Biochemical reaction networks

Biochemical reaction networks describe the interaction of chemical species in a biological setting, for example different proteins or ions in a cell. The ingredients for the modelling are the molecular species and their interactions. Furthermore, if an external stimulus such as light stimulation plays a role, we need to define it and its interaction with the species of the model. For further reading on the basics explaining in this section, please refer to (Ingalls,2012).

We describe the time-dependent concentration of each molecular species as a vari-able and their interaction by reaction equations, which can be formulated as dif-ferential equations. This is best understood using an example, so let us consider the following reactions:

→ A (2.1) A+B → C

C → D+E

D ↔ F

E →

E → B.

The reaction network described by these reactions is shown in figure2.1. It consists of six species labelled A to F, which can interact with each other. The possible reactions are characterized by their interaction partners, reactants which react to produce products, as well as the reaction rate constants k0 to k5.

k0 A

B

k1 C k2

D

E

k3f k3b F

k4 k5

Figure 2.1: An exemplary chemical reaction network. The letters A, B, C, D, E and F stand for the interacting species, while k1 to k5 are the reaction rate constants.

Let us investigate some of the possible reactions in this scheme. The reaction characterized by the rate k0 produces A, seemingly out of thin air. This is the case when a species of the model is produced by some process whose exact details are outside the scope of the model. The result will be an influx of species A.

Next, let us look at the reaction characterized by k1. It consumes A and B in equal parts and produces C. The reaction rate constant is k1, but the reaction rate v_f itself is proportional to the amounts of A and B, since they have to be available for the reaction to occur:

v_f =k1·A·B. (2.2)

This is called the law of mass action: the reaction rate is proportional to the re-action rate constant and the amount of the reactant species.

Species C can then go on to produce D and E with the rate constant k2. The rate of this reaction is proportional to the concentration of C. D can react to F with the rate constantk3f, and F can react back to Dwith the rate constantk3b.

This is a reversible reaction: it can occur in both directions.

E can react to nothing with the rate constant k4. For the purpose of our model, it is destroyed. More realistically, it would react to some reactant that is not in-cluded in the model, and can therefore be ignored.

Finally, the reaction with the rate constantk5 convertsE toB. This introduces a feedback into our model. Without this reaction, a clear up- and downstream could be defined: The species go from A and B through C to D, E and F. Now, we have introduced a cycle in our model.

Without the reactions characterized byk0 and k4, our model would be considered closed: there would be no reactants or products outside the network. The conse-quence of this is that we could define conservation laws for the species involved.

The steady state of a closed reaction network is thethermal equilibrium, where all reaction rates have gone to zero.

However, since we have included these reactions, the reaction network is open. In this type of reaction networks, we can also achieve a steady state, the dynamic equilibrium: in this state, all reaction rates are constant but not necessarily zero, resulting in a steady state for the species and a constant flux through the network..

In this reaction network, there is no reaction that requires more than two reactants.

This is typically the case, and it is due to the fact that the reactant species need to meet at the same time and same place and at the correct spatial configuration in order for the reaction to occur. If three or more reactants were involved in a re-action, this would be prohibitively improbable. If a reaction requires