. For most subsequent analysis it is ideal to restrict the problem to only twoor three variables. The regulatory interactions between the variables need then to berepresented by approximations similar to those discussed in the previous chapter. Herewe discussed Michaelis-Menten kinetics of the form

dPdt

= vmaxX

X +K(2.1)

19

20 CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS

to describe the rate at which a substrate X is converted into a product P by an enzyme Ewhose total concentration ET determined the value of vmax. Moreover, we consideredHill kinetics of the form

dPdt

= vmaxXn

Xn +Kn (2.2)

for the case when the substrate would bind a multimeric enzyme cooperatively. Weshould note that a switch-like response becomes steeper the larger n, i.e. for larger n asmall change in X can shift the system from not producing P to producing P at maxi-mal rate (and vice versa). While such behaviour typically facilitates the mathematicaldescription of biological processes we should note that for most biological signalinginteractions n ≤ 2, and further processes need to be taken into account to explain theobserved biological sensitivtiy. More generally speaking we can model an activatingimpact of a component Y on X as

dXdt

= vY n

Y n +Kn . (2.3)

Such process could represent the expression of X in response to changes in the ligandor transcription factor Y . Other possible interactions are the Y -induced phosphorylationof X and many others. An inhibitory impact of Y on X can be captured as

dXdt

= vKn

Y n +Kn . (2.4)

If Y interferes with an activating action of some other protein Z on X by competing forthe active site we write

dXdt

= vZn

Zn +Kn(

1+(

YKI

)nI) (2.5)

or if Y reduces the overall speed of the catalysis independently of Z we write

dXdt

= vZn1

Zn1 +Kn11

Kn2

Y n2 +Kn22. (2.6)

2.2 Non-dimensionalisationBefore we continue with the analysis of a greatly simplified model it is sensible tofurther simplify the mathematical formulation. To that end, all variables and parametersshould be rescaled in a way that the new variables and parameters have no physicaldimensions and can therefore more easily be compared. Moreover, the total number ofparameters in the model should be reduced by combining parameter values. There isno standard method to non-dimensionalize a model, and some consider the procedurean art because a clever non-dimensionalisation can sometimes greatly facilitate thesubsequent analysis of the problem. There are some guidelines, however: 1) If thereis a maximal value that a variable can attain it is sensible to normalize with respect to

2.3. PHASE PLANE ANALYSIS 21

this maximal value. 2) If a variable is linked to a certain parameter inclusion of thisparameter in the normalization can reduce the total number of parameters in the model.3) If possible, parameters should be combined to obtain small and large parameters asthis enables the use of perturbation methods.

2.3 Phase Plane AnalysisPhase planes are a graphical tool to understand how a dynamic system evolves intime from some given initial conditions. Phase planes are typically constructed for2-component dynamical systems because in this case the two variables can be repre-sented on the two axes. We will illustrate this for the two-component network

k1 // X //

k3�� // Y.

k2

OO

k4 //

X and Y are produced and degraded based on mass action. X , which is produced atconstant rate, enhances the production of Y , which in turn enhances the degradation ofX . The corresponding system of equations is then

dxdt

= f (x,y) = k1− k2xy (2.7)

dydt

= g(x,y) = k3x− k4y. (2.8)

where f and g could also be non-linear functions in case of non-linear reaction kinetics.Figure 2.1A shows some typical kinetics of the dynamic system from one set of initialconditions; based on the plot it seems that the system quickly reaches a steady state.With the help of the phase plane we can directly inspect the dynamic behaviour fromany set of initial conditions within the phase plane.

Panel B and C demonstrate the construction of the phase plane. The two axes of thephase plane represent the state variables x and y. We first determine the set of points inthe phase plane on which either x, y or both do not change in time. This can be doneby setting the time derivatives to zero, i.e. dx

dt = f (u,v) = 0 and dydt = g(u,v) = 0, and

solving for x or y. We obtain

y =k1

k2x(2.9)

y =k3xk4

. (2.10)

On these curves, the nullclines, at least one time derivative is zero. Since our kineticsare rather simple we can determine the unique steady state algebraically as

xss =

√k1k4

k2k3, yss =

k3

k4

√k1k4

k2k3. (2.11)


Figure 2.1: Constructing a phase plane. (A) Evolution of X and Y in time startingfrom an initial point (X0,Y0), until they reach the steady state (Xss,Yss). (B) The null-clines of X (grey line) and of Y (black line) intersect at the steady state (Xss,Yss). Thenullclines split the phase plane into a part where one of the state variables increases(positive time derivative) and another part where it decreases (negative time derivative).The two areas are indicated as different shades of grey for Y and as different texturesfor X . (C) Based on the sign of the time derivatives of X and Y the direction of thephase vectors can be inferred. The phase vectors indicate the direction in which thesystem develops from a given initial point. The black line indicates a sample trajectoryfrom (X0,Y0) to the steady state (Xss,Yss).

An algebraic calculation of the steady state values of x and y is typically tedious. It iseasier to determine the steady state graphically as the intersection of the two nullclinesas shown in Figure 2.4b. Typically the nullclines divide the phase plane into a side witha positive time derivative and one with a negative time derivative (Fig. 2.1B). We cancalculate the sign of the derivative from Eq. 2.7 by checking the impact of a value ofthe state variable that is larger or a smaller than the steady state value. If we start ata point in the sub-plane with the negative time derivative then the state variable willincrease with time - and vice versa (Fig. 2.1C). We can be more precise and plot smallarrows into the phase plane that give the direction and speed with which the systemevolves (Fig. 2.4b). The vector field has the form

~θ =

(du/dtdv/dt

)(2.12)

and is tangent to the trajectory

~x(t) =(

u(t)v(t)

), (2.13)

a parametric representation of x and y with parameter t, in the x-y plane over time t(Fig. 2.4b, grey lines). As we can see all trajectories meet in a common point, theintersection point of the two nullclines. These tangent vectors ~θ point in the directionin which the system develops from a point (x∗, y∗). The length of the tangent vectorindicates the speed with which the system will change. Accordingly the arrows are of

2.4. LINEAR STABILITY ANALYSIS 23

zero length in the steady state and they cross nullclines perpendicularly as one compo-nent of the vector (one time derivative) is zero. If all vectors point to the steady statethen the steady state is globally stable, i.e. the system returns to the steady state aftereach perturbation. If only the arrows in the vicinity of the steady state all point to thesteady state then the steady state is said to be locally stable. The phase plane with thetrajectories and phase vector field is called a phase portrait.

Apart from the compact representation, there is another important feature of thephase plane analysis: although we have not solved the system we already have a ratheraccurate graphical representation of its solution. This is very important for the under-standing of nonlinear systems where analytical solution might not even exist.

2.4 Linear Stability Analysis

For sufficiently simple systems a graphical analysis can be employed to judge the sta-bility of a steady state and to reveal oscillations, adaptation, or switches. In the fol-lowing we discuss a more generally applicable method, the linear stability analysis, toevaluate the (local) stability of steady states. The idea behind a linear stability anal-ysis is to introduce a small perturbation at the steady state and to study whether theperturbation grows or decays over time. In the first case the steady state is said to beunstable while in the latter case the steady state is stable. Linearization of the system ofdifferential equations at the steady state greatly facilitates the analysis but means thatour results apply only locally, i.e. in the vicinity of the studied steady state.

