From stochastic to deterministic models of gene expression
2.1 From stochastic to deterministic descriptions of molec ular systems
The discrete state approach to genetic regulation has been reviewed by Kepler and Elston [40] illustrated by examples of simple activating factors. Details on the derivation and expansion of the Master equation are described in the books by van Kampen [83] and Gardiner [25]. The theory for molecular systems with finite states and infinite copy numbers has been studied in detail by Sbano and Kirkilionis in [69, 66, 70, 67, 68].
Molecular systems with finite states and infinite copy numbers
Modelling gene expression involves the description of the evolution of the transcrip- tion and translation products, which in turn depend on the activation state of the promoters of the genes in question. On a more abstract level and disregarding the biochemical details one can simplify these complex processes in terms of discrete states and stochastic transitions between them. More precisely, assume there is a fixed number of macromolecucles, for example DNA carrying genetic code or mem- brane channels, which can reside in a finite number of different states - a gene can be transcriptionally active or inactive, a membrane channel can be open or closed. Further there are smaller molecules that are characterized only by their number, and which are produced or destroyed depending on the state of the macromolecules. As reviewed in [66] the evolution of the species numbers can be described by a Master equation: ∂P(s,n, t) ∂t = X s,s0∈S L∗ss0(P(s0,n, t)) + 1 X s0∈S KssT0(n)P(s0,n, t), (2.1)
where sdenotes the state of the macro-molecules, s∈S is the corresponding finite
state space, and N is the number of species of the smaller molecules. Further
n= (n1, . . . , nN)∈NN is the vector that describes the number of particles for each of the N species, and P(s,n, t) is the probability that at time t the system has
ni particles or the i-th species, i = 1, . . . , N and is in state s. The matrix K is the infinitesimal generator of the Markov chain that contains the transition rates between the states in S. The small parameter >0 describes the time scale of the
Chapter 2. From stochastic to deterministic models of gene expression 42
evolution of the Markov chain that is defined byK. This parameter is motivated by
the notion that the Markov chain K evolves much faster than the production and
destruction events of the species. L∗ is a matrix of difference operators describing the birth-death process of the species. In our context the matrix L∗ will always be diagonal with diagonal elementsL∗s fors∈S. Hence equation (2.1) reduces to
∂P(s,n, t) ∂t =L ∗ s(P(s,n, t)) +1 X s0∈S KssT0(n)P(s0,n, t). (2.2)
The structure of the state space S is determined by the geometry of the states that the macro-molecules can reside in, for example conformational states for membrane channels, or binding states for genetic promoter regions. In particular, in the case where the states sare defined by the binding state of a number of binding sites on the macro-molecules, which the communicating molecules can bind to, the structure of S is typically a product space, that is formed by the Cartesian product of the state spaces corresponding to the individual binding sites.
The average dynamics
When the amount of particles is given in terms of concentrationsx∈RN+, then the system’s dynamics are described by a corresponding Focker-Planck equation (FPE). Also, the FPE can be regarded as an approximation of the master equation (2.1),
for when the number of particles is approximated by the concentration x = δn,
where δ is the inverse of the system size (volume or average particle numbers).
The probability distributionP(s,n, t) becomes a densityp(s,x, t) and the difference operators L∗s turn into differential operators ˆL∗s. The FPE is given by
∂p(s,x, t) ∂t = ˆL ∗ s(p(s,x, t)) +1ε X s0∈S KssT0(x)p(s0,x, t), (2.3)
where the parameterεis related toby the time and size scales as discussed in [66]. In chapter 3 the explicit from of the matrices L∗s and ˆL∗s are given for an example system (see tables 3.1 and 3.2).
From different perspective, the FPE (2.3) can be understood as describing a dynamical system on X = RN+ ×S with elements (x, s) ∈ X, whose dynamics are given by a hybrid set of laws. For each s ∈ S the evolution on RN+ is given by a
Chapter 2. From stochastic to deterministic models of gene expression 43
deterministic vector field,
dx(t)
dt =X
(s)(x(t)). (2.4)
The vector field is deterministic, because we neglect diffusion of the particles, which otherwise would yield a corresponding stochastic differential equation. Rather we assume that particles are homogeneously distributed in the system. So the Fokker- Planck equation (2.3) does not contain a diffusion term.
