Detailed balance steady states of binding networks

Chapter II: Polyhedral constraints enable holistic analysis of bioregulation

3.4 Detailed balance steady states of binding networks

Since binding reactions between molecules tend to happen at a faster time scale than the phenomena we are interested in, e.g. production and degradation of molecules, it is natural to assume that the binding reactions have reached a steady state. In fact, we can further assume that the steady states the binding reactions reach are at equilibrium, based on the intuition that nonequilibrium steady states require continuous energy input at the fast time scale of binding reactions, which is costly and therefore unlikely. In other words, although a mathematically specified CRN may not have equilibrium steady states, and may have steady states that are not at equilibrium, it is physically plausible to assume that a binding network on a fast time scale should have equilibrium steady states and practically stay in equilibrium steady states. Therefore, in this work we focus our study of equilibrium steady states in a way that does not concern whether they exist from mathematical specifications.

In this section, we describe the manifold of equilibrium steady states, which are also called detailed balance steady states in chemical reaction network theory.

Let𝑥∈R^𝑛_>0 denote a vector of species concentrations in a binding network. Then detailed balance steady states are ones that satisfy the condition that the forward (dissociation) and backward (association) fluxes of each reaction balances out. Let𝑘_𝑗 = ^𝑘

− 𝑗

𝑘⁺_𝑗 , the ratio between the forward (dissociation) and backward (association) reaction rates, be the equilibrium constant of the𝑗th binding reaction (in the dissociation direction, so also called dissociation constants). Assuming mass-action kinetics, then the balancing of forward and backward fluxes corresponds to𝑘_𝑗⁻𝑥^𝛼𝑓𝑗 =𝑘⁺_𝑗 𝑥^𝛽𝑓𝑗, where𝛼^𝑓_𝑗,𝛽^𝑓_𝑗 ∈ Z^𝑛 are the reactant and product vectors of the𝑗th forward (dissociation) reaction. This can be written as𝑥^𝛾𝑗𝑓 =𝑘_𝑗, where 𝛾_𝑗^𝑓 =𝛽_𝑗^𝑓−𝛼^𝑓_𝑗 ∈Z^𝑛is the stoichiometry vector of the𝑗th forward (dissociation) reaction. Let 𝑟denote the rank of the stoichiometry matrixΓ ∈R^𝑛×𝑚, which is also the rank ofΓ^𝑓 ∈R^𝑛×𝑚2, the stoichiometry of only the forward reactions. Then we can always select 𝑟 forward reactions with linearly independent stoichiometry vector to form the transpose-reduced stoichiometry matrix𝑁, whose rows are selected columns ofΓ^𝑓. Similarly, we take the equilibrium constants for the𝑟reactions selected to form vector𝑘∈R^𝑟_>0. Taking log and write in matrix form, we have that the detailed balanced condition becomes

𝑁log𝑥= log𝑘. (3.11)

Since there are𝑛variables and𝑟equations, we see the solutionlog𝑥of this equation for a fixed𝑘has𝑑=𝑛−𝑟degrees of freedom.

What variables could represent this𝑑degrees of freedom for the detailed balance solutions log𝑥? One natural choice are the conserved quantities. Let𝐿∈R^𝑑×𝑛be the conservation law matrix, then the conserved quantities are defined as𝑡=𝐿𝑥. The conserved quantities are denoted as𝑡for totals since in binding networks they often correspond physically to the total of some species in various forms. So we see the𝑑total concentrations 𝑡 ∈ R^𝑑_>0 form a natural representation of the remaining𝑑degrees of freedom. If we further require the binding network to be isomer-atomic, then alternatively we can represent the𝑑degrees of freedom as the concentrations of atomic species. We will see this explicitly as alternative coordinate charts for the manifold of detailed balance steady states.

Formally, we define the manifold for the detailed balanced solutions of a binding network.

We consider this manifold as the set of all possible values the variables of interest can take for a given binding network. The variables of interest from our above discussion are the species concentration 𝑥 ∈ R^𝑛_>0, total concentrations𝑡 ∈ R^𝑑_>0, and equilibrium constants

𝑘∈R^𝑟_>0. The manifold is then defined by the constraint on these variables imposed by the detailed balance condition of a particular binding network.