2.4.1 Linearization around the steady state

Since we will deal with linear systems we start by combining our set of state variablesin a vector, i.e.

~x =(

u(t)v(t)

)and

d~xdt

=

(u(t)v(t)

)=

(f (u,v)g(u,v)

), (2.14)

where the dots denote time derivatives. The functions f and g are in general non-linearfunctions. We therefore first need to linearise f and g at the steady states. To this end,we introduce a small perturbation from the steady state, ~xs = (us,vs)

T , and write for theperturbation ~w =~x−~xs = (u− us,v− vs)

T . We can approximate the values of f (u,v)and g(u,v) close to this steady state using Taylor series expansion, i.e.

f (u,v) = f (us,vs) +11!

∂ f∂u

∣∣∣ss(u−us)+

12!

∂ 2 f∂u2

∣∣∣ss(u−us)

2 + . . .

+11!

∂ f∂v

∣∣∣ss(v− vs)+

12!

∂ 2 f∂v2

∣∣∣ss(v− vs)

2 + . . . (2.15)

and likewise for g(u,v). We next linearise the system by ignoring all terms that are of


order two and higher, i.e.

f (u,v) ∼ f (us,vs)+∂ f∂u


∂ f∂v

∣∣∣ss(v− vs)

g(u,v) ∼ g(us,vs)+∂g∂u


∂g∂v

∣∣∣ss(v− vs) (2.16)

We emphasise that f (u,v) and g(u,v) are now only approximations of the original func-tions f and g.

The steady state is stable if the perturbation decays to zero for long times t, i.e.~w→ 0 as t→∞. To express the differential equation in terms of ~w we use the linearisedsystem of equations, and we write

d~xdt

=d~xs

dt+ J(~x−~xs) → d~w

dt= J~w. (2.17)

J is referred to as Jacobian and is the matrix of all first-order partial derivatives of thevector-valued function at the steady state (us,vs), i.e.

J =

(d fdu |us,vs

d fdv |us,vs

dgdu |us,vs

dgdv |us,vs

)=

(fu fvgu gv

). (2.18)

2.4.2 Solution to the linearised ODEWe seek to determine the solution ~w(t) of the linear set of ODEs

d~wdt

= J~w. (2.19)

This can be achieved with the help of some concepts from Linear Algebra, in partic-ular by the use of eigenvectors and eigenvalues. Eigenvectors of J are those vectors~vi which are only stretched by a factor λi, the eigenvalue, but NOT rotated when thematrix J is applied to them, i.e. J~v = λ~v. The set of all eigenvectors of a matrix, eachpaired with its corresponding eigenvalue, is called the eigensystem of that matrix. If ann×n matrix has n linearly independent eigenvectors, then these eigenvectors constitutean eigenbasis for this matrix, and the matrix can be diagnolized. Much as the axes ofthe cartesian coordinate system the linear independent eigenvectors ~vi are orthogonalto each other, i.e. the product of two eigenvectors is zero. They thus constitute analternative, but much more appropriate, basis for the vector space that we analyse here.

To make use of this new basis space we need to express the vector ~w in terms of thenew basis vectors. In the cartesian coordinate system we write for our 2-componentvector ~w

~w(t) =(

wx(t)wy(t)

)= wx(t)

(10

)+wy(t)

(01

). (2.20)


Similarly, ~w can be written as a linear combination of the two eigenvectors, i.e.

~w(t) = ∑i

γi(t)~vi. (2.21)

The set of ODEs then becomes

d~wdt

= ∑i

dγi(t)dt

~vi = J~w = ∑i

γi(t)J~vi = ∑i

γi(t)λi~vi. (2.22)

To solve for a particular γ j, we can multiply with the eigenvector v j, and obtain

∑i

dγi(t)dt

~vi~v j = ∑i

γi(t)λi~vi~v j. (2.23)

Since the basis vectors {~v1, . . . ,~vn} are linearly independent (orthogonal),~vi~v j = 0 un-less i = j and ~vi~v j = 1 if i = j if the eigenvectors are normalized appropriately. Wethen have

dγi(t)dt

= γi(t)λi. (2.24)

By separation of variables we then have

γi(t) = γi(0)exp(λit). (2.25)

Using Eq. 2.21 the differential equation for ~w(t), d~wdt = J~w (Eq. 2.19), can be solved as

~w(t) = ∑i

γi(0)~vi exp(λit). (2.26)

γi(0) is a constant that is determined by the initial perturbation, i.e. ~w(0) = ∑i γi(0)~vi.If the real parts of all eigenvalues are negative then the exponential functions decay tozero for large times and so does the perturbation ~w. We thus require that all real partsof the eigenvalues are negative for the steady state to be linearly (locally) stable.


Diagonalize Matrices using Similar MatricesWe could have uncoupled the ODEs also by diagonalizing the matrix J,using the concept of similar matrices. In case we have a full set of linearlyindependent eigenvectors {~v1, . . . ,~vn} that form a complete basis for then×n matrix J, we can construct an invertible matrix C= (~v1, . . . ,~vn), wherethe ith column of C is the vector~vi such that

J = C−1ΛC (2.27)

with

Λ =

λ1 0 · · · 0... λ2

......

......

. . ....

0 0 · · · λn

. (2.28)

J and Λ are similar matrices.

We can then write

d~w(t)dt

= J~w(t) = C−1ΛC~w(t)

Cd~w(t)

dt= ΛC~w(t)

(~v1, . . . ,~vn)∑i

d~γi(t)dt

~vi = Λ(~v1, . . . ,~vn)∑i

γi(t)~vi.

We can now use that~vi~v j = 1 if i = j and~vi~v j = 0 if i 6= j and obtain

γi(t)dt

= λiγi(t)

2.4.3 Eigenvalues and Eigenvectors of the Jacobian

The eigenvalues λ and eigenvectors ~v follow from the relation J~v = λ~v which can berewritten as (J− λ I)~v = 0 (where I is the identity matrix). There is only the trivialsolution~v =~0 unless

det(J−λ I) = 0. (2.29)

For the 2-component system that we have studied so far we thus have

det(J−λ I) = det

(d fdu |us,vs−λ

d fdv |us,vs

dgdu |us,vs

dgdv |us,vs−λ

)=

(fu−λ fv

gu gv−λ

)= λ

2− ( fu +gv)λ + fugv− fvgu = λ2− tr(J)λ +det(J) = 0,


where tr(J) refers to the trace of J and det(J) to the determinant of J. This can besolved for λi as

λ1,2 =tr(J)±

√tr(J)2−4det(J)

2. (2.30)

For a system with n coupled ODEs Eq. 2.29 leads to the characteristic polynomial ofdegree n,

P(λ ) =n

∑i

αiλi = 0. (2.31)

Even though ~w(t) is a real-valued vector we notice from Eq. (2.30) that the eigenvalues(as well as the eigenvectors and the βi) can have imaginary parts, i.e. λi = ℜ(λi)+iℑ(λi) and thus

~w(t) = ∑i

βi~vi exp(λit) = ∑i

βi~vi exp(ℜ(λi)t)exp(iℑ(λi)t). (2.32)

Figure 2.2: The stability of fixed points for 2-dimensional ODE systems.

2.4.4 Stability of the steady-states

The stability of the steady-state depends on whether or not the initial perturbation~w(0) decays to zero as t → ∞. The absolute value of exp(iℑ(λi)t) = cos(ℑ(λi)t)+isin(ℑ(λi)t is one for all times t. The long-term behaviour of ~w(t) therefore dependson exp(ℜ(λi)t).


