Mathematical Modelling for Systems Biology
March 7, 2016
2
Contents
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Biochemical Reaction Modelling
1.1 An Introduction to Modelling . . . . . . . . . . . . . . .
1.2 Basic Reaction Types . . . . . . . . . . . . . . . . . . .
1.2.1 0th Order Reactions - Constant Reaction Rates .
1.2.2 1st Order Reactions - Monomolecular Reactions
1.2.3 2nd Order Reactions - Bimolecular Reactions . .
1.3 Rule-based modeling . . . . . . . . . . . . . . . . . . .
1.3.1 Rule-based modeling concepts . . . . . . . . . .
1.3.2 Contact maps . . . . . . . . . . . . . . . . . . .
1.3.3 Simple example . . . . . . . . . . . . . . . . . .
1.4 Simplifying Approximations . . . . . . . . . . . . . . .
1.4.1 Michaelis-Menten Kinetics . . . . . . . . . . . .
1.4.2 Hill Kinetics - Cooperativity . . . . . . . . . . .
1.4.3 Inhibitory interactions . . . . . . . . . . . . . .
1.4.4 Goldbeter-Koshland Kinetics . . . . . . . . . . .
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Model Development and Analysis
2.1 Development of simplified models . . . . . . . . . .
2.2 Non-dimensionalisation . . . . . . . . . . . . . . . .
2.3 Phase Plane Analysis . . . . . . . . . . . . . . . . .
2.4 Linear Stability Analysis . . . . . . . . . . . . . . .
2.4.1 Linearization around the steady state . . . . .
2.4.2 Solution to the linearised ODE . . . . . . . .
2.4.3 Eigenvalues and Eigenvectors of the Jacobian
2.4.4 Stability of the steady-states . . . . . . . . .
2.5 A worked example: A model for TGF-β signaling . .
2.5.1 Model Development . . . . . . . . . . . . .
2.5.2 Non-dimensionalisation . . . . . . . . . . .
2.5.3 Equilibrium concentrations . . . . . . . . . .
2.5.4 Phase Plane Analysis . . . . . . . . . . . . .
2.5.5 Linear Stability Analysis . . . . . . . . . . .
2.5.6 Linearization around the steady state . . . . .
2.6 Limit Cycles . . . . . . . . . . . . . . . . . . . . . .
2.7 Delay Equations . . . . . . . . . . . . . . . . . . . .
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4
CONTENTS
2.8
Bifurcation Analysis . . . . . .
2.8.1 Transcritical Bifurcation
2.8.2 Saddle-Node Bifurcation
2.8.3 Hopf bifurcation . . . .
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Chapter 1
Biochemical Reaction Modelling
This chapter provides an introduction to the formulation and analysis of differentialequation-based models for biological regulatory networks. We will discuss basic
reaction types and the use of mass action kinetics and of simplifying approximations 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 predict how a dynamical system evolves over time and what equilibrium it assumes we
can formulate differential equations that describe the state of the system. Each system
component is represented by a state variable, x. Typically x is a concentration, but x
could also represent a density or the number of molecules. The change of x, ∆x, per
time interval ∆t depends on the rate v+ at which x is generated, the ”gain” rate, and on
the loss rate v− at which x is removed. Here the rate of generation reflects all processes
that lead to an increase in x, i.e. synthesis, change of chemical modification and many
more while the loss rate includes all processes that lead to a decrease in the value of x.
More formally we can write
∆x
= gain rate - loss rate = v+ − v− .
∆t
(1.1)
Instead of considering finite time intervals ∆t we will consider the change in an infisitimal small time interval dt, such that
dx
= gain rate - loss rate = v+ − v− .
dt
(1.2)
In many cases there are multiple components and multiple compartments (i.e. cytoplasm and nucleus) with distinct pools of x so that instead of a single x we have many
state variables xi ∈ {x1 (t), x2 (t), . . .}. The value of each xi then also changes due to the
formation 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
= synthesis - degradation±shuttling±complex formation±chemical modification±· · ·
dt
(1.3)
The values of all state variables at a given time point t constitute the state of the system
at time t. When gain and loss rates balance the variable no longer changes with time,
i
i.e. dx
dt = 0. When the gain and loss rates of all variables balance then the system
reaches 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 the
basis of all feedback regulation. Mathematically, we say that the system of ODEs is
coupled, i.e. the differential equations for the different variables depend on each other
dxi
= f (x1 , x2 , ..., xn ).
dt
(1.4)
This means that we need to consider the entire set of equations simultaneously and
cannot solve the different equations separately. Software packages (such as Matlab
or Mathematica) are available that provide algorithms to solve these sets of equations
numerically. In the following we will discuss how the rates of some typical biological reaction types are affected by changes in the values (i.e. concentrations) of state
variables.
1.2
Basic Reaction Types
The most accurate model can be obtained when the law of mass action is used to formulate kinetic laws for all elementary reactions in Eq. 1.3.
Mass Action Kinetics According to the law of mass action, the rate of a
reaction is proportional to the probability of a collision of reactants. This
probability, in turn, is proportional to the concentrations of the participating
molecules to the power of the molecularity, i.e. the number in which they
enter the specific reaction.
Thus, if the molecules participating in the gain reaction have concentrations ci and
molecularites mi , then the gain rate according to the general mass action rate law is
given by
i
v+ = k+ ∏ cm
(1.5)
i .
i
Similarly, if the molecules participating in the loss reaction have concentrations c j and
molecularites m j , then the loss rate according to the general mass action rate law is
given by
m
v− = k− ∏ c j j .
(1.6)
j
1.2. BASIC REACTION TYPES
3
The equilibrium constant Keq is then given by
mj
∏j cj
k+
Keq =
=
m
k−
∏i ci i
(1.7)
where the concentrations are those in equilibrium. Here, it should be noted that this
only holds as long as there are no parallel inputs to the gain or loss rates. Parallel inputs
would have to be included as sums, i.e.
m
v+ = ∑ v+ j = ∑ k+ j ∏ ci ji .
j
j
(1.8)
i
We will continue with the simple version. Here, we can distinguish between different
order reactions and relate these to the basic reaction types that are frequently found in
biological settings. Thus, for
dci
i
= kcm
(1.9)
i
dt
the order of the reaction depends on mi . For ∑i mi = 0, we speak of a zero-order
reaction, for ∑i mi = 1 of a first-order reaction, and for ∑i mi = 2 of a second order
reaction.
1.2.1
0th Order Reactions - Constant Reaction Rates
0th order reactions are the simplest of all reactions because the rate of the reaction does
not depend on the state variables. This kinetic law is used frequently to describe the
synthesis of a molecular component.
Constant Synthesis Assuming that the species X is produced at a constant rate k prod
we write for the concentration of X, [X],
d[X]
= k prod .
(1.10)
dt
This equation can be solved as [X(t)] = [X(t0 )] + k prod (t −t0 ) and we note that the concentration of X at time t depends only on the initial value of X at time t0 and on the
time interval t − t0 that has passed. Accordingly, the rate at which X is produced does
not change when the concentration of X is changed (Fig. 1.1A, a).
MATLAB Exercise
Simulate equation 1.10 with k prod = 1 and [X(t0 )]=0 for t ∈ [0, 10] and compare the
solution to the analytical solution:
function ODE_model1_prod()
x0 = 0; % initial concentration
4
CHAPTER 1. BIOCHEMICAL REACTION MODELLING
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 concentrations vary with time. Reactions that only depend on one such component are referred
to as monomolecular reactions. Important examples include the decay of a molecular
species or its transport between compartments, i.e. cytoplasm and nucleus. Mathematically the dynamics of the state variable linearly depends on the state variable in 1st
order reactions.
Linear Degradation The rate at which a protein, mRNA or similar is removed or
inactivated is often proportional to its own abundance, i.e. it changes linearly with its
own concentration (Fig. 1.1A, b). We write for the concentration of such a component
X
d[X]
= −kdeg [X].
(1.11)
dt
This equation can be solved as [X(t)] = [X(t0 )] exp (−kdeg (t − t0 )) and we note that
the concentration of X decays exponentially over time. An important measure is the
characteristic time t1/2 = lnk (2) by which the initial concentration [X(t0 )] has decreased
deg
by half.
MATLAB Exercise
Simulate equation 1.11 with kdeg = 1 and [X(t0 )]=1 for t ∈ [0, 10] and compare the
solution to the analytical solution:
1.2. BASIC REACTION TYPES
5
function ODE_model2_degradation()
x0 = 1; % initial concentration
k_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 compartments (i.e. nucleus and cytoplasm) can be described by two coupled differential equations for the concentrations of X in the nucleus, [Xn ], and in the cytoplasm [Xc ]. Importantly, we need to take the volume difference between the two compartments into
account. Let us denote the different volumes of cytoplasm and nucleus as Vc and Vn . If
a concentration [Xc ] shuttles from the cytoplasm to the nucleus then Vc [Xc ] molecules
of Xc leave the cytoplasm and enter the nucleus. In the nucleus, these Vc [Xc ] molecules
correspond to a concentration Vc /Vn · [Xc ], where Vn is the volume of the nucleus. Accordingly, if X is exported from the nucleus at rate kout and is imported from the cytoplasm at rate kin then we have
d[Xn ]
dt
d[Xc ]
d[Xn ]
=−
dt
dt
Vc
[Xc ] − kout [Xn ]
Vn
Vn
= −kin [Xc ] + kout [Xn ] .
Vc
= kin
(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
n ]Vn
and obtain a differential equation that is similar to Eq. 1.11 except for
[Xc ] = T −[X
Vc
an additional constant term, kin VTn ,
d[Xn ]
Vc T − [Xn ]Vn
T
= kin
− kout [Xn ] = kin − (kin + kout )[Xn ].
dt
Vn
Vc
Vn
(1.13)
6
CHAPTER 1. BIOCHEMICAL REACTION MODELLING
MATLAB Exercise
Combine the previous MATLAB examples (production and liner decay) to simulate
equation 1.13.