Definition 3.4.1. Given a binding network with reduced stoichiometry matrix𝑁 ∈R^𝑟×𝑛. Let𝑘∈R^𝑟_>0be the equilibrium constants for the𝑟reactions selected. Let𝐿∈R^𝑑×𝑛,𝑑=𝑛−𝑟 be the unique conservation law matrix of the network. Then the manifold of detailed balance steady statesof the binding network, also calledequilibrium manifoldof the binding network for short, is

ℳ=^{︁(𝑥,𝑡,𝑘)∈R^2𝑛_>0 :𝑡 =𝐿𝑥,𝑁log𝑥= log𝑘^}︁. (3.12) Note thatℳis𝑛-dimensional and immersed inR^2𝑛_>0. We denote a point in ℳas vector 𝑝∈R^2𝑛_>0. When convenient, we could also consider thelogmanifold immersed inR^2𝑛.

logℳ:=^{︁(log𝑥,log𝑡,log𝑘)∈R^2𝑛: (𝑥,𝑡,𝑘)∈ ℳ^}︁. (3.13) We caution that this approach of defining ℳconsiders a given binding network as a detailed-balanceconstrainton the values(𝑥,𝑡,𝑘)can take. The detailed balance condition takes higher priority than any specification of rates of a CRN. This is different from the typical approaches in CRN theory where the specification of network stoichiometry and rates take precedence, and the existence of detailed balance steady state is then determined later from the CRN specification. Because of our taking detailed balance as higher priority than CRN specifications, several CRNs can be equivalent to the same detailed balance behavior. In other words, if detailed balance is guaranteed, then specifying the full CRN may be redundant. For example, consider 3-state transition𝐶1 ⇌𝐶2 ⇌ 𝐶3 ⇌𝐶1. With detailed balance given, this network is equivalent to the same network with one reaction deleted. Because of this, any nonequilibrium steady states are also automatically excluded.

Coordinate charts of the equilibrium manifold

With the equilibrium manifoldℳdefined for a given binding networks, we would like to describe it and characterize its properties. The first thing to do with a smooth manifold is to specify coordinate charts on it, so that we have a parameterization of points on the manifold, and can talk about its tangent bundle for how to move around the manifold.

ℳis an𝑛-dimensional manifold. It is naturally immersed in2𝑛-dimensional space, where each point is specified as𝑝= (𝑥,𝑡,𝑘)∈R²𝑛_>0. But these variables are further related by𝑛 equations, namely𝑡 =𝐿𝑥and𝑁log𝑥= log𝑘. So we would like a parameterization ofℳ that maps its points toR^𝑛in a one-to-one or invertible way, so that the𝑛degrees of freedom

inℳare now explicitly some variable inR^𝑛. Such parameterizations are calledcoordinate charts, and a collection of charts that together cover the whole manifold is called anatlas.

Forℳ, we will see below that the natural choices of atlas we discuss have just one chart.

The most natural choice to begin with is of course the species concentrations log𝑥. By slight abuse of notation, this corresponds to the maplog𝑥 :ℳ → R^𝑛that takes a point specified as𝑝= (𝑥,𝑡,𝑘)∈ ℳand output the log of the first𝑛variableslog𝑥. This map is invertible because givenlog𝑥, we can find𝑡and𝑘 uniquely. Also, this chart is an atlas because every point in ℳcan be represented in this way. So instead of using a vector 𝑝 ∈ R^2𝑛_>0 to denote a point on ℳ, we could equivalently use 𝑥 ∈ R^𝑛_>0. We can use the diagram below to represent this mapping:

𝑝= (𝑥,𝑡,𝑘)∈ ℳ−−−−−−−−↽−−−−−−−−^log𝑥 ⇀

(𝑥,𝐿𝑥,𝑁log𝑥)

log𝑥∈R^𝑛. (3.14)

We also want alternative charts for different purposes. For example, if we consider changing the steady states of a given binding network by adjusting the concentrations. In this case, although𝑥varies, the equilibrium constants𝑘are not modified by such changes, therefore should remain constant. So to study changes in concentrations in this case, we would like𝑘to appear explicitly as variables in our parameterization. Another reason we may want this is to study what changes to the system would be caused by modifying the equilibrium constants𝑘, which is natural when asking questions about energy of molecules or temperature.