Stable steady states If the real parts of all eigenvalues are negative (ℜ(λi) < 0∀i)then the exponential functions exp(ℜ(λi)t) decay to zero for large times and so doesthe perturbation ~w. We thus require that the real parts of all eigenvalues are negativefor the steady state to be linearly (locally) stable. In case of a 2-component model werequire tr(J) < 0 and det(J) > 0 for the real parts of all eigenvalues to be negative(Eq. (2.30)). Stable steady states are therefore located in the upper left hand plane(tr(J)< 0, det(J)> 0) in Figure 2.2.

Unstable and half-stable steady states Contrary, if all real parts of the eigenvaluesare positive (ℜ(λi) > 0∀i) then the steady state is (locally) unstable. If some of thereal parts are positive and others are negative then we speak of a (locally) half-stablesteady state. Unstable steady states are therefore located in the upper right hand plane(tr(J) > 0, det(J) > 0), while half-stable steady states (det(J) < 0) are located in thelower plane in Figure 2.2.

Center solutions and Spirals If at least one eigenvalue has a non-zero imaginarypart then, with λi = φi + iωi, the solution for ~w(t) (Eq. 2.26) can be written as

~w(t) = ∑i

βi~vi exp((φi + iωi)t)

= ∑i

βi~vi exp(φit)exp(iωit) = ∑i

βi~vi exp(φit)(cos(ωit)+ isin(ωit)).(2.33)

The perturbation around the steady state thus oscillates in amplitude with time. Theperiod of the oscillations is thus determined by the imaginary part, ωi, of the eigenval-ues. If ~w(t) is a periodic solution, so is c~w(t) for any constant c 6= 0. Hence ~w(t) issurrounded by a one-parameter family of closed orbits. The amplitude of the oscilla-tions is determined by the pre-factor βi~vi exp(φit). The βis are determined by the initialconditions, i.e.

~w(0) =~x(0)−~xs =V~b (2.34)

where the columns of V are the eigenvectors ~vi and the entries of~b are the βis. Con-sequently, the amplitude of a linear(ized) oscillation is set entirely by its initial condi-tions; any slight disturbance to the amplitude will persist forever. If the real part, φi, ofthe complex eigenvalue is negative then these oscillations will be dampened, while theoscillations will grow with time if the real part of the eigenvalue is positive. Sustainedoscillations exist if eigenvalues have zero real and non-zero imaginary part.


Example: Linear 2-component Model

dxdt

= −y+ x = f (x,y)

dydt

= −2x− y = g(x,y)

with initial conditions x = 0, y = 1. The nullclines are given by

dxdt

= 0⇒ x = y

dydt

= 0⇒ y =−2x

and intersect in the unique steady state (xs,ys) = (0,0).

Perturbation around the steady stateWe consider a perturbation around the steady state (xs,ys) = (0,0)

~w =

(x− xsy− ys

)(2.35)

We can write

d~wdt

= J~w

where

J =

(1 −1−2 −1

)(2.36)

is the Jacobian.

Eigenvalues & Eigenvectors We next determine the eigenvalues, λ , andeigenvectors,~v, of the Jacobian J according to

J~v = λ~v. (2.37)

This yields

λ1,2 =12

(tr(J)±

√tr(J)2−4det(J)

). (2.38)

with tr(J) = 0, det(J) = 1. Thus, λ1 = i, λ2 = −i, ~v1 =

(1

1− i

),

~v2 =

(1

1+ i

).

Type of Steady State For λi = φ + iω

~w(t) = ∑i

βi~vi exp(φit)(cos(ωit)+ isin(ωit))

The real part of the eigenvalue is zero, φ = 0, as tr(J) =0. Accordingly,there is a center solution at the steady state.


Example: Linear 2-component Model - continued

Period of OscillationsThe imaginary part of the eigenvalue, ω = 1, determines the period of theoscillation as 2π .

Amplitude of OscillationsThe amplitude follows from the initial conditions, x = 0, y = 1, and thus

~w(t = 0) =(

01

), given that xs = 0, ys = 0.

Inserting the initial conditions with t = 0 into the solution ~w(t) =

∑i βi~vi exp(λit) we obtain

~w(t = 0) =(

01

)= ∑

iβi~vi = β1

(1

1− i

)+β2

(1

1+ i

)(2.39)

and thus β1 = 0.5i and β2 =−0.5i.

In summary, the full solution is given by

~w(t) =12

i[(

11− i

)exp(it)−

(1

1+ i

)exp(−it)

]=

(01

)cos(t)+

(11

)sin(t)

From this we see immediately that x oscillates between −1 and 1. Theamplitude is therefore 2.

Higher dimensional systems

For a higher dimensional system of equations it becomes increasingly difficult if notimpossible to determine all the roots of the polynomial. However, in order to determinethe local stability of the steady state we do not need the exact values of the roots butonly the sign of their real parts. There are two helpful techniques: Descartes’ Rule ofSigns and the Ruth-Hurwitz criterion.

Descartes’ Rule of Signs Consider the polynomial P(λ ) = ∑ni αiλ

i = 0. Let N bethe number of sign changes in the sequence of coefficients {αn,αn−1, . . . ,α0}, ignoringany that are zero. Then there are at most N roots of P(λ ) which are real and positive,and further, there are N,N−2 or N−4, . . . real positive roots. By setting ω =−λ andagain applying this rule, information is obtained about possible negative roots.


Example: Descartes’ Rule of SignsConsider the characteristic polynomial

P(λ ) = (λ −1)(λ −2)(λ +5) = λ3 +2λ

2−13λ +10 = 0. (2.40)

The polynomial has two sign changes. Accordingly, there are at most tworeal positive roots. Now set ω =−λ ,

P(ω) =−ω3 +2ω

2 +13ω +10 = 0. (2.41)

There is now one sign change and thus at most one real positive root for ω ,that is at most one negative real root for λ , as is indeed the case (λ =−5).

Now consider the case

P(λ ) = (λ − i)(λ + i)(λ +1) = λ3 +λ

2 +λ +1 = 0. (2.42)

The polynomial has no sign changes. Accordingly, there are at most zeroreal positive roots, i.e. no real positive roots, as is indeed the case. In thecontext of a stability analysis, we would conclude that the steady state mustbe stable. Now set ω =−λ ,

P(ω) =−ω3 +ω

2−ω +1 = 0. (2.43)

There are now three sign change and thus either three or one real positiveroots for ω , that is either three or one negative real roots for λ , as is indeedthe case (λ = −1). In combination, we can conclude that there are eitherthree negative real roots for λ (and thus a stable node), or one negative realroot and two complex roots with zero real part. The latter implies (stable)oscillations. Descartes’ Rule of Signs does not allow us to distinguish be-tween the latter two possibilities.