1.2.3
2nd Order Reactions - Bimolecular Reactions
Most reactions in biology involve some form of complex formation and therefore depend on the interaction of more than one time-varying component. Here it is important
to distinguish between homo- and heterodimerization.
Complex formation - heterodimers The formation of heterodimers, XY , is the result
of the interaction of two components X and Y (Fig. 1.1A, c). The reaction rate depends
linearly on both the concentrations of X and of Y . Assuming that the reaction proceeds
at rate kon and that the total concentrations of both components is constant we have
d[XY ]
= kon [X][Y ] = kon (XT − [XY ])(YT − [XY ])
dt
(1.14)
where XT = [X] + [XY ] and YT = [Y ] + [XY ] are the total concentrations of X and Y reYT −XT
spectively. This equation can be solved to give [X](t) = XT −[XY (t)] = YT
XT
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] and
compare the solution to the analytical solution.
Complex formation - homodimers Similarly, the kinetics of homodimer formation
between two X components can be described by the following quadratic rate law
d[X2 ]
= kon [X]2 = kon (XT − 2[X2 ])2
dt
(1.15)
where XT is the total amount of X which we again assume to be constant. Here the
rate of homodimer X2 formation depends non-linearly on the concentration of the
monomers X (Fig. 1.1A, d). The dynamics of X can be described by
d[X]
d[X2 ]
= −2
= −2kon [X]2
dt
dt
(1.16)
.
1.2. BASIC REACTION TYPES
7
Figure 1.1: Basic Reaction Types (A): (a) Constant Synthesis. (b) Monomolecular
Reactions: 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 or
competitive inhibition. (d) Goldbeter-Koshland Kinetics. The reaction scheme and a
plot of the representative reaction rate versus the concentration of the reactant X (A) or
the enzyme E (B) are depicted. In (B, d) the steady state concentrations of X p and X
are plotted versus the signal strength S.
8
CHAPTER 1. BIOCHEMICAL REACTION MODELLING
The equation can be solved to give [X](t) =
XT
2XT 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 the
monomers and the dimers would then need to be solved. In case of higher order complexes the formation can, in general, be modeled as a sequential step of bimolecular
reactions.
1.3
Rule-based modeling
Based on mass action, accurate models can be formulated even for large networks, as
long as both the components (i.e. the proteins, compartments, complexes etc) and the
rules and kinetics of their interactions are known. Typically such models are based on
network cartoons of the form shown in Fig 1.2A. To translate such cartoons into mathematical models we assign a single state variable xi (t) to each icon. One state variable
would be the unbound ligand x1 (t). Another state variable would be the ligand-receptor
complex, x2 (t), and so on. The set of values of all state variables {x1 (t), x2 (t), ...} at a
given 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 dynamic systems that are based on hundreds of ODEs to describe the interactions between less than 10 components, a problem referred to as ”combinatorical complexity”.
The problem arises because proteins typically have multiple binding sites such that
even a simple network as shown in Figure 1.2A translates into a much larger network
as shown in Figure 1.2B when these are taken into account. The technical problem
of 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 it
is easy to generate a comprehensive system and avoid making any errors due to missing interactions and/or unjustified assumptions. There are different software that have
been designed to enable rule-based modeling. Among other softwares, most popular
are 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 concepts
We can summarize the possible rules in rule-based modeling in the following five
basic transformations. (1) Complex formation: a bond can be formed to link two
molecules through their available binding sites. (2) Complex dissociation: an existing bond between two molecules can be removed (3) Change the state-label of a
component: a molecule undergoes a certain post-translational state modification (e.g.
become phosphorylated), or alters the state-label of its functional shape or conformation (e.g. open/closed conformation of integrins). (4) Add a molecule: production of
a species. (5) Delete a molecule: degradation of a species. The above stated rules can
be expressed as unidirectional transformations, but some of them can be also bidirectional (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 regulatory
interactions in the network that controls σ F during sporulation in Bacillus subtilis. The
image was reproduced from Figure 1 in [?]. (Down-left) Manually created network of
interactions (Right) Contact map
Based on these concepts, already with a very small set of rules and molecule types we
can generate enormous systems.
1.3.2
Contact maps
Large regulatory networks are difficult to visualise. On the one hand, the conceptual
cartoons that are usually found in text books are too abstract to reflect the full system
that needs to be modeled mathematically. On the other hand, detailed network maps
are 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 of
the system; the actual transitions from reactants to products are not given explicitly. In
Figure 1.2, an example of a signaling pathway is illustrated both in all three ways.
1.3.3
Simple example
The following figure illustrates some basic elements with a small example. Assume
there are two interacting molecules, a ligand L and a receptor R. Let the ligand have one
binding site for the receptor, and the receptor to have two binding sites, one for ligandbinding and one that enables homodimerization with another receptor. Also assume
that the receptor has a tyrosine site Y with two state-labels, phosphorylated P and
unphosphorylated U (Figure 1.3(a)). Having defined the molecules and their properties
we can start defining patterns, e.g. free receptors irrespective of their state-label (Figure
1.3(b)). Now we can write down a set of simple rules that will summarize our set of
interactions: the ligand and the receptor can form a heterodimer complex, this complex
can further homodimerize, and this allows the receptors to further trans-phosphorylate
each other (Figure 1.3(c)). This scenario of interactions could be further continued
by just binding of adapter molecules on the phosphorylated site, or by enabling the
modulation of other interactions upon phosphorylation.
10
CHAPTER 1. BIOCHEMICAL REACTION MODELLING
Figure 1.3: Rule-based modeling concepts using BioNetGen language (BNGL). (a)
Molecules (b) Patterns (c) Simple example of interactions
1.4
Simplifying Approximations
If we formulate the kinetics of large networks based on first principles then the description becomes very complex and will be accurate only if we are able to determine
a large number of parameters with high accuracy. In particular, in case of cooperative enzymes it can be very difficult to obtain accurate data on the reaction rates of all
intermediate complexes. Most of the times we do not know all elementary/molecular
interactions that regulate a particular reaction. Therefore there are many situations in
which simplifications are sufficient and in fact preferable. Even from a computational
point 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 different time scales are exploited on which reactions proceed. If some reactions proceed
much faster than others then certain concentrations are constant early on while other
concentrations barely change at a later time. This is used in the derivation of MichaelisMenten kinetics for the enzymatic turn-over of a substrate (Fig. 1.1B (a)). In a basic
enzymatic reaction a substrate X binds to an enzyme E to form a substrate-enzyme
complex C. Complex formation is a reversible reaction while the formation of the
product P is irreversible,
k1
k
2
GGGGGG
B
X +E F
G C −→ E + P.
k−1
(1.17)
1.4. SIMPLIFYING APPROXIMATIONS
11
Figure 1.4: The Kinetics of the Michaelis Menten Reaction (a, c)The kinetics of
substrate X and product P on linear and log scale. (b,d) The kinetics of enzyme E and
substrate-enzyme complex C on linear and log scale.
The elementary reaction rates for the enzymatic turn-over of a substrate are:
d[X]
dt
d[E]
dt
d[C]
dt
d[P]
dt
= k−1 [C] − k1 [X][E]
(1.18)
= (k−1 + k2 )[C] − k1 [E][X]
(1.19)
= −(k−1 + k2 )[C] + k1 [E][X]
(1.20)
= k2 [C]
(1.21)
with initial conditions [X(0)] = XT , [E(0)] = ET , [C(0)] = [P(0)] = 0. We notice that
d[E]
d[C]
dt + 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 from
the other differential equations since P does not impact on X, C, or E. We can therefore
reduce the set of 4 differential equations to a set of 2 coupled differential equations:
d[X]
dt
d[C]
dt
= k−1 [C] − k1 [X](ET − [C])
(1.22)
= −(k−1 + k2 )[C] + k1 (ET − [C])[X]
(1.23)
12
CHAPTER 1. BIOCHEMICAL REACTION MODELLING
Non-dimensionalization To simplify all subsequent analyses, we first non-dimensionalize
the equations. As such, each variable and each parameter needs to be transformed into
a 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 can
attain then it is sensible to normalize the variable with respect to this maximal value.
[C]
We therefore write s = [X]
XT , c = ET . (2) Parameters should be grouped so as to reduce
k
+k
k
2
−1
the total number of parameters. We write τ = k1 ET t, κ1 = −1
k1 XT , and κ2 = k1 XT . (3)
If possible, very small and very large parameters should be generated so as to enable
the use of perturbation methods. Here we exploit that the substrate concentration, XT ,
is much larger than the total enzyme concentration, [ET ] and thus ε = EXTT 1. We then
obtain
ds
dτ
dc
ε
dτ
= −s + c(s + κ2 )
= 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 used
when processes occur on very different timescales such that within a certain time interdc
≈0
val a variable barely changes in value. In case of the Michaelis-Menten model ε dτ
dc
dc
once dτ ≤ O(1). In that case ε dτ = s − c(s + κ1 ) ≈ 0 and the quasi-state approxis
mation thus yields as quasi-steady state c = s+κ
. In dimensional form we then have
1
k1 [X]
, and thus for the rate at which the product (P) is formed the well[C] = ET k [X]+k
1
−1 +k2
known Michaelis-Menten kinetics
d[P]
k1 [X]
[X]
= k2 [C] = k2 ET
= vmax
.
dt
k1 [X] + k−1 + k2
[X] + Km
(1.25)
k2 ET is the maximal rate, vmax , at which this reaction can proceed when the substrate
k +k
concentration is large ([X] Km ). Km = −1k1 2 is the Michaelis-Menten constant and
specifies the substrate concentration at which the reaction proceeds at half-maximal
rate. Importantly, the rate at which product is formed versus the substrate concentration yields a hyperbolic graph (Fig. 1.1B (a) RHS). While the conditions for MichaelisMenten kinetics do not always strictly apply, such dependency of the reaction rate on
the substrate concentration is observed more generally. In such cases the reaction rate
[X]
.