To obtain coordinate charts with𝑘as variables, we first note that𝑘∈R^𝑟_>0is of𝑟dimensions, so to parameterize the𝑛dimensions ofℳ, we need another𝑑variables. A simple choice that is still close to thelog𝑥chart is to take 𝑑of the𝑥variables. Assuming our binding network is isomer-atomic, then one natural choice for this is𝑥^𝑎∈R^𝑑_>0, concentrations for the𝑑atomic species. This yields chart(log𝑥^𝑎,log𝑘).

We can investigate properties of this chart by how it is mapped from thelog𝑥chart. To write the map, we re-order the species so that the first𝑑species are atomic. So𝑥= (𝑥^𝑎,𝑥^𝑐) is split into two parts, the𝑑atomic species𝑥^𝑎 ∈R^𝑑, and𝑟complex species𝑥^𝑐∈R^𝑟. Using detailed balanced condition Eq (3.11), we have

⎡

⎣

log𝑥^𝑎 log𝑘

⎤

⎦=

⎡

⎣

I_𝑑 0 𝑁₁ 𝑁₂

⎤

⎦log𝑥.

Invert this expression and use that𝐿^⊺₂ =−𝑁₂⁻¹𝑁1 yields an explicit alternative parameter-

ization of detailed balance steady states:

{log𝑥∈R^𝑛:𝑁log𝑥= log𝑘}

=

⎧

⎨

⎩

log𝑥=𝐿^⊺log𝑥^𝑎+

⎡

⎣

0 𝑁₂⁻¹

⎤

⎦log𝑘: log𝑥^𝑎 ∈R^𝑑,log𝑘∈R^𝑟

⎫

⎬

⎭

=

⎧

⎨

⎩

log𝑥=

⎡

⎣

I_𝑑 0 𝐿^⊺₂ 𝑁₂⁻¹

⎤

⎦

⎡

⎣

log𝑥^𝑎 log𝑘

⎤

⎦: log𝑥^𝑎∈R^𝑑,log𝑘∈R^𝑟

⎫

⎬

⎭

.

(3.15)

So we see that the map between chartlog𝑥and chart(log𝑥^𝑎,log𝑘)is linear, so we represent this map explicitly as matrices in the following diagram:

log𝑥

⎡

⎣

I_𝑑 0 𝑁₁ 𝑁₂

⎤

⎦

−−−−−−−−⇀

↽−−−−−−−−

⎡

⎣

I_𝑑 0 𝐿^⊺₂ 𝑁₂⁻¹

⎤

⎦

(log𝑥^𝑎,log𝑘). (3.16)

Since this map is invertible, we know(log𝑥^𝑎,log𝑘)is also a one-chart atlas forℳ.

Although chart(log𝑥^𝑎,𝑘)contains𝑘as explicit variables, there are still scenarios where we may want an alternative. For one, we need the isomer-atomic assumption to have the atomic species. This may not hold in general. More importantly, in many scenarios when the concentrations in a binding network is adjusted, it is not by adjusting the atomic species’

concentration, but by adjusting the conserved quantities or the totals (see Section3.1). This is especially often the case for time-scale separation for binding and catalysis reactions, where binding reaches equilibrium steady state while catalysis produces or degrades molecules to change concentrations. Here when a molecule is produced or degraded by catalysis, whether it is bound or not, or in which state among the state transitions, is not distinguishable from the slow time scale of catalysis. Once a molecule is produced or degraded in a particular form or state, the binding network quickly equilibrates and the net change of one molecule is then on the total, not any particular form or state.

From this reason, we would like to have the chart (log𝑡,log𝑘), where 𝑡 = 𝐿𝑥 is the totals or conserved quantities. To describe the map between this chart and chatlog𝑥is more involved, as can be seen from the mixing of linear map𝑡 =𝐿𝑥and log-linear map log𝑘=𝑁 log𝑥. We delve into this in the next section.