Ruth-Hurwitz criterion The Ruth-Hurwitz criterion is derived using complex vari-able methods and it provides necessary and sufficient conditions that the real parts of alleigenvalues are negative. The real parts of all roots of the polynomial P(λ )=∑

ni αn−iλ

i

with an > 0 are negative as long as the following condition is met for all k = 1, ...,n

Dk =

a1 a3 · · · ·1 a2 a4 · · ·0 a1 a3 · · ·0 1 a2 · · ·· · · · · ·0 0 · · · ak

> 0. (2.44)


Example: Ruth-Hurwitz criterionConsider the characteristic polynomial

P(λ ) = (λ −1)(λ −2)(λ +5) = λ3 +2λ

2−13λ +10 = 0. (2.45)

We thus have a0 = 1, a1 = 2, a2 = −13, a3 = 10. The first determinantthat we have to consider is D1 = a1 = 2 > 0, which meets the Ruth-Hurwitzcriterion. The second determinant that we need to consider is

D2 =

(a1 a31 a2

)= a1a2−a3 =−26−10 < 0. (2.46)

This violates the Ruth-Hurwitz criterion and we can conclude that not allreal parts of the eigenvalues are negative. Indeed, there are two eigenvalueswith positive real part, λ1 = 1, λ2 = 2.

Now consider the case

P(λ ) = (λ − i)(λ + i)(λ +1) = λ3 +λ

2 +λ +1 = 0. (2.47)

Here we have a0 = 1, a1 = 1, a2 = 1, a3 = 1. The first determinant thatwe have to consider is D1 = a1 = 1 > 0, which meets the Ruth-Hurwitzcriterion. The second determinant that we need to consider is

D2 =

(a1 a31 a2

)= a1a2−a3 = 1−1 = 0. (2.48)

This violates the Ruth-Hurwitz criterion and we can conclude that atleast one eigenvalue has a real part that is not negative. Indeed, for twoeigenvalues the real part is zero. In combination with the result fromDescartes’ Rule of Signs, we can thus conclude that the polynomial musthave one negative real root and two complex roots with zero real part.


Figure 2.3: A cartoon pathway of TGF-β signaling. The ligand TGF-β reversiblybinds to the TGF-β receptor, which associates with its co-receptor and is then phos-phorylated to become fully active (1). The active receptor induces phosphorylation ofR-Smad (2), which in turn can reversibly dimerize (3) or form a complex with Co-Smad(4). Those two reactions can take place either in the cytoplasm or in the nucleus andthe five species Smad, phosphorylated Smad, Co-Smad, homodimers and heterodimerscan shuttle from the cytoplasm to the nucleus and back (5). Nuclear Smad/Co-Smadcomplexes act as transcription factors and trigger the transcription of I-Smad mRNA inthe nucleus (6). The I-Smad mRNA then shuttles to the cytoplasm, where it can be de-graded or translated into I-Smad. I-Smad mediates a negative feedback by sequesteringthe active receptor (7) and can be degraded.


2.5 A worked example: A model for TGF-β signaling

2.5.1 Model Development

Figure 2.3 shows a typical depiction of the TGF-β network. TGF-β is a soluble se-creted protein, that signals by binding to the TGF-β receptor (1). The ligand-boundreceptor phosphoryates the regulatory Smad (R-Smad) (2). After dimerization (3),phosphorylated R-Smads bind a Co-Smad (4) and enter the nucleus (5) where theyregulate a wide range of genes. One of the genes that is up-regulated encodes an in-hibitory Smad (I-Smad) (6) that downregulates TGF-β signaling by interfering withthe receptor-dependent phosphorylation of the R-Smads (7). The network clearly has anegative feedback via the I-Smad that may lead to a stable steady state or oscillations,but the network in Fig. 2.3 is too complex to easily analyse its qualitative behaviour.We will therefore start with a much simpler model of the TGF-β network as graphicallysummarized in Fig. 2.4a. Here we focus on the core signaling proteins, the regulatorySmads (R-Smads, [R]) and the inhibitory Smad, [I]. All other regulatory factors areincluded only indirectly. The advantage of such simplistic model is that it is amenableto a range of mathematical techniques. However, there are also important limitationsthat can lead to misleading conclusions regarding the regulatory dynamics of the fullnetwork. These will be discussed below.

Our simplest model considers two time-dependent variables that describe the dy-namics of the concentrations of phosphorylated R-Smad, [R], and for the concentrationof the inhibitory I-Smad, [I]. These two components are part of five reactions: (1)the signal-dependent activation of the R-Smad, (2) the subsequent induction of I-Smadproduction, and (3) the negative feedback of the I-Smad back on the R-Smad. Bothproteins are turned over in reactions (4) and (5). After having decided the topologyand the reactions of the network we need to define the kinetic laws for the gain and theloss rates. We assume that the rate at which phosphorylated R-Smad is formed dependslinearly on the concentration of unphosphorylated R-Smad, [Ru].

ν+ ∝ [Ru]

If we assume that the total concentration of R-Smad, RT , does not change on the con-sidered time-scale, i.e. there is no expression of degradation of R-Smads or the twoprocesses perfectly balance, then we can write [R] + [Ru] = RT . The gain rate couldthen be formulated as ν+ ∝ (RT − [R]). If we further assume that the maximal rate ofR-Smad phosphorylation depends linearly on the signal strength, S, then we need toextend our formulation of the gain rate and write

ν+ = k1S(RT − [R]). (2.49)

The I-Smad inhibits phosphorylation of the R-Smad by binding to the free receptoras well as to the receptor-ligand complex. Since the I-Smad and the ligand bind thereceptor at different sites we assume allosteric cooperative inhibition of the ligand-induced activation. In analogy to section 1.4.3 we then have for the rate of R-Smad

2.5. A WORKED EXAMPLE: A MODEL FOR TGF-β SIGNALING 35

phosphorylation,

ν+ =k1S

1+(

[I]KI

)p (RT − [R]). (2.50)

As regards the loss rate, we assume that the rate of R-Smad dephosphorylation dependslinearly on the concentration of phosphorylated R-Smad

ν− = k2[R]. (2.51)

This implies that it is the availability of the substrate [R] rather than the availabilityof the phosphatase that is limiting. The concentration of the phosphatase can there-fore be considered to be constant and can be lumped into the reaction constant. If thephosphatase was limiting we would need to use a Michaelis-Menten kinetics or Hillkinetics for the reaction as discussed in connection with the Goldbeter-Koshland ki-netics. If both concentrations were not limiting then the reaction would proceed at aconstant rate over time. In summary, we have for the kinetics of the active R-Smad, R,

d[R]dt

= ν+−ν− =k1S

1+(

[I]KI

)p (RT − [R])− k2[R]. (2.52)

The production of the I-Smad depends on the concentration of the active R-Smad,a transcription factor. R-Smad most likely binds to DNA in a cooperative fashion.Accordingly, we model the rate of I-Smad production (gain rate) by a Hill function,

[R]q

[R]q+KqR

, with Hill constant KR and Hill factor q. k3 is the maximal rate at which the

I-Smad can be produced when R-Smad is abundant ([R]� KR). We assume lineardecay of the I-Smad at rate k4, i.e. we assume that the concentration of the proteasethat degrades the I-Smad is not limiting. We then have

d[I]dt

= k3[R]q

[R]q +KqR− k4[I]. (2.53)

2.5.2 Non-dimensionalisation

Before we continue with the analysis of this model it is sensible to simplify the mathe-matical formulation. We will rescale all variables and parameters in a way that the newvariables and parameters have no physical dimensions and can therefore more easilybe compared. Moreover, we will reduce the total number of parameters in the modelby combining parameter values. There is no standard method to non-dimensionalize amodel, and some consider the procedure an art because a clever non-dimensionalisationcan sometimes greatly facilitate the subsequent analysis of the problem. There aresome guidelines, however: 1) If there is a maximal value that a variable can attain it issensible to normalize with respect to this maximal value. 2) If a variable is linked to acertain parameter inclusion of this parameter in the normalization can reduce the totalnumber of parameters in the model. 3) If possible, parameters should be combinedto obtain small and large parameters as this enables the use of perturbation methods.