ν can be approximated by ν = νmax [X]+K
m
Note that this approximation is only valid at sufficiently long times. If we check
the 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 a
relatively stable quasi-steady state value (Fig. 1.4D). We can estimate the length of
1.4. SIMPLIFYING APPROXIMATIONS
13
the relevant time scales. The first time scale Tc on which C changes rapidly while X
remains 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) = ET
XT
(1 − exp (−k1 (XT + Km )t)) .
XT + Km
(1.28)
[X]
The quasi-steady-state concentration of C, [C]qstst = ET [X]+K
, is thus reached expom
nentially fast and
Tc =
1
k1 (XT + Km )
(1.29)
represents the time within which the concentration of C reached its quasi-steady state
value up to 1 − exp(−1) ∼ 63%. This characteristic time can thus be used as the first
time scale Tc over which [C] is changing rapidly. The subsequent timescale on which
X changes significantly can be estimated as
Ts =
XT
dX
| dt |max
∼
XT + Km
.
k2 ET
(1.30)
[X]
which applies once [C] has reaches
Here we used d[X]/dt = −d[P]/dt = −k2 ET [X]+K
m
a quasi-steady state; the reaction speed is maximal initially while [X] ∼ XT .
14
CHAPTER 1. BIOCHEMICAL REACTION MODELLING
Quasi-stationarity requires
1. a separation of the timescales on which C and X change rapidly, i.e.
Tc Ts and thus
k2 ET
1
(1.31)
k1 (XT + Km )2
2. that thesubstrate
concentration is almost constant in the time interval
Tc , i.e. ∆[X]
1
with |∆[X]| ∼ k1 ET XT Tc since in this time interval
XT
[X] is changing mainly due to binding of ET . Therefore
∆[X] k1 ET XT
ET
1
XT ∼ XT k1 (XT + Km ) = XT + Km 1.
(1.32)
The second condition is more restrictive and quasi-stationarity thus applies
if XT ET , as characteristic for most metabolic reactions, but typically
not valid for protein signaling networks where both substrate and enzyme
are typically proteins and XT ∼ ET . Alternatively, if XT ≤ ET we can have
ET , XT KM such that the enzymatic reaction is limited by the formation
of the complex relative to the processing rate (and C is therefore low, but
almost constant) and the reaction then proceeds at speed v vmax .
While the quasi-steady state approximation yields the very useful Michaelis-Menten
equation we may still want a full solution. It is not possible to get analytically a closed
form solution for Eq. 1.24, but singular perturbation methods can be used to obtain
approximate solutions.
1.4.2
Hill Kinetics - Cooperativity
Many 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 the
binding characteristics at all binding sites (Fig. 1.1B (b) 2nd column). The detailed
modeling of all interactions and transitions is tedious. It can be shown [?] that if the
k +k
first ligand binds with very low affinity (i.e. large K1 = −1k1 2 ), and all subsequent
ligands i = 2...n binds with an increasing affinity (i.e. smaller Ki ), then
d[P]
[X]n
= vmax n
.
dt
K + [X]n
(1.33)
Strictly speaking this formula is obtained in the limit K1 → ∞ and Kn → 0 while keeping
n
1
n
n
K1 Kn finite. K n[X]
+[X]n is referred to as Hill function with Hill constant K = (∏i=1 Ki )
and Hill coefficient n. If we plot the rate at which product is formed versus the substrate
concentration we obtain a sigmoid graph (Fig. 1.1B (b) RHS). The Hill constant K
corresponds 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
1.4. SIMPLIFYING APPROXIMATIONS
15
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 interactions
Inhibitors of a chemical reaction either fully prevent a reaction or reduce the reaction
rate. When the effect of an inhibitor is reversible, the steady state of the inhibited
species 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 paradigm
is the use of inhibitors and activators to modulate the speed of reactions. Inhibitors
can either compete with the substrate for the catalytic cleft (competitive inhibition) or
alternatively inhibitors can induce a conformational change that alters the activity of
the enzyme (allosteric inhibition).
Competitive Inhibition Inhibitors that bind to the active site of an enzyme and compete with substrate for access are termed competitive inhibitors (Fig.1.1B (c)). The
set of differential equations which describes the system is (with C2 referring to the CI
complex):
d[X]
dt
d[E]
dt
d[I]
dt
d[C1 ]
dt
d[C2 ]
dt
d[P]
dt
= k−1 [C1 ] − k1 [E][X]
(1.34)
= (k−1 + k2 )[C1 ] − k1 [E][X]
(1.35)
= −k3 [E][I] + k−3 [C2 ]
(1.36)
= −(k−1 + k2 )[C1 ] + k1 [E][X] + k−3 [C2 ] − k3 [E][I]
(1.37)
= −k−3 [C2 ] + k3 [E][I]
(1.38)
= k2 [C1 ]
(1.39)
R
d[C1 ]
d[C2 ]
We have d[E]
dt + dt + dt = 0 ⇒ [E] + [C1 ] + [C2 ] = ET . Using again a quasi steadystate approximation for C1 and C2 we have
C1
=
C2
=
ET [X]KI
[X]KI + KI Km + [I]Km
ET [I]Km
[X]KI + KI Km + [I]Km
(1.40)
(1.41)
(1.42)
where Km =
k−1 +k2
k1
d[P]
dt
and KI =
= k2 [C1 ] =
k−3
k3 .
The product is then produced according to
k2 ET [X]KI
[X]
= Vmax
[X]KI + KI Km + [I]Km
[X] + Km (1 + [I] )
KI
(1.43)
16
CHAPTER 1. BIOCHEMICAL REACTION MODELLING
A higher amount of substrate is therefore required to achieve a particular reaction rate:
the half-saturation constant increases from Km to Km (1 + K[I]I ), where KI is the dissociation constant for the enzyme-inhibitor interaction. Similarly, in case of Hill kinetics
competitive inhibition is modelled by an increase in the Hill constant K by a factor of
1 + K[I]I .
Allosteric Inhibition Allosteric inhibitors do not bind to the substrate binding site
but affect the reaction rate by binding to a different site where they may induce a conformational 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, reduce the maximal velocity of the reaction vmax (i.e. k20 k2 in Fig. 1.1B (c)), and we
have
[X]
vmax
.
(1.44)
v=
[I] K + [X]
m
1+
KI
1.4.4
Goldbeter-Koshland Kinetics
The biological activity of signaling proteins is often controlled by a reversible chemical
transformation, i.e. phosphorylation, methylation etc. If we were to model all steps
explicitly the models would again be complex (Fig. 1.1B (d)), and experimental data
may lack to parameterize the model. These enzymatic reactions are therefore often
approximated with Michaelis-Menten reactions, i.e.
k1
k
2
GGGGGG
B
X + E1 F
G C1 −→ E1 + X p
k−1
p1
p2
GGGGGGB
X p + E2 F
GG C2 −→ E2 + X.
p−1
We then have for the kinetics of the phosphorylated and unphosphorylated forms, X p
and X respectively,
XT − [X p ]
[X p ]
d[X p ]
d[X]
=−
= k phos S
− kdephos
dt
dt
KM1 + XT − [X p ]
KM2 + [X p ]
(1.45)
Here k phos and kdephos are the vmax of the enzymatic reactions. S refers to an external
signal that is assumed to only affect the kinase and thus the phosphorylation reaction.
In equilibrium
d[X p ] d[X]
=
=0
(1.46)
dt
dt
and we obtain the Goldbeter-Koshland formula
X p∗ =
[X p ]
2u1 J2
p
= G(u1 , u2 , J1 , J2 ) =
.
XT
B + B2 − 4(u2 − u1 )u1 J2
(1.47)
where u1 = k phos S,u2 = kdephos ,J1 = KXM1
,J2 = KXM2
, and B = u2 − u1 + J1 u2 + J2 u1 . XT
T
T
refers to the total concentration of the signal protein X, i.e XT = [X] + [X p ]. In the
1.4. SIMPLIFYING APPROXIMATIONS
17
context of larger regulatory networks with such regulatory motif (Fig. 1.1B (d)), the
Goldbeter-Koshland formula can be used to approximate the fraction of active enzyme
dependent on the input signal S as long as quasi-stationarity for the reaction that regulates the enzyme relative to the rest of the network is a reasonable assumption.
18
CHAPTER 1. BIOCHEMICAL REACTION MODELLING
Chapter 2
Model Development and
Analysis
In this chapter we will discuss tools to identify the long-term attractors of dynamic
systems. This will allow us to predict the time evolution of ODE systems and to
evaluate the sensitivity and robustness of dynamic systems to perturbations. We
will start by describing the development and non-dimensionalization of mathematical models based on biological network cartoons, and then introduce phase
plane, 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 what
details are essential to obtain a model given the goal(s) of the analysis and which
details only increase the problem size and thus render the analysis difficult without
contributing to the studied effects. Conceptually it is easiest to define the components
to be included in the model (i.e. the proteins, compartments, complexes etc) as well
as the dynamical options of each component, and to then translate these into a set of
ODEs based on the laws of mass action as discussed in the previous Chapter. However, even if we can generate such systems of ODEs efficiently, simulations may not be
sufficiently informative to grasp the regulatory complexity and possibilities in such a
network. Therefore simple, meaningful approximations to such networks are typically
studied first to better understand and predict the qualitative behaviour of such complex
networks. In developing simple models it is important to identify the key variables in
the problem. For most subsequent analysis it is ideal to restrict the problem to only two
or three variables. The regulatory interactions between the variables need then to be
represented by approximations similar to those discussed in the previous chapter. Here
we discussed Michaelis-Menten kinetics of the form
dP
X
= vmax
dt
X +K
19
(2.1)
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 E
whose total concentration ET determined the value of vmax . Moreover, we considered
Hill kinetics of the form
dP
Xn
= vmax n
dt
X + Kn
(2.2)
for the case when the substrate would bind a multimeric enzyme cooperatively. We
should note that a switch-like response becomes steeper the larger n, i.e. for larger n a
small change in X can shift the system from not producing P to producing P at maximal rate (and vice versa). While such behaviour typically facilitates the mathematical
description of biological processes we should note that for most biological signaling
interactions n ≤ 2, and further processes need to be taken into account to explain the
observed biological sensitivtiy. More generally speaking we can model an activating
impact of a component Y on X as
Yn
dX
.