For each x∈RN+ the evolution on S is a finite Markov chain, whose probability distribution evolves according to the FPE
dP(t, s)
dt =
X
s0∈S
KssT0(x)P(t, s0). (2.5)
The vector field in (2.4) is related to the FPE (2.3) by the equation
ˆ L∗s(p(s,x, t)) =−∇ X(s)(x)p(s,x, t) . (2.6)
If we assume that the Markov chain on S defined by K reaches its equilibrium
probability distribution on a time scale that is shorter than the time scale of the evolution of the concentrations x ∈RN+, defined by (2.4), we can separate the two processes by an adiabatic approximation of (2.3). In fact, in [66, 69, 70] it is shown that the FPE (2.3) can be solved by looking for an asymptotical solution inε, with
p(s,x, t) =
∞
X
n=0
εnpn(s,x, t).
The leading order term of this expansion is the marginal distribution
f(x, t) =X
s∈S
p(s,x, t),
whose evolution is given by
∂f(x, t) ∂t = X s∈S ˆ L∗s(µs(x)f(x, t)). (2.7)
Chapter 2. From stochastic to deterministic models of gene expression 44
by
KT(x)µ(x) = 0, X s∈S
µs(x) = 1. (2.8)
For the systems studied here the Markov chain always evolves to a unique station- ary measure, although it is possible also to treat the case of non unique invariant measures [66, 69, 70].
Plugging the expression for ˆL∗s (2.6) into (2.7) yields
∂f(x, t) ∂t =− X s∈S ∇X(s)(x)µs(x)f(x, t) . (2.9)
The preceding equation is a Liouville equation for the deterministic dynamical sys- tem in Rn, whose trajectories are described by the system of differential equations
dx(t)
dt =
X
s∈S
X(s)(x)µs(x). (2.10)
For details of the correspondence of the FPE and ordinary and stochastic differential equations see the books of Gardiner and van Kampen [25, 83].
Decoupling of Markov chains
In the case of a gene, whose expression activity is governed by two types of regulatory factors binding to two distinct operator sites, the state space of the corresponding MC has the structure of a product of the two state spaces corresponding to the two operator sites. More generally let the matrix K generate the Markov chain on the state space S=S1×. . .×Sd. Then K can be written as a sum of tensor products
K =
d
X
l=1
I⊗(l−1)⊗Kl⊗I⊗(d−l),
whereKlgenerates a Markov chain onSl, andI⊗mis a shorthand forI|⊗I⊗{z. . .⊗I}
m times
. It can be shown that the unique stationary measureµof K can be written as
µ=µ1⊗. . .⊗µd,
whereµlKl = 0 for alll. The decoupling of Markov chains in this fashion is described in more detail in [68]. Definitions and properties for the tensor product can be found
Chapter 2. From stochastic to deterministic models of gene expression 45
in [48]. This theory will be applied to decompose the system of the engineered clock circuit described in the introductory chapter, which consists of two genes governed by multiple operator sites.
Summary: From stochastic to deterministic descriptions of molecular systems
In this section it was shown how gene regulation and expression can be described as a stochastic system of finite states and infinite copy numbers. From the stochastic description deterministic equations can be derived by taking appropriate limits - an adiabatic limit for fast operator state changes and a continuum limit to move from single particles to concentrations.
The deterministic rate laws derived this way have the advantage to be based on the underlying discrete description and can thus discriminate between different assumption of the dynamics at the molecular scale rather than having to rely on heuristics. From the analytical perspective deterministic equations are preferable to stochastic ones because more can be said about their dynamics.
The main task for setting up a model for a given genetic regulatory network is the generation of the Markov chain. In this section it was shown how Markov chains can be broken up into independent modules with the help of the tensor product. Yet the construction of the Markov chain can be tedious even for smaller systems, especially if it is done by hand. This motivated the fabrication of an algorithm that automatically generates the Markov chain for general models for genetic regulation of operons with any number of binding sites and regulatory DNA looping. This algorithm is presented in the next section.