Keeping all this in mind we rewrite the model in dimensionless form by making thefollowing substitutions

R =[R][RT ]

⇔ [R] = R[RT ] (2.54)

I =

([I]KI

)p

⇔ [I] = IKI (2.55)

τ = k2t⇔ t =τ

k2(2.56)

The non-dimensionalized model then reads

dRdτ

= σ1−R1+ Ip −R = f (R, I) (2.57)

dIdτ

= κ1Rq

Rq +Kq −κ2I = g(R, I) (2.58)

with σ = k1k2

S, κ1 =k3

k2KI, κ2 =

k4k2

and K = KRRT

.

2.5.3 Equilibrium concentrationsThe concentrations of the R-Smad and the I-Smad in the limit of long times, i.e. whenthe system has attained its equilibrium, are obtained by setting the time derivatives tozero, i.e.

dRdt

= 0⇒ R(I) =σ

σ +1+ Ip R-nullcline (2.59)

dIdt

= 0⇒ I(R) =κ1

κ2

Rq

Rq +Kq I-nullcline (2.60)

The two functions R(I) and I(R) are referred to as nullclines. On the R-nullcline R doesnot change, while on the I-nullcline I does not change with time. The R-nullcline canthus be interpreted as the signal-response curve for the active R-Smad concentration asa function of the inhibitor concentration. Similarly, the I-nullcline can be interpreted asa dose-response curve for the inhibitor concentration as a function of the concentrationof active R-Smad. An algebraic calculation of the steady state values of R and I istedious. It is easier to determine the steady state graphically as the intersection of thetwo nullclines (Fig. 2.4b).

2.5.4 Phase Plane AnalysisFig. 2.4b is referred to as phase plane. We can use the phase plane to understand howthe dynamic system evolves in time. For this we plot the trajectories

~x(t) =(

R(t)I(t)

), (2.61)

a parametric representation of R and I with parameter t, in the I-R plane over time t(Fig. 2.4b, grey lines). As we can see all trajectories meet in a common point, the


Figure 2.4: Model Analysis The wiring diagram (a), the phase plane (b), and the time-dependent evolution of activated R-Smad levels (R) in response to signal pulses (c) fora 2-component model of TGF-beta signaling.

intersection point of the two nullclines. To see how the system develops from anypoint (R∗, I∗) in the phase plane we plot small arrows that represent the vector field oftangents

~t =(

dR/dtdI/dt

)(2.62)

to the trajectory~x(t). These tangent vectors~t point in the direction in which the systemdevelops from a point (R∗, I∗). The length of the tangent vector indicates the speedwith which the system will change. Accordingly the arrows are of zero length in thesteady state and they cross nullclines perpendicularly. If all vectors point to the steadystate then the steady state is globally stable, i.e. the system returns to the steady stateafter each perturbation. If only the arrows in the vicinity of the steady state all point tothe steady state then the steady state is said to be locally stable. The phase plane withthe trajectories and phase vector field is called a phase portrait.

In the phase portrait for the 2-dimensional TGF-β model (Fig. 2.4ba) all trajec-tories converge in the steady state (Fig. 2.4b). The steady state is therefore stable. Itshould be noted that despite this stability, an increase in the signaling strength leadsto an increase in the steady state concentration of the active R-Smad. The steady statemoves in the phase plane (the R-nullcline is being shifted to the right), but it remainsstable (Fig. 2.4c).

2.5.5 Linear Stability AnalysisFor sufficiently simple systems a graphical analysis can be employed to judge the sta-bility of a steady state and to reveal oscillations, adaptation, or switches. In the follow-ing, we will discuss a more generally applicable method, the linear stability analysis,to evaluate the (local) stability of steady states. The idea behind a linear stability anal-ysis is to introduce a small perturbation at the steady state and to study whether theperturbation grows or decays over time. In the first case, the steady state is said to beunstable, while in the latter case the steady state is stable. Linearization of the systemof differential equations at the steady state greatly facilitates the analysis but meansthat our results apply only locally, i.e. in the vicinity of the studied steady state.


2.5.6 Linearization around the steady state

Since we will deal with linear systems we start by combining our set of state variablesin a vector, i.e.

~x =(

R(t)I(t)

)and

d~xdt

=

(R(t)I(t)

)=

(f (R, I)g(R, I)

), (2.63)

where the dots denote time derivatives The functions f and g are in general non-linearfunctions. We therefore first need to linearise f and g at the steady states. To this end,we introduce a small perturbation from the steady state, ~xs = (Rs, Is)

T , and write for theperturbation ~w =~x−~xs = (R−Rs, I− Is)

T . We can approximate the values of f (R, I)and g(R, I) close to this steady state using Taylor series expansion, i.e.:

f (R, I) = f (Rs, Is) +11!

∂ f∂R

∣∣∣ss(R−Rs)+

12!

∂ 2 f∂R2

∣∣∣ss(R−Rs)

2 + . . .

+11!

∂ f∂ I

∣∣∣ss(I− Is)+

12!

∂ 2 f∂ I2

∣∣∣ss(I− Is)

2 + . . . (2.64)

and likewise for g(R, I). We next linearise the system by ignoring all terms that are oforder two and higher, i.e.

f (R, I) ∼ f (Rs, Is)+∂ f∂R


∂ f∂ I

∣∣∣ss(I− Is)

g(R, I) ∼ g(Rs, Is)+∂g∂R


∂g∂ I

∣∣∣ss(I− Is) (2.65)

The steady state is stable if the perturbation decays to zero for long times t, i.e.~w→ 0 as t→∞. To express the differential equation in terms of ~w we use the linearisedsystem of equations, and we write

d~xdt

=d~xs

dt+ J(~x−~xs) ⇒ d~w

dt= J~w. (2.66)

J is referred to as Jacobian, and is the matrix of all first-order partial derivatives of thevector-valued function at the steady state (Rs, Is), i.e.

J =

(d fdR |Rs,Is

d fdI |Rs,Is

dgdR |Rs,Is

dgdI |Rs,Is

)=

(fR fIgR gI

). (2.67)

The simple dimensionless 2-component TGF-β model (Eq. 2.57, 2.58) reads

dRdτ

= σ1−R1+ Ip −R = f (R, I)

dIdτ

= κ1Rq

Rq +Kq −κ2I = g(R, I)


with σ = k1k2

S, κ1 =k3

k2KI> 0, κ2 =

k4k2

> 0 and K = KRRT

> 0. The Jacobian at the steadystate is given as

J =

− σ

1+Ips−1 −σ

pIp−1s

(1+Ips )2

κ1qRq−1

s Kq

(Rqs+Kq)2 −κ2

=

(− −+ −

). (2.68)

The sign of the entries in the Jacobian reflect the type of interactions. Since both Rand I affect their own concentration negatively, d f

dR |Rs,Is < 0, dgdI |Rs,Is < 0. Furthermore

R has a positive impact on I, and accordingly the entry J21 is positive, while I has anegative impact of R, and accordingly J12 < 0. We require tr(J) < 0 and det(J) > 0for the real parts of all eigenvalues to be negative (Eq. (2.30)). By inserting the entriesfrom the Jacobian in Eq. (2.68) we obtain for our system of interest

tr(J) = fR +gI =−σ

1+ Ips−1−κ2 < 0

det(J) = fRgI− fIgR =

(− σ

1+ Ips−1)(−κ2)−

(−σ

pIp−1s

(1+ Ips )2

)(κ1

qRq−1s Kq

(Rqs +Kq)2

)> 0.