=v n
dt
Y + Kn
(2.3)
Such process could represent the expression of X in response to changes in the ligand
or transcription factor Y . Other possible interactions are the Y -induced phosphorylation
of X and many others. An inhibitory impact of Y on X can be captured as
Kn
dX
=v n
.
dt
Y + Kn
(2.4)
If Y interferes with an activating action of some other protein Z on X by competing for
the active site we write
dX
Zn
nI =v
dt
Z n + K n 1 + KYI
(2.5)
or if Y reduces the overall speed of the catalysis independently of Z we write
Z n1
dX
K n2
=v n
.
n
dt
Z 1 + K1 1 Y n2 + K2n2
2.2
(2.6)
Non-dimensionalisation
Before we continue with the analysis of a greatly simplified model it is sensible to
further simplify the mathematical formulation. To that end, all variables and parameters
should be rescaled in a way that the new variables and parameters have no physical
dimensions and can therefore more easily be compared. Moreover, the total number of
parameters in the model should be reduced by combining parameter values. There is
no standard method to non-dimensionalize a model, and some consider the procedure
an art because a clever non-dimensionalisation can sometimes greatly facilitate the
subsequent analysis of the problem. There are some guidelines, however: 1) If there
is 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 this
parameter 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 as
this enables the use of perturbation methods.
2.3
Phase Plane Analysis
Phase planes are a graphical tool to understand how a dynamic system evolves in
time from some given initial conditions. Phase planes are typically constructed for
2-component dynamical systems because in this case the two variables can be represented on the two axes. We will illustrate this for the two-component network
k1
/X
k3
/
O
k2
/ Y.
k4
/
X and Y are produced and degraded based on mass action. X, which is produced at
constant rate, enhances the production of Y , which in turn enhances the degradation of
X. The corresponding system of equations is then
dx
= f (x, y) = k1 − k2 xy
dt
dy
= g(x, y) = k3 x − k4 y.
dt
(2.7)
(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 initial
conditions; 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 from
any set of initial conditions within the phase plane.
Panel B and C demonstrate the construction of the phase plane. The two axes of the
phase plane represent the state variables x and y. We first determine the set of points in
the phase plane on which either x, y or both do not change in time. This can be done
dy
by setting the time derivatives to zero, i.e. dx
dt = f (u, v) = 0 and dt = g(u, v) = 0, and
solving for x or y. We obtain
k1
k2 x
k3 x
y=
.
k4
y=
(2.9)
(2.10)
On these curves, the nullclines, at least one time derivative is zero. Since our kinetics
are rather simple we can determine the unique steady state algebraically as
s
s
k1 k4
k3 k1 k4
xss =
, yss =
.
(2.11)
k2 k3
k4 k2 k3
22
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
Figure 2.1: Constructing a phase plane. (A) Evolution of X and Y in time starting
from an initial point (X0 ,Y0 ), until they reach the steady state (Xss ,Yss ). (B) The nullclines of X (grey line) and of Y (black line) intersect at the steady state (Xss ,Yss ). The
nullclines 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 textures
for X. (C) Based on the sign of the time derivatives of X and Y the direction of the
phase vectors can be inferred. The phase vectors indicate the direction in which the
system develops from a given initial point. The black line indicates a sample trajectory
from (X0 ,Y0 ) to the steady state (Xss ,Yss ).
An algebraic calculation of the steady state values of x and y is typically tedious. It is
easier to determine the steady state graphically as the intersection of the two nullclines
as shown in Figure 2.4b. Typically the nullclines divide the phase plane into a side with
a positive time derivative and one with a negative time derivative (Fig. 2.1B). We can
calculate the sign of the derivative from Eq. 2.7 by checking the impact of a value of
the state variable that is larger or a smaller than the steady state value. If we start at
a point in the sub-plane with the negative time derivative then the state variable will
increase with time - and vice versa (Fig. 2.1C). We can be more precise and plot small
arrows into the phase plane that give the direction and speed with which the system
evolves (Fig. 2.4b). The vector field has the form
~θ = du/dt
(2.12)
dv/dt
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, the
intersection point of the two nullclines. These tangent vectors ~θ point in the direction
in which the system develops from a point (x∗ , y∗ ). The length of the tangent vector
indicates 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 component of the vector (one time derivative) is zero. If all vectors point to the steady state
then the steady state is globally stable, i.e. the system returns to the steady state after
each perturbation. If only the arrows in the vicinity of the steady state all point to the
steady state then the steady state is said to be locally stable. The phase plane with the
trajectories and phase vector field is called a phase portrait.
Apart from the compact representation, there is another important feature of the
phase plane analysis: although we have not solved the system we already have a rather
accurate graphical representation of its solution. This is very important for the understanding 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 stability of a steady state and to reveal oscillations, adaptation, or switches. In the following we discuss a more generally applicable method, the linear stability analysis, to
evaluate the (local) stability of steady states. The idea behind a linear stability analysis is to introduce a small perturbation at the steady state and to study whether the
perturbation grows or decays over time. In the first case the steady state is said to be
unstable while in the latter case the steady state is stable. Linearization of the system of
differential equations at the steady state greatly facilitates the analysis but means that
our 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 variables
in a vector, i.e.
~x =
u(t)
v(t)
and
d~x
=
dt
u̇(t)
v̇(t)
=
f (u, v)
g(u, v)
,
(2.14)
where the dots denote time derivatives. The functions f and g are in general non-linear
functions. 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 the
perturbation ~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 ) +
+
1 ∂ 2 f 1 ∂ f (u − us ) +
(u − us )2 + . . .
1! ∂ u ss
2! ∂ u2 ss
1 ∂ 2 f 1 ∂ f (v − vs ) +
(v − vs )2 + . . .
1! ∂ v ss
2! ∂ v2 ss
(2.15)
and likewise for g(u, v). We next linearise the system by ignoring all terms that are of
24
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
order two and higher, i.e.
∂ f ∂ f (u − us ) +
(v − vs )
∂ u ss
∂ v ss
∂g
∂g
g(u, v) ∼ g(us , vs ) + (u − us ) + (v − vs )
∂ u ss
∂ v ss
f (u, v) ∼
f (us , vs ) +
(2.16)
We emphasise that f (u, v) and g(u, v) are now only approximations of the original functions 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 linearised
system of equations, and we write
d~x d~xs
=
+ J(~x − ~xs )
dt
dt
→
d~w
= J~w.
dt
(2.17)
J is referred to as Jacobian and is the matrix of all first-order partial derivatives of the
vector-valued function at the steady state (us , vs ), i.e.
! df
df
fu fv
|
|
us,vs
us,vs
du
dv
J=
=
.
(2.18)
dg
dg
gu gv
du |us,vs
dv |us,vs
2.4.2
Solution to the linearised ODE
We seek to determine the solution ~w(t) of the linear set of ODEs
d~w
= J~w.
dt
(2.19)
This can be achieved with the help of some concepts from Linear Algebra, in particular 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 the
matrix J is applied to them, i.e. J~v = λ~v. The set of all eigenvectors of a matrix, each
paired with its corresponding eigenvalue, is called the eigensystem of that matrix. If an
n × n matrix has n linearly independent eigenvectors, then these eigenvectors constitute
an eigenbasis for this matrix, and the matrix can be diagnolized. Much as the axes of
the cartesian coordinate system the linear independent eigenvectors ~vi are orthogonal
to each other, i.e. the product of two eigenvectors is zero. They thus constitute an
alternative, 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 the
new basis vectors. In the cartesian coordinate system we write for our 2-component
vector ~w
wx (t)
1
0
~w(t) =
= wx (t)
+ wy (t)
.
(2.20)
wy (t)
0
1
2.4. LINEAR STABILITY ANALYSIS
25
Similarly, ~w can be written as a linear combination of the two eigenvectors, i.e.
~w(t) = ∑ γi (t)~vi .
(2.21)
dγi (t)
d~w
=∑
~vi = J~w = ∑ γi (t)J~vi = ∑ γi (t)λi~vi .
dt
dt
i
i
i
(2.22)
i
The set of ODEs then becomes
To solve for a particular γ j , we can multiply with the eigenvector v j , and obtain
∑
i
dγi (t)
~vi~v j = ∑ γi (t)λi~vi~v j .
dt
i
(2.23)
Since the basis vectors {~v1 , . . . ,~vn } are linearly independent (orthogonal), ~vi~v j = 0 unless i = j and ~vi~v j = 1 if i = j if the eigenvectors are normalized appropriately. We
then have
dγi (t)
= γi (t)λi .
dt
(2.24)
By separation of variables we then have
γi (t) = γi (0) exp (λit).
Using Eq. 2.21 the differential equation for ~w(t),
d~w
dt
(2.25)
= J~w (Eq. 2.19), can be solved as
~w(t) = ∑ γi (0)~vi exp (λit).
(2.26)
i
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 to
zero for large times and so does the perturbation ~w. We thus require that all real parts
of the eigenvalues are negative for the steady state to be linearly (locally) stable.
26
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
Diagonalize Matrices using Similar Matrices
We 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 linearly
independent eigenvectors {~v1 , . . . ,~vn } that form a complete basis for the
n ×n matrix J, we can construct an invertible matrix C = (~v1 , . . . ,~vn ), where
the ith column of C is the vector ~vi such that
J = C−1 ΛC
with



Λ=


λ1
..
.
..
.
0
0
λ2
..
.
0
···
..
.
..
.
···
(2.27)
0
..
.
..
.
λn



.


(2.28)
J and Λ are similar matrices.
We can then write
d~w(t)
dt
d~w(t)
C
dt
d~γi (t)
~vi
(~v1 , . . . ,~vn ) ∑
dt
i
= J~w(t) = C−1 ΛC~w(t)
= ΛC~w(t)
= Λ(~v1 , . . . ,~vn ) ∑ γi (t)~vi .
i
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
2.4.3
= λi γi (t)
Eigenvalues and Eigenvectors of the Jacobian
The eigenvalues λ and eigenvectors ~v follow from the relation J~v = λ~v which can be
rewritten as (J − λ I)~v = 0 (where I is the identity matrix). There is only the trivial
solution ~v = ~0 unless
det(J − λ I) = 0.