We have tr(J) < 0 and det(J) > 0 for all σ and the steady state is therefore stable forall signal strengths σ ≥ 0. To obtain oscillations we would require tr(J)2 < 4det(J).If fR and gI balance to give tr(J) = fR + gI = 0 and det(J) > 0 then the oscillationsare sustained. Since all molecular species decay, a positive entry on the diagonal ispossible only if at least one component is autocatalysing its own production.

The dynamic possibilities are limited in the 2D plane. We will therefore now con-sider a 3-component model (Fig. 2.5) with three state variables, the phosphorylatedR-Smad (R), the I-Smad (I), and the I-Smad mRNA concentration (M). The R-Smadnow affects the production of the I-Smad mRNA, and the I-Smad protein is producedfrom the mRNA. We use the same kinetics as before and further assume that the rateof protein production depends linearly on the concentration of mRNA, [M], i.e. weassume that the mRNA rather than the translation machinery is limiting. We then have

d[R]dt

= k1(S)(RT − [R])1

1+( IKI)p−d1[R]

d[M]

dt= νmaxET

[R]q

[R]q +KqR−d2M

d[I]dt

= k5[M]−d3[I]. (2.69)

Nondimensionalisation As before we simplify the model and use dimensionlessvariables and parameters. We write dτ = k5dt, k5δi = di, k5σ = k1, k5µ = νmaxE,E = ET

KI, K = KR

RT, R = [R]

RT, M = [M]

KI, I = I

KI. Time is thus scaled with respect to mRNA

stability, the I-Smad mRNA and protein concentrations are scaled relative to the affin-ity constant KI , and the R-Smad concentration is scaled relative to the total R-Smad


Figure 2.5: A 3-component model for TGF-beta signaling The wiring diagram (a),the phase plane (b), and the time-dependent evolution of activated R-Smad levels (R)in response to signal pulses (c) for a 3-component model for TGF-beta signaling.

concentration. The model then readsdRdτ

= σ(1−R)1

1+ Ip −δ1R = F(R,M, I)

dMdτ

= µRq

Rq +Kq −δ2M = G(R,M, I)

dIdτ

= M−δ3I = H(R,M, I). (2.70)

Phase Plane We start by plotting the 2-dimensional R− I phase plane. We notice thatthe steady state values of M and I are linearly related as I = κ3

κ4M. We then have the

R-nullcline R(I) = κ1κ1+κ2(1+( I

KIp))

and the I-nullcline I(R) = κ3κ4

νRq

Rq+Hq which intersect

in the steady state. When we plot the phase vectors they appear to all point to the staedystate yet the trajectories all converge on a limit cycle. This inconsistency is due to ourmapping of the phase plane to 2D. In reality the phase vectors point out of the R− Iplane and away from the steady state as can be seen in the 3 dimensional plot of thephase plane (Fig. 2.5b). The phase vector field strongly depends on the parametersused, and in the following we will determine the parameter sets for which we obtainsustained oscillations (center solutions) with the help of a linear stability analysis asdescribed above.

Linear Stability Analysis Center solutions arise in linear(ized) systems if the eigen-values have zero real part and non-zero imaginary part. Center solutions arise in 2-component systems if tr(J) = 0 and tr(J)2 < 4det(J). In a 3-component system wehave as characteristic polynomial

P(λ ) = λ3 +α2λ

2 +α1λ +α0 = 0. (2.71)

The coefficients are determined by the Jacobian of the dynamical system. To findparameter sets with center solutions we substitute λ = iω (zero real part) and solve thecharacteristic polynomial. In case of a 3-component network, this yields

P(iω) =−iω3−α2ω2 + iα1ω +α0 = 0. (2.72)


By matching real and complex parts, i.e −iω3 + iα1ω = 0 and −α2ω2 +α0 = 0, weobtain ω2 = α1 =

α0α2

. The eigenvalues follow from ω2 = α1 as

λ1,2 =±i√

α1, λ3 =−α2 (2.73)

Therefore we require α1 =α0α2

for the existence of center solutions that would lead tosustained oscillations. We can determine such parameter sets numerically that matchthe condition α1 =

α0α2

.

Period and Amplitude of Center Oscillations The period of center oscillations isdetermined by the eigenvalues as 2π√

α1. Because of the third negative eigenvalue, the

solution is initially dampened until it reaches the center solution. To calculate the am-plitude of the oscillations we need to determine the solution~x(t) of the set of equations.The eigenvectors that correspond to λ1,2 are also conjugate complex and of the form

~v1,2 =~u± iφ . (2.74)

The general solution~x(t) = ~xs +∑

iβi~vi expλit (2.75)

can be simplified to

~x(t) = ~xs +(β1 +β2)(~ucos(√

α1t)+~φ sin(√

α1t)) (2.76)+ i(β1−β2)(~φ cos(

√α1t)+~usin(

√α1t))+β3~v3 exp(−α2t), (2.77)

where ~xs represents the steady-state vector. Since α1,α2 > 0 holds for all parametersthis system exhibits sustained oscillations. The βi (and thus the amplitude) depend onthe initial conditions,~x(0) =~x0, and can be determined from the linear equation in βi,

~η = ~x0−~xs = (β1 +β2)~u+ i(β1−β2)~φ +β3~v3. (2.78)

Once the βis have been determined, we need to determine the times t at which thevariables R, M, I are maximal and minimal. The difference yields the amplitude of theoscillations. Note that the relative amplitudes of R, M, I do not depend on the initialconditions and thus can be determined without knowledge of the βis.

Oscillations arise in the 3-component TGF-β model because of a slow intermediatestep, the slow(!) formation of I-Smad mRNA (M). The gene for the I-Smad indeed con-tains large introns so that the pre-mRNA is about 14kb long although the protein itselfis less than 1kb of length. Transcription is relatively fast, but splicing events introducethe required delay as shown experimentally for HES oscillations [?, ?]. Oscillationsin the TGF-β response are therefore physiologically plausible, but would have beenmissed if we had analysed only a 2-component model.

For appropriate parameter values we can obtain a center solution around the steadystate (Fig. 2.5b) and sustained oscillations emerge (Fig. 2.5c). As we vary parametervalues, we notice that the Linear Stability Analysis does not predict all oscillations thatwe observe, and that the observed oscillations typically do not depend on the initialconditions. This short-coming is due to the linearization of the equations. Non-linearequations can give rise to Limit Cycle behaviour as discussed next.


2.6 Limit Cycles

Limit cycles are inherently nonlinear phenomena; they cannot occur in linear systems.A limit cycle is an isolated closed trajectory; isolated means that neighbouring trajec-tories are not closed - they spiral either towards or away from the limit cycle. If allneighbouring trajectories approach the limit cycle, we say the limit cycle is stable orattracting. Otherwise the limit cycle is unstable, or in exceptional cases half-stable.Unlike for center solutions, the amplitude of a limit cycle oscillator is determined bythe structure of the system itself and not by the initial conditions. An important methodto establish that there exists such an orbit is presented by the Poincare-Bendixson The-orem as illustrated in Fig. 2.6.

Poincare-Bendixson TheoremSuppose that

1. R is a closed bounded subset of the plane

2. d~x/dt = f (~x) with~x =(

xy

)is a continuously differentiable vector

field on an open set containing R

3. R does not contain any fixed points FP

4. There exists a trajectory C that is confined in R in the sense that itstarts in R and stays in R for all future time

Then either C is a closed orbit or it spirals towards a closed orbit as t→ ∞.So, R contains a closed orbit.

Figure 2.6: The Poincare-Bendixson Theorem All phase vectors point into a closedbounded subset R of the phase plane (grey area). The fixed point (FP) is excluded.

2.6. LIMIT CYCLES 43

Negative criterion of BendixsonIf the trace of the Jacobian does not change sign within a region of the phaseplane, then there is no closed trajectory in this area. Hence a necessarycondition for the existence of a limit cycle is a change in the sign of thetrace of the Jacobian.

Example: The Glycolytic Oscillator The famous Schnakenberg kinetics [?] giverise to limit cycle oscillations for suitable parameter values

x = a− x2y

y = b− y+ x2y. (2.79)

x and y represent normalized concentrations and a,b > 0 are normalized reaction con-stants. x is produced at a constant rate a, and y is produced at a constant rate b, anddecays linearly. Two molecules of x interact with one molecule of y to enhance pro-duction of y and removal of x. This model could also describe binding of a dimericligand y to a monomeric receptor x. The complex would then induce production of xand removal of the ligand y.

First we need to find the nullclines and construct a phase plane. The nullclines aregiven by y = a/x2 and y = b/(1− x2) and cross in the positive fixed point (xs,ys) =(√

a(a+b) ,a+b

)as depicted in Figure 2.7. Depending on where the nullclines cross,

the fixed point is either stable (Fig. 2.7A) or half-stable (Fig. 2.7B). A limit cycle canexist only if the fixed point is not stable. To access how the stability of the steady statesdepends on the parameters a and b we carry out a linear stability analysis. The Jacobian

Figure 2.7: The phase plane of the Schnakenberg model. (A) The phase plane witha stable fixed point. (B) The phase plane with an unstable fixed point and a limit cycle.


is given as

J =

(−2xsys −x2

s2xsys −1+ x2

s

)s. (2.80)

with determinant det(J) = 2xsys = 2√

a(a+b)> 0 and trace tr(J)=−2xsys−1+x2s =

−2√

a(a+b)− 1+ a(a+b) . The fixed point is unstable if tr(J) > 0. This is the case

if 3a+ b < (a+ b)3. This defines a curve in (a,b) space that separates the parameterspace with limit cycle solution from the one is a stable steady state solution. We finallyneed to define a trapping region that excludes the fixed point and where all phase vec-tors point inwards. Such trapping region can indeed be constructed and a limit cyclemust thus exist when the nullclines cross such that the fixed point is not stable.

Lienard systemA Lienard equation is a second order differential equation of the form

x+ f (x)x+g(x) = 0 (2.81)

where f and g are two continuously differentiable functions on R, withf an even function (i.e. f (x) = f (−x)) and g an odd function (i.e.g(x) =−g(−x)).

The equation can be transformed into an equivalent two-dimensional systemof ordinary differential equations, the so-called Lienard system, of the form[

x1x2

]= h(x1,x2) :=

[x2−F(x1)−g(x1)

](2.82)

where

F(x) :=∫ x

0f (ζ )dζ

x1 := x

x2 := dx/dt +F(x).

A Lienard system has a unique and stable limit cycle surrounding the originif it satisfies the following additional properties:

• g(x)> 0 for all x > 0

• limx→∞ F(x) := limx→∞

∫ x0 f (ζ )dζ = ∞

• F(x) has exactly one positive root at some value p, where F(x) < 0for 0 < x < p and F(x)> 0 and monotonic for x > p.

2.6. LIMIT CYCLES 45

Example: The van-der-Pol equation We can transform the van-der-Pol equation

x+µ(x2−1)x+ x = 0 (2.84)

into a Lienard system if we recognize that

x+µ(x2−1)x =ddt

(x+µ(

13

x3− x)). (2.85)

So if we letF(x) =

13

x3− x w =dxdt

+µF(x) (2.86)

the van-der-Pol equation (Eq. 2.84) implies that

dwdt

= d2x/dt2 +µ(x2−1)dx/dt =−x. (2.87)

Hence the van-der-Pol equation is equivalent to

dxdt

= w−µF(x)

dwdt

= −x. (2.88)

One further change of variables further simplifies the analysis. If we let y = wµ

then wehave

dxdt

= µ(y−F(x))

dydt

= − 1µ

x. (2.89)

This set of equations is similar to the Fitzhugh-Nagumo model (Eq. ??) that modelsthe generation of action potentials in nerve cells.

The period of the van-der-Pol oscillator So far we have mainly considered quali-tative questions. We are now interested to determine the period of this oscillator. Byinspection of the phase plane we notice that there are two fast branches and two slowbranches (Fig. 2.8A). The slow branches are those that follow the nullclines while thefast branches are those that are far away from the nullclines. The period T is essen-tially the time required to travel along the two slow branches, since the time spent inthe jumps is negligible for large µ . By symmetry the time spent on each branch is thesame. Note that on the slow branches y∼ F(x) and thus

dy/dt = F ′(x)dx/dt = (x2−1)dx/dt. (2.90)

But since dy/dt =−x/µ we have dx/dt =− xµ(x2−1) . Therefore

dt ∼ µx2−1

xdx (2.91)


Figure 2.8: The van-der-Pol oscillator. (A) A phase plane of the van der Pol oscillator.(B) A characteristic time course.

on a slow branch, and accordingly

T =∮

dt ∼ 2∫ xB

xA

−µx2−1

xdx = µ[3−2ln2]. (2.92)

where xB = 1 is the positive x for which y ∼ F(x) assumes its minimum. xA = 2 fol-lows by determining the maximum of F on the negative x-axis, i.e. F(xD =−1) = 2/3,and by subsequently determining the positive x for which this value of F is reached,i.e. F(xA = 2) = 2/3. With much more work a refined solution can be obtained, i.e.T = µ[3−2ln2]+2αµ−1/3 + . . . where α = 2.383 is the smallest root of Ai(−α) = 0where Ai(x) is a special function called the Airy function.

ChaosThe dynamical possibilities in the 2D plane are very limited: if a trajectoryis confined to a closed, bounded region that contains no fixed points, thenthe trajectory must eventually approach a closed orbit - nothing more com-plicated is possible! In higher dimensional systems trajectories may wanderaround forever in a bounded region without settling down to a fixed pointor a closed orbit.Trajectories may also be attracted to a strange attractor, afractal set, on which the motion is aperiodic and sensitive to tiny changes inthe initial conditions, leading to chaotic behaviour.