(2.29)
For the 2-component system that we have studied so far we thus have
! df
df
fu − λ
|us,vs − λ
|us,vs
du
dv
det(J − λ I) = det
=
dg
dg
gu
|
|
−
λ
us,vs
us,vs
du
dv
fv
gv − λ
= λ 2 − ( fu + gv )λ + fu gv − fv gu = λ 2 − tr(J)λ + det(J) = 0,
2.4. LINEAR STABILITY ANALYSIS
27
where tr(J) refers to the trace of J and det(J) to the determinant of J. This can be
solved for λi as
p
tr(J) ± tr(J)2 − 4 det(J)
λ1,2 =
.
(2.30)
2
For a system with n coupled ODEs Eq. 2.29 leads to the characteristic polynomial of
degree n,
n
P(λ ) = ∑ αi λ i = 0.
(2.31)
i
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) = ∑ βi~vi exp (λit) = ∑ βi~vi exp (ℜ(λi )t) exp (iℑ(λi )t).
i
(2.32)
i
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) +
i sin (ℑ(λi )t is one for all times t. The long-term behaviour of ~w(t) therefore depends
on exp (ℜ(λi )t).
28
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
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 does
the perturbation ~w. We thus require that the real parts of all eigenvalues are negative
for the steady state to be linearly (locally) stable. In case of a 2-component model we
require 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 eigenvalues
are positive (ℜ(λi ) > 0 ∀i) then the steady state is (locally) unstable. If some of the
real parts are positive and others are negative then we speak of a (locally) half-stable
steady 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 the
lower plane in Figure 2.2.
Center solutions and Spirals If at least one eigenvalue has a non-zero imaginary
part then, with λi = φi + iωi , the solution for ~w(t) (Eq. 2.26) can be written as
~w(t) =
∑ βi~vi exp ((φi + iωi )t)
i
=
(2.33)
∑ βi~vi exp (φit) exp (iωit) = ∑ βi~vi exp (φit)(cos(ωit) + i sin(ωit)).
i
i
The perturbation around the steady state thus oscillates in amplitude with time. The
period of the oscillations is thus determined by the imaginary part, ωi , of the eigenvalues. If ~w(t) is a periodic solution, so is c~w(t) for any constant c 6= 0. Hence ~w(t) is
surrounded by a one-parameter family of closed orbits. The amplitude of the oscillations is determined by the pre-factor βi~vi exp (φit). The βi s are determined by the initial
conditions, 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 βi s. Consequently, the amplitude of a linear(ized) oscillation is set entirely by its initial conditions; any slight disturbance to the amplitude will persist forever. If the real part, φi , of
the complex eigenvalue is negative then these oscillations will be dampened, while the
oscillations will grow with time if the real part of the eigenvalue is positive. Sustained
oscillations exist if eigenvalues have zero real and non-zero imaginary part.
2.4. LINEAR STABILITY ANALYSIS
29
Example: Linear 2-component Model
dx
dt
dy
dt
= −y + x = f (x, y)
= −2x − y = g(x, y)
with initial conditions x = 0, y = 1. The nullclines are given by
dx
dt
dy
dt
= 0⇒x=y
= 0 ⇒ y = −2x
and intersect in the unique steady state (xs , ys ) = (0, 0).
Perturbation around the steady state
We consider a perturbation around the steady state (xs , ys ) = (0, 0)
x − xs
~w =
y − ys
(2.35)
We can write
d~w
dt
= J~w
where
J=
1
−2
−1
−1
(2.36)
is the Jacobian.
Eigenvalues & Eigenvectors We next determine the eigenvalues, λ , and
eigenvectors, ~v, of the Jacobian J according to
J~v = λ~v.
(2.37)
q
1
2
tr(J) ± tr(J) − 4 det (J) .
λ1,2 =
2
(2.38)
This yields
with tr(J) = 0, det (J) = 1. Thus, λ1 = i, λ2 = −i, v~1 =
1
v~2 =
.
1+i
1
1−i
,
Type of Steady State For λi = φ + iω
~w(t) =
∑ βi~vi exp (φit)(cos(ωit) + i sin(ωit))
i
The real part of the eigenvalue is zero, φ = 0, as tr(J) =0. Accordingly,
there is a center solution at the steady state.
30
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
Example: Linear 2-component Model - continued
Period of Oscillations
The imaginary part of the eigenvalue, ω = 1, determines the period of the
oscillation as 2π.
Amplitude of Oscillations
The amplitude
from the initial conditions, x = 0, y = 1, and thus
follows
0
~w(t = 0) =
, given that xs = 0, ys = 0.
1
Inserting the initial conditions with t = 0 into the solution ~w(t) =
∑i βi~vi exp (λit) we obtain
0
1
1
~w(t = 0) =
= ∑ βi~vi = β1
+ β2
(2.39)
1
1−i
1+i
i
and thus β1 = 0.5i and β2 = −0.5i.
In summary, the full solution is given by
1
1
1
i
exp (it) −
exp (−it)
~w(t) =
1−i
1+i
2
0
1
=
cos (t) +
sin (t)
1
1
From this we see immediately that x oscillates between −1 and 1. The
amplitude is therefore 2.
Higher dimensional systems
For a higher dimensional system of equations it becomes increasingly difficult if not
impossible to determine all the roots of the polynomial. However, in order to determine
the local stability of the steady state we do not need the exact values of the roots but
only the sign of their real parts. There are two helpful techniques: Descartes’ Rule of
Signs and the Ruth-Hurwitz criterion.
Descartes’ Rule of Signs Consider the polynomial P(λ ) = ∑ni αi λ i = 0. Let N be
the number of sign changes in the sequence of coefficients {αn , αn−1 , . . . , α0 }, ignoring
any 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 and
again applying this rule, information is obtained about possible negative roots.
2.4. LINEAR STABILITY ANALYSIS
31
Example: Descartes’ Rule of Signs
Consider the characteristic polynomial
P(λ ) = (λ − 1)(λ − 2)(λ + 5) = λ 3 + 2λ 2 − 13λ + 10 = 0.
(2.40)
The polynomial has two sign changes. Accordingly, there are at most two
real positive roots. Now set ω = −λ ,
P(ω) = −ω 3 + 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) = λ 3 + λ 2 + λ + 1 = 0.
(2.42)
The polynomial has no sign changes. Accordingly, there are at most zero
real positive roots, i.e. no real positive roots, as is indeed the case. In the
context of a stability analysis, we would conclude that the steady state must
be stable. Now set ω = −λ ,
P(ω) = −ω 3 + ω 2 − ω + 1 = 0.
(2.43)
There are now three sign change and thus either three or one real positive
roots for ω, that is either three or one negative real roots for λ , as is indeed
the case (λ = −1). In combination, we can conclude that there are either
three negative real roots for λ (and thus a stable node), or one negative real
root and two complex roots with zero real part. The latter implies (stable)
oscillations. Descartes’ Rule of Signs does not allow us to distinguish between the latter two possibilities.
Ruth-Hurwitz criterion The Ruth-Hurwitz criterion is derived using complex variable methods and it provides necessary and sufficient conditions that the real parts of all
eigenvalues are negative. The real parts of all roots of the polynomial P(λ ) = ∑ni αn−i λ i
with an > 0 are negative as long as the following condition is met for all k = 1, ..., n




Dk = 



a1
1
0
0
·
0
a3
a2
a1
1
·
0
·
a4
a3
a2
·
·
·
·
·
·
·
·
· ·
· ·
· ·
· ·
· ·
· ak




 > 0.



(2.44)
32
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
Example: Ruth-Hurwitz criterion
Consider the characteristic polynomial
P(λ ) = (λ − 1)(λ − 2)(λ + 5) = λ 3 + 2λ 2 − 13λ + 10 = 0.
(2.45)
We thus have a0 = 1, a1 = 2, a2 = −13, a3 = 10. The first determinant
that we have to consider is D1 = a1 = 2 > 0, which meets the Ruth-Hurwitz
criterion. The second determinant that we need to consider is
a1 a3
D2 =
= a1 a2 − a3 = −26 − 10 < 0.
(2.46)
1 a2
This violates the Ruth-Hurwitz criterion and we can conclude that not all
real parts of the eigenvalues are negative. Indeed, there are two eigenvalues
with positive real part, λ1 = 1, λ2 = 2.
Now consider the case
P(λ ) = (λ − i)(λ + i)(λ + 1) = λ 3 + λ 2 + λ + 1 = 0.
(2.47)
Here we have a0 = 1, a1 = 1, a2 = 1, a3 = 1. The first determinant that
we have to consider is D1 = a1 = 1 > 0, which meets the Ruth-Hurwitz
criterion. The second determinant that we need to consider is
a1 a3
D2 =
= a1 a2 − a3 = 1 − 1 = 0.
(2.48)
1 a2
This violates the Ruth-Hurwitz criterion and we can conclude that at
least one eigenvalue has a real part that is not negative. Indeed, for two
eigenvalues the real part is zero. In combination with the result from
Descartes’ Rule of Signs, we can thus conclude that the polynomial must
have one negative real root and two complex roots with zero real part.
2.4. LINEAR STABILITY ANALYSIS
33
Figure 2.3: A cartoon pathway of TGF-β signaling. The ligand TGF-β reversibly
binds to the TGF-β receptor, which associates with its co-receptor and is then phosphorylated to become fully active (1). The active receptor induces phosphorylation of
R-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 and
the five species Smad, phosphorylated Smad, Co-Smad, homodimers and heterodimers
can shuttle from the cytoplasm to the nucleus and back (5). Nuclear Smad/Co-Smad
complexes act as transcription factors and trigger the transcription of I-Smad mRNA in
the nucleus (6). The I-Smad mRNA then shuttles to the cytoplasm, where it can be degraded or translated into I-Smad. I-Smad mediates a negative feedback by sequestering
the active receptor (7) and can be degraded.
34
2.5
2.5.1
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
A worked example: A model for TGF-β signaling
Model Development
Figure 2.3 shows a typical depiction of the TGF-β network. TGF-β is a soluble secreted protein, that signals by binding to the TGF-β receptor (1). The ligand-bound
receptor phosphoryates the regulatory Smad (R-Smad) (2). After dimerization (3),
phosphorylated R-Smads bind a Co-Smad (4) and enter the nucleus (5) where they
regulate a wide range of genes. One of the genes that is up-regulated encodes an inhibitory Smad (I-Smad) (6) that downregulates TGF-β signaling by interfering with
the receptor-dependent phosphorylation of the R-Smads (7). The network clearly has a
negative 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 graphically
summarized in Fig. 2.4a. Here we focus on the core signaling proteins, the regulatory
Smads (R-Smads, [R]) and the inhibitory Smad, [I]. All other regulatory factors are
included only indirectly. The advantage of such simplistic model is that it is amenable
to a range of mathematical techniques. However, there are also important limitations
that can lead to misleading conclusions regarding the regulatory dynamics of the full
network. These will be discussed below.
Our simplest model considers two time-dependent variables that describe the dynamics of the concentrations of phosphorylated R-Smad, [R], and for the concentration
of 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-Smad
production, and (3) the negative feedback of the I-Smad back on the R-Smad. Both
proteins are turned over in reactions (4) and (5). After having decided the topology
and the reactions of the network we need to define the kinetic laws for the gain and the
loss rates. We assume that the rate at which phosphorylated R-Smad is formed depends
linearly 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 considered time-scale, i.e. there is no expression of degradation of R-Smads or the two
processes perfectly balance, then we can write [R] + [Ru ] = RT . The gain rate could
then be formulated as ν+ ∝ (RT − [R]). If we further assume that the maximal rate of
R-Smad phosphorylation depends linearly on the signal strength, S, then we need to
extend our formulation of the gain rate and write
ν+ = k1 S(RT − [R]).
(2.49)
The I-Smad inhibits phosphorylation of the R-Smad by binding to the free receptor
as well as to the receptor-ligand complex. Since the I-Smad and the ligand bind the
receptor at different sites we assume allosteric cooperative inhibition of the ligandinduced 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+ =
1+
k1 S
p (RT − [R]).
[I]
KI
(2.50)
As regards the loss rate, we assume that the rate of R-Smad dephosphorylation depends
linearly 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 availability
of the phosphatase that is limiting. The concentration of the phosphatase can therefore be considered to be constant and can be lumped into the reaction constant. If the
phosphatase was limiting we would need to use a Michaelis-Menten kinetics or Hill
kinetics for the reaction as discussed in connection with the Goldbeter-Koshland kinetics. If both concentrations were not limiting then the reaction would proceed at a
constant rate over time. In summary, we have for the kinetics of the active R-Smad, R,
k1 S
d[R]
p (RT − [R]) − k2 [R].
= ν+ − ν− =
dt
1 + K[I]I
(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
q , with Hill constant KR and Hill factor q. k3 is the maximal rate at which the
[R]q +K
R
I-Smad can be produced when R-Smad is abundant ([R] KR ). We assume linear
decay of the I-Smad at rate k4 , i.e. we assume that the concentration of the protease
that degrades the I-Smad is not limiting. We then have
d[I]
[R]q
− k4 [I].
= k3 q
dt
[R] + KRq
2.5.2
(2.53)
Non-dimensionalisation
Before we continue with the analysis of this model it is sensible to simplify the mathematical formulation. We will rescale all variables and parameters in a way that the new
variables and parameters have no physical dimensions and can therefore more easily
be compared. Moreover, we will reduce the total number of parameters in the model
by combining parameter values. There is no standard method to non-dimensionalize a
model, and some consider the procedure an art because a clever non-dimensionalisation
can sometimes greatly facilitate the subsequent analysis of the problem. There are
some guidelines, however: 1) If there is a maximal value that a variable can attain it is
sensible to normalize with respect to this maximal value. 2) If a variable is linked to a
certain parameter inclusion of this parameter 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 as this enables the use of perturbation methods.
36
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
Keeping all this in mind we rewrite the model in dimensionless form by making the
following substitutions
[R]
⇔ [R] = R[RT ]
[RT ]
p
[I]
=
⇔ [I] = IKI
KI
τ
= k2t ⇔ t =
k2
=
R
I
τ
(2.54)
(2.55)
(2.56)
The non-dimensionalized model then reads
dR
dτ
dI
dτ
with σ =
2.5.3
k1
k2 S,
κ1 =
k3
k2 KI ,
1−R
− R = f (R, I)
1 +Ip
Rq
= κ1 q
− κ2 I = g(R, I)
R + Kq
= σ
κ2 =
k4
k2
and K =
(2.57)
(2.58)
KR
RT .
Equilibrium concentrations
The concentrations of the R-Smad and the I-Smad in the limit of long times, i.e. when
the system has attained its equilibrium, are obtained by setting the time derivatives to
zero, i.e.
dR
dt
dI
dt
σ
σ +1 +Ip
κ1 Rq
= 0 ⇒ I(R) =
κ2 Rq + K q
= 0 ⇒ R(I) =
R-nullcline
(2.59)
I-nullcline
(2.60)
The two functions R(I) and I(R) are referred to as nullclines. On the R-nullcline R does
not change, while on the I-nullcline I does not change with time. The R-nullcline can
thus be interpreted as the signal-response curve for the active R-Smad concentration as
a function of the inhibitor concentration. Similarly, the I-nullcline can be interpreted as
a dose-response curve for the inhibitor concentration as a function of the concentration
of active R-Smad. An algebraic calculation of the steady state values of R and I is
tedious. It is easier to determine the steady state graphically as the intersection of the
two nullclines (Fig. 2.4b).
2.5.4
Phase Plane Analysis
Fig. 2.4b is referred to as phase plane. We can use the phase plane to understand how
the dynamic system evolves in time. For this we plot the trajectories
R(t)
~x(t) =
,
(2.61)
I(t)
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
2.5. A WORKED EXAMPLE: A MODEL FOR TGF-β SIGNALING
37
Figure 2.4: Model Analysis The wiring diagram (a), the phase plane (b), and the timedependent evolution of activated R-Smad levels (R) in response to signal pulses (c) for
a 2-component model of TGF-beta signaling.
intersection point of the two nullclines. To see how the system develops from any
point (R∗ , I ∗ ) in the phase plane we plot small arrows that represent the vector field of
tangents
dR/dt
~t =
(2.62)
dI/dt
to the trajectory ~x(t). These tangent vectors~t point in the direction in which the system
develops from a point (R∗ , I ∗ ). The length of the tangent vector indicates the speed
with which the system will change. Accordingly the arrows are of zero length in the
steady state and they cross nullclines perpendicularly. If all vectors point to the steady
state then the steady state is globally stable, i.e. the system returns to the steady state
after each perturbation. If only the arrows in the vicinity of the steady state all point to
the steady state then the steady state is said to be locally stable. The phase plane with
the 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 trajectories converge in the steady state (Fig. 2.4b). The steady state is therefore stable. It
should be noted that despite this stability, an increase in the signaling strength leads
to an increase in the steady state concentration of the active R-Smad. The steady state
moves in the phase plane (the R-nullcline is being shifted to the right), but it remains
stable (Fig. 2.4c).
2.5.5
Linear Stability Analysis
For sufficiently simple systems a graphical analysis can be employed to judge the stability of a steady state and to reveal oscillations, adaptation, or switches. In the following, 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 analysis is to introduce a small perturbation at the steady state and to study whether the
perturbation grows or decays over time. In the first case, the steady state is said to be
unstable, while in the latter case the steady state is stable. Linearization of the system
of differential equations at the steady state greatly facilitates the analysis but means
that our results apply only locally, i.e. in the vicinity of the studied steady state.
38
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
2.5.6
Linearization around the steady state
Since we will deal with linear systems we start by combining our set of state variables
in a vector, i.e.
d~x
R(t)
Ṙ(t)
f (R, I)
~x =
=
=
and
,
(2.63)
˙
I(t)
I(t)
g(R, I)
dt
where the dots denote time derivatives The functions f and g are in general non-linear
functions. 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 the
perturbation ~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 ) +
+
1 ∂ 2 f 1 ∂ f (R − Rs ) +
(R − Rs )2 + . . .
1! ∂ R ss
2! ∂ R2 ss
1 ∂ f 1 ∂ 2 f (2.64)
(I − Is ) +
(I − Is )2 + . . .
1! ∂ I ss
2! ∂ I 2 ss
and likewise for g(R, I). We next linearise the system by ignoring all terms that are of
order two and higher, i.e.
∂ f ∂ f (R − Rs ) +
(I − Is )
∂ R ss
∂ I ss
∂g ∂g
g(R, I) ∼ g(Rs , Is ) +
(R − Rs ) + (I − Is )
∂ R ss
∂ I ss
f (R, I) ∼
f (Rs , 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 linearised
system of equations, and we write
d~x d~xs
=
+ J(~x − ~xs )
dt
dt
⇒
d~w
= J~w.
dt
(2.66)
J is referred to as Jacobian, and is the matrix of all first-order partial derivatives of the
vector-valued function at the steady state (Rs , Is ), i.e.
! df
df
fR fI
|
|
Rs,Is
Rs,Is
dR
dI
J=
=
.
(2.67)
dg
dg
gR gI
dR |Rs,Is
dI |Rs,Is
The simple dimensionless 2-component TGF-β model (Eq. 2.57, 2.58) reads
dR
dτ
dI
dτ
1−R
− R = f (R, I)
1 +Ip
Rq
= κ1 q
− κ2 I = g(R, I)
R + Kq
= σ
2.5. A WORKED EXAMPLE: A MODEL FOR TGF-β SIGNALING
with σ = kk12 S, κ1 =
state is given as
k3
k2 KI

J=
> 0, κ2 =
k4
k2
σ
− 1+I
p −1
κ
s
q−1
qRs K q
1 (Rq +K q )2
s
> 0 and K =
p−1
pIs
−σ (1+I
p 2
)
s
−κ2
KR
RT
39
> 0. The Jacobian at the steady

=
− −
+ −
.
(2.68)
The sign of the entries in the Jacobian reflect the type of interactions. Since both R
df
|Rs,Is < 0, dg
and I affect their own concentration negatively, dR
dI |Rs,Is < 0. Furthermore
R has a positive impact on I, and accordingly the entry J21 is positive, while I has a
negative impact of R, and accordingly J12 < 0. We require tr(J) < 0 and det(J) > 0
for the real parts of all eigenvalues to be negative (Eq. (2.30)). By inserting the entries
from the Jacobian in Eq. (2.68) we obtain for our system of interest
tr(J) =
det(J) =
σ
− 1 − κ2 < 0
1 + Isp
!
!
q
σ
pIsp−1
qRq−1
s K
> 0.
fR gI − fI gR = −
− 1 (−κ2 ) − −σ
κ1 q
1 + Isp
(1 + Isp )2
(Rs + K q )2
fR + gI = −
We have tr(J) < 0 and det(J) > 0 for all σ and the steady state is therefore stable for
all signal strengths σ ≥ 0. To obtain oscillations we would require tr(J)2 < 4 det(J).
If fR and gI balance to give tr(J) = fR + gI = 0 and det(J) > 0 then the oscillations
are sustained. Since all molecular species decay, a positive entry on the diagonal is
possible only if at least one component is autocatalysing its own production.
The dynamic possibilities are limited in the 2D plane. We will therefore now consider a 3-component model (Fig. 2.5) with three state variables, the phosphorylated
R-Smad (R), the I-Smad (I), and the I-Smad mRNA concentration (M). The R-Smad
now affects the production of the I-Smad mRNA, and the I-Smad protein is produced
from the mRNA. We use the same kinetics as before and further assume that the rate
of protein production depends linearly on the concentration of mRNA, [M], i.e. we
assume that the mRNA rather than the translation machinery is limiting. We then have
d[R]
dt
d[M]
dt
d[I]
dt
= k1 (S)(RT − [R])
= νmax ET
1
− d1 [R]
1 + ( KII ) p
[R]q
− d2 M
[R]q + KRq
= k5 [M] − d3 [I].
(2.69)
Nondimensionalisation As before we simplify the model and use dimensionless
variables and parameters. We write dτ = k5 dt, k5 δi = di , k5 σ = k1 , k5 µ = νmax E,
I
E = EKTI , K = RKTR , R = R[R]T , M = [M]
KI , I = KI . Time is thus scaled with respect to mRNA
stability, the I-Smad mRNA and protein concentrations are scaled relative to the affinity constant KI , and the R-Smad concentration is scaled relative to the total R-Smad
40
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
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 reads
1
dR
= σ (1 − R)
− δ1 R = F(R, M, I)
dτ
1 +Ip
q
dM
R
= µ q
− δ2 M = G(R, M, I)
dτ
R + Kq
dI
= M − δ3 I = H(R, M, I).
dτ
(2.70)
Phase Plane We start by plotting the 2-dimensional R − I phase plane. We notice that
the steady state values of M and I are linearly related as I = κκ34 M. We then have the
R-nullcline R(I) =
κ1
p
κ1 +κ2 (1+( KI ))
I
and the I-nullcline I(R) =
κ3
Rq
κ4 ν Rq +H q
which intersect
in the steady state. When we plot the phase vectors they appear to all point to the staedy
state yet the trajectories all converge on a limit cycle. This inconsistency is due to our
mapping of the phase plane to 2D. In reality the phase vectors point out of the R − I
plane and away from the steady state as can be seen in the 3 dimensional plot of the
phase plane (Fig. 2.5b). The phase vector field strongly depends on the parameters
used, and in the following we will determine the parameter sets for which we obtain
sustained oscillations (center solutions) with the help of a linear stability analysis as
described above.
Linear Stability Analysis Center solutions arise in linear(ized) systems if the eigenvalues have zero real part and non-zero imaginary part. Center solutions arise in 2component systems if tr(J) = 0 and tr(J)2 < 4 det(J). In a 3-component system we
have as characteristic polynomial
P(λ ) = λ 3 + α2 λ 2 + α1 λ + α0 = 0.
(2.71)
The coefficients are determined by the Jacobian of the dynamical system. To find
parameter sets with center solutions we substitute λ = iω (zero real part) and solve the
characteristic polynomial. In case of a 3-component network, this yields
P(iω) = −iω 3 − α2 ω 2 + iα1 ω + α0 = 0.
(2.72)
2.5. A WORKED EXAMPLE: A MODEL FOR TGF-β SIGNALING
41
By matching real and complex parts, i.e −iω 3 + iα1 ω = 0 and −α2 ω 2 + α0 = 0, we
obtain ω 2 = α1 = αα02 . The eigenvalues follow from ω 2 = α1 as
√
λ1,2 = ±i α1 ,
λ3 = −α2
(2.73)
Therefore we require α1 = αα02 for the existence of center solutions that would lead to
sustained oscillations. We can determine such parameter sets numerically that match
the condition α1 = αα02 .
Period and Amplitude of Center Oscillations The period of center oscillations is
determined by the eigenvalues as √2πα1 . Because of the third negative eigenvalue, the
solution is initially dampened until it reaches the center solution. To calculate the amplitude 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)
~x(t) = ~xs + ∑ βi~vi exp λit
(2.75)
The general solution
i
can be simplified to
√
√
~x(t) = ~xs + (β1 + β2 )(~u cos ( α1t) + ~φ sin ( α1t))
(2.76)
√
√
~
+ i(β1 − β2 )(φ cos ( α1t) +~u sin ( α1t)) + β3~v3 exp (−α2t), (2.77)
where ~xs represents the steady-state vector. Since α1 , α2 > 0 holds for all parameters
this system exhibits sustained oscillations. The βi (and thus the amplitude) depend on
the initial conditions, ~x(0) =~x0 , and can be determined from the linear equation in βi ,
~η = x~0 − ~xs = (β1 + β2 )~u + i(β1 − β2 )~φ + β3~v3 .
(2.78)
Once the βi s have been determined, we need to determine the times t at which the
variables R, M, I are maximal and minimal. The difference yields the amplitude of the
oscillations. Note that the relative amplitudes of R, M, I do not depend on the initial
conditions and thus can be determined without knowledge of the βi s.
Oscillations arise in the 3-component TGF-β model because of a slow intermediate
step, the slow(!) formation of I-Smad mRNA (M). The gene for the I-Smad indeed contains large introns so that the pre-mRNA is about 14kb long although the protein itself
is less than 1kb of length. Transcription is relatively fast, but splicing events introduce
the required delay as shown experimentally for HES oscillations [?, ?]. Oscillations
in the TGF-β response are therefore physiologically plausible, but would have been
missed if we had analysed only a 2-component model.
For appropriate parameter values we can obtain a center solution around the steady
state (Fig. 2.5b) and sustained oscillations emerge (Fig. 2.5c). As we vary parameter
values, we notice that the Linear Stability Analysis does not predict all oscillations that
we observe, and that the observed oscillations typically do not depend on the initial
conditions. This short-coming is due to the linearization of the equations. Non-linear
equations can give rise to Limit Cycle behaviour as discussed next.
42
2.6
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
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 trajectories are not closed - they spiral either towards or away from the limit cycle. If all
neighbouring trajectories approach the limit cycle, we say the limit cycle is stable or
attracting. 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 by
the structure of the system itself and not by the initial conditions. An important method
to establish that there exists such an orbit is presented by the Poincare-Bendixson Theorem as illustrated in Fig. 2.6.
Poincare-Bendixson Theorem
Suppose that
1. R is a closed bounded subset of the plane
x
2. d~x/dt = f (~x) with ~x =
is a continuously differentiable vector
y
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 it
starts 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 closed
bounded subset R of the phase plane (grey area). The fixed point (FP) is excluded.
2.6. LIMIT CYCLES
43
Negative criterion of Bendixson
If the trace of the Jacobian does not change sign within a region of the phase
plane, then there is no closed trajectory in this area. Hence a necessary
condition for the existence of a limit cycle is a change in the sign of the
trace of the Jacobian.
Example: The Glycolytic Oscillator The famous Schnakenberg kinetics [?] give
rise to limit cycle oscillations for suitable parameter values
ẋ
= a − x2 y
ẏ
= b − y + x2 y.
(2.79)
x and y represent normalized concentrations and a, b > 0 are normalized reaction constants. x is produced at a constant rate a, and y is produced at a constant rate b, and
decays linearly. Two molecules of x interact with one molecule of y to enhance production of y and removal of x. This model could also describe binding of a dimeric
ligand y to a monomeric receptor x. The complex would then induce production of x
and removal of the ligand y.
First we need to find the nullclines and construct a phase plane. The nullclines are
2
2
given
q by y = a/x
and y = b/(1 − x ) 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 can
exist only if the fixed point is not stable. To access how the stability of the steady states
depends 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 with
a stable fixed point. (B) The phase plane with an unstable fixed point and a limit cycle.
44
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
is given as
J=
−xs2
−1 + xs2
−2xs ys
2xs ys
.
(2.80)
s
p
a(a + b) > 0 and trace tr(J)= −2xs ys − 1 + xs2 =
with
determinant
det(J)
=
2x
y
=
2
s
s
p
a
−2 a(a + b) − 1 + (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 parameter
space with limit cycle solution from the one is a stable steady state solution. We finally
need to define a trapping region that excludes the fixed point and where all phase vectors point inwards. Such trapping region can indeed be constructed and a limit cycle
must thus exist when the nullclines cross such that the fixed point is not stable.
Liénard system
A Liénard equation is a second order differential equation of the form
ẍ + f (x)ẋ + g(x) = 0
(2.81)
where f and g are two continuously differentiable functions on R, with
f 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 system
of ordinary differential equations, the so-called Liénard system, of the form
x˙1
x2 − F(x1 )
= h(x1 , x2 ) :=
(2.82)
x˙2
−g(x1 )
where
Z x
f (ζ )dζ
F(x) :=
0
x1
:= x
x2
:= dx/dt + F(x).
A Liénard system has a unique and stable limit cycle surrounding the origin
if it satisfies the following additional properties:
• g(x) > 0 for all x > 0
• limx→∞ F(x) := limx→∞
Rx
0
f (ζ )dζ = ∞
• F(x) has exactly one positive root at some value p, where F(x) < 0
for 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
ẍ + µ(x2 − 1)ẋ + x = 0
(2.84)
into a Liénard system if we recognize that
ẍ + µ(x2 − 1)ẋ =
d
dt
1
ẋ + µ( x3 − x) .
3
(2.85)
So if we let
1
dx
F(x) = x3 − x
w=
+ µF(x)
3
dt
the van-der-Pol equation (Eq. 2.84) implies that
(2.86)
dw
= d 2 x/dt 2 + µ(x2 − 1)dx/dt = −x.
dt
(2.87)
Hence the van-der-Pol equation is equivalent to
dx
dt
dw
dt
= w − µF(x)
= −x.
One further change of variables further simplifies the analysis. If we let y =
have
dx
dt
dy
dt
=
(2.88)
w
µ
then we
µ(y − F(x))
1
= − x.
µ
(2.89)
This set of equations is similar to the Fitzhugh-Nagumo model (Eq. ??) that models
the generation of action potentials in nerve cells.
The period of the van-der-Pol oscillator So far we have mainly considered qualitative questions. We are now interested to determine the period of this oscillator. By
inspection of the phase plane we notice that there are two fast branches and two slow
branches (Fig. 2.8A). The slow branches are those that follow the nullclines while the
fast branches are those that are far away from the nullclines. The period T is essentially the time required to travel along the two slow branches, since the time spent in
the jumps is negligible for large µ. By symmetry the time spent on each branch is the
same. Note that on the slow branches y ∼ F(x) and thus
dy/dt = F 0 (x)dx/dt = (x2 − 1)dx/dt.
(2.90)
But since dy/dt = −x/µ we have dx/dt = − µ(x2x−1) . Therefore
dt ∼ µ
x2 − 1
dx
x
(2.91)
46
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
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
I
T=
dt ∼ 2
Z xB
xA
−µ
x2 − 1
dx = µ[3 − 2ln2].
x
(2.92)
where xB = 1 is the positive x for which y ∼ F(x) assumes its minimum. xA = 2 follows 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] + 2α µ −1/3 + . . . where α = 2.383 is the smallest root of Ai(−α) = 0
where Ai(x) is a special function called the Airy function.
Chaos
The dynamical possibilities in the 2D plane are very limited: if a trajectory
is confined to a closed, bounded region that contains no fixed points, then
the trajectory must eventually approach a closed orbit - nothing more complicated is possible! In higher dimensional systems trajectories may wander
around forever in a bounded region without settling down to a fixed point
or a closed orbit.Trajectories may also be attracted to a strange attractor, a
fractal set, on which the motion is aperiodic and sensitive to tiny changes in
the initial conditions, leading to chaotic behaviour.
2.7
Delay Equations
Delays can have important consequences as we have seen by example of the TGF-β
network where the delay enabled oscillations. In the TGF-β model this was achieved
by 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 introduce delay equations to incorporate an explicit delay into our differential equations.
Consider the differential equation
dx
π
= − x(t − T )
dt
2T
(2.93)
where x decays at a rate that is proportional to its value at an earlier time t − T . The
equation can be solved to give
π x(t) = A cos
t
(2.94)
2T
where A is a constant. This simple delay equation produces oscillations, though we
should note that x can assume both positive and negative values which would not make
sense in many biological settings. We can consider a slightly more sophisticated model
that is similar to a logarithmic growth model
dx
= x(t) (1 − x(t − T ))
dt
(2.95)
Here, x enhances its own production and decay, but the negative effect occurs with a
delay T and thus depends on the concentration of x at an earlier time t −T . Even though
the equation still appears simple solutions can only be found numerically. To compute
the solutions for t > 0 we now require as initial conditions x(t) for all −T ≤ t ≤ 0.
2.8
Bifurcation Analysis
Key to all regulatory control is how the stability of a steady state is effected by regulatory control. Bifurcation diagrams are employed to analyse how the values and the
stability of equilibrium points depend on a regulatory control parameter, the bifurcation
48
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
parameter. Those points at which the stability of an equilibrium point changes or new
steady state solutions appear or disappear are called bifurcation points. Particularly important bifurcation behaviours in biological models include transcritical bifurcations,
saddle-node bifurcations and Hopf bifurcations.
Generating Bifurcation Diagrams
To generate a bifurcation diagram the stability of the steady states needs
to be evaluated as the bifurcation parameter is changed. While the stable steady state branches can also be determined by solving the set of
equations numerically as the bifurcation parameter is changed this does
not allow the determination of the unstable steady state branches. To
determine also the unstable steady state branches a linear stability analysis has to be carried out. This can be laborious and several software
packages 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 whose
real part passes through zero. Both before and after the bifurcation, there is one unstable and one stable fixed point (Fig. 2.10a). However, their stability is exchanged when
they 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 saddlenode bifurcation (Fig. 2.10b). Saddle-node bifurcations generate switches that are
robust to fluctuations because they enable hysteresis, a term which is used to describe
systems which have memory. The bifurcation diagram in Fig. 2.10b contain a region
with three steady states (2 stable and 1 unstable). If the system is started on the lower
branch at low signaling strength (point 0) it will follow this branch as the signal strength
S 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. As
the two steady states are annihilated at the bifurcation point a further increase in the
signal strength S results in a jump to the remaining equilibrium (point 2). If the signal
strength 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 equilibrium
that the system attains at a given signal strength S therefore depends on the history
of the system, a phenomenon referred to as hysteresis. The new stable equilibrium
branch 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 stable
equilibrium branch.
2.8. BIFURCATION ANALYSIS
49
Figure 2.10: Bifurcation Behaviour. (a) Transcritical Bifurcation (b) Saddle-Node
Bifurcation: Toggle switch. (c) Negative Feedback Oscillator (left) Network motifs.
(center) Phase plane. (right) Bifurcation diagrams with bifurcation parameter S. Stable steady states are denoted by solid lines, unstable steady states by dotted lines. Hopf
bifurcations are denoted by H, saddle-node bifurcations by SN. For details please see
the main text.
2.8.3
Hopf bifurcation
At Hopf bifurcations oscillatory solutions alter their stability (Fig. 2.10c), i.e. the real
part of the conjugate complex eigenvalues changes its sign or visually: the two complex
conjugate eigenvalues simultaneously cross the imaginary axis. We can distinguish two
different types of Hopf bifurcations: supercritical and subcritical Hopf bifurcations. In
case of a supercritical Hopf bifurcation the amplitudes of the oscillations grow slowly
as the bifurcation parameter passes the bifurcation point while in case of a subcritical Hopf bifurcation the amplitudes of the oscillations are large immediately as the
bifurcation point is passed.
50
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS
Figure 2.11: Hopf Bifurcations. (A) Supercritical Hopf Bifurcation. (B) Subcritical
Hopf Bifurcation.
Supercritical Hopf Bifurcation
In terms of the flow in phase space, a supercritical Hopf bifurcation occurs when a
stable spiral changes into an unstable spiral surrounded by a small, nearly elliptical
limit cycle. Hopf bifurcations can occur in phase spaces of any dimension n ≥ 2. A
simple example is provided by the following set of equations
ṙ
= r(µ − r2 )
θ̇
= ω + br2 .
(2.96)
µ controls the stability of the fixed point at the origin, ω gives the frequency of infinitesimal oscillations, and b determines the dependence of frequency on amplitude
for larger amplitude oscillations. To analyse the bifurcation behaviour we rewrite the
system in Cartesian coordinates, x = r cos θ , y = r sin θ . Then
µx − ωy + cubic terms
ẋ
=
ẏ
= 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 left
to right as µ increases from the negative to positive values. Thus when µ < 0 the origin
r = 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 at
√
r = µ. Finally, the linear stability analysis wrongly predicts a center at the origin - in
fact for µ = 0 the origin is still a stable spiral!
2.8. BIFURCATION ANALYSIS
51
Subcritical Hopf Bifurcation
In case of a subcritical Hopf bifurcation the trajectories jump to a more distant attractor, which may be a fixed point, another limit cycle, infinity, or in three and higher
dimensions, a chaotic attractor.
A simple example is provided by the following set of equations
ṙ
= r(µ + r2 − r4 )
θ̇
= ω + br2 .
(2.98)
An analytical criterion exists to distinguish subcritical and supercritical Hopf bifurcations, but it is typically difficult to use. Numerical solutions can be used to a certain
extend though computer experiments are NOT proofs, and the code needs to be carefully checked before arriving at any firm conclusion. In case of a supercritical Hopf
bifurcation a small, attracting limit cycle should appears immediately after the fixed
point goes unstable and the amplitude should shrink back to zero as the parameter is
reversed.
Degenerate Bifurcation
A good example is provided by the dampened pendulum, i.e.
ẍ + µ ẋ + sin(x) = 0.
(2.99)
As µ is changed from positive to negative the fixed point at the origin changes from a
stable to an unstable spiral. However, at µ = 0 we do not have a true Hopf bifurcation
because there are no limit cycles on either side of the bifurcation. Instead at µ = 0 we
have a continuous band of closed orbits surrounding the origin.
The degenerate case typically arises when a nonconservative system suddenly becomes
conservative at the bifurcation point. Then the fixed point becomes a nonlinear center,
rather than a weak spiral required by a Hopf bifurcation.
52
CHAPTER 2. MODEL DEVELOPMENT AND ANALYSIS