2.7 Delay EquationsDelays can have important consequences as we have seen by example of the TGF-βnetwork where the delay enabled oscillations. In the TGF-β model this was achievedby adding a slow step of mRNA transcription. Because of the delay requirement we

2.8. BIFURCATION ANALYSIS 47

Figure 2.9: The van-der-Pol oscillator.

so far had to consider at least two components to create an oscillator. We will now in-troduce delay equations to incorporate an explicit delay into our differential equations.Consider the differential equation

dxdt

=− π

2Tx(t−T ) (2.93)

where x decays at a rate that is proportional to its value at an earlier time t−T . Theequation can be solved to give

x(t) = Acos(

π

2Tt)

(2.94)

where A is a constant. This simple delay equation produces oscillations, though weshould note that x can assume both positive and negative values which would not makesense in many biological settings. We can consider a slightly more sophisticated modelthat is similar to a logarithmic growth model

dxdt

= x(t)(1− x(t−T )) (2.95)

Here, x enhances its own production and decay, but the negative effect occurs with adelay T and thus depends on the concentration of x at an earlier time t−T . Even thoughthe equation still appears simple solutions can only be found numerically. To computethe solutions for t > 0 we now require as initial conditions x(t) for all −T ≤ t ≤ 0.

2.8 Bifurcation AnalysisKey to all regulatory control is how the stability of a steady state is effected by regu-latory control. Bifurcation diagrams are employed to analyse how the values and thestability of equilibrium points depend on a regulatory control parameter, the bifurcation


parameter. Those points at which the stability of an equilibrium point changes or newsteady state solutions appear or disappear are called bifurcation points. Particularly im-portant bifurcation behaviours in biological models include transcritical bifurcations,saddle-node bifurcations and Hopf bifurcations.

Generating Bifurcation DiagramsTo generate a bifurcation diagram the stability of the steady states needsto be evaluated as the bifurcation parameter is changed. While the sta-ble steady state branches can also be determined by solving the set ofequations numerically as the bifurcation parameter is changed this doesnot allow the determination of the unstable steady state branches. Todetermine also the unstable steady state branches a linear stability anal-ysis has to be carried out. This can be laborious and several softwarepackages are available to draw bifurcation diagrams, including matcont(http://www.matcont.ugent.be) and auto (http://cmvl.cs.concordia.ca/auto).

2.8.1 Transcritical Bifurcation

A transcritical bifurcation is characerised by an equilibrium with an eigenvalue whosereal part passes through zero. Both before and after the bifurcation, there is one unsta-ble and one stable fixed point (Fig. 2.10a). However, their stability is exchanged whenthey collide. So the unstable fixed point becomes stable and vice versa.

2.8.2 Saddle-Node Bifurcation

When two equilibrium points collide and annihilate each other we speak of a saddle-node bifurcation (Fig. 2.10b). Saddle-node bifurcations generate switches that arerobust to fluctuations because they enable hysteresis, a term which is used to describesystems which have memory. The bifurcation diagram in Fig. 2.10b contain a regionwith three steady states (2 stable and 1 unstable). If the system is started on the lowerbranch at low signaling strength (point 0) it will follow this branch as the signal strengthS is increased (point 1) until the system reaches the saddle-node bifurcation point (SN1)where the stable equilibrium branch collides with the unstable equilibrium branch. Asthe two steady states are annihilated at the bifurcation point a further increase in thesignal strength S results in a jump to the remaining equilibrium (point 2). If the signalstrength is reduced again the system continues to follow the new equilibrium branch(point 4) and does not switch back at the previous bifurcation point. The equilibriumthat the system attains at a given signal strength S therefore depends on the historyof the system, a phenomenon referred to as hysteresis. The new stable equilibriumbranch meets the unstable branch at yet another saddle-node bifurcation point (SN2)where both equilibria are annihilated and the system jumps back to the initial stableequilibrium branch.


Figure 2.10: Bifurcation Behaviour. (a) Transcritical Bifurcation (b) Saddle-NodeBifurcation: Toggle switch. (c) Negative Feedback Oscillator (left) Network motifs.(center) Phase plane. (right) Bifurcation diagrams with bifurcation parameter S. Sta-ble steady states are denoted by solid lines, unstable steady states by dotted lines. Hopfbifurcations are denoted by H, saddle-node bifurcations by SN. For details please seethe main text.

2.8.3 Hopf bifurcation

At Hopf bifurcations oscillatory solutions alter their stability (Fig. 2.10c), i.e. the realpart of the conjugate complex eigenvalues changes its sign or visually: the two complexconjugate eigenvalues simultaneously cross the imaginary axis. We can distinguish twodifferent types of Hopf bifurcations: supercritical and subcritical Hopf bifurcations. Incase of a supercritical Hopf bifurcation the amplitudes of the oscillations grow slowlyas the bifurcation parameter passes the bifurcation point while in case of a subcrit-ical Hopf bifurcation the amplitudes of the oscillations are large immediately as thebifurcation point is passed.


Figure 2.11: Hopf Bifurcations. (A) Supercritical Hopf Bifurcation. (B) SubcriticalHopf Bifurcation.

Supercritical Hopf Bifurcation

In terms of the flow in phase space, a supercritical Hopf bifurcation occurs when astable spiral changes into an unstable spiral surrounded by a small, nearly ellipticallimit cycle. Hopf bifurcations can occur in phase spaces of any dimension n ≥ 2. Asimple example is provided by the following set of equations

r = r(µ− r2)

θ = ω +br2. (2.96)

µ controls the stability of the fixed point at the origin, ω gives the frequency of in-finitesimal oscillations, and b determines the dependence of frequency on amplitudefor larger amplitude oscillations. To analyse the bifurcation behaviour we rewrite thesystem in Cartesian coordinates, x = r cosθ , y = r sinθ . Then

x = µx−ωy+ cubic termsy = ωx+µy+ cubic terms

So the Jacobian at the origin is

J =

(µ −ω

ω µ

), (2.97)

which has eigenvalues λ = µ± iω . The eigenvalues cross the imaginary axis from leftto right as µ increases from the negative to positive values. Thus when µ < 0 the originr = 0 is a stable spiral, though a very weak one, i.e. the decay is only algebraically fast.For µ > 0 there is an unstable spiral at the origin and a stable circular limit cycle atr =√

µ . Finally, the linear stability analysis wrongly predicts a center at the origin - infact for µ = 0 the origin is still a stable spiral!


Subcritical Hopf Bifurcation

In case of a subcritical Hopf bifurcation the trajectories jump to a more distant attrac-tor, which may be a fixed point, another limit cycle, infinity, or in three and higherdimensions, a chaotic attractor.

A simple example is provided by the following set of equations

r = r(µ + r2− r4)

θ = ω +br2. (2.98)

An analytical criterion exists to distinguish subcritical and supercritical Hopf bifur-cations, but it is typically difficult to use. Numerical solutions can be used to a certainextend though computer experiments are NOT proofs, and the code needs to be care-fully checked before arriving at any firm conclusion. In case of a supercritical Hopfbifurcation a small, attracting limit cycle should appears immediately after the fixedpoint goes unstable and the amplitude should shrink back to zero as the parameter isreversed.

Degenerate Bifurcation

A good example is provided by the dampened pendulum, i.e.

x+µ x+ sin(x) = 0. (2.99)

As µ is changed from positive to negative the fixed point at the origin changes from astable to an unstable spiral. However, at µ = 0 we do not have a true Hopf bifurcationbecause there are no limit cycles on either side of the bifurcation. Instead at µ = 0 wehave a continuous band of closed orbits surrounding the origin.

The degenerate case typically arises when a nonconservative system suddenly becomesconservative at the bifurcation point. Then the fixed point becomes a nonlinear center,rather than a weak spiral required by a Hopf bifurcation.

mathematical modelling for systems biology - eth zürich · mathematical modelling for systems...

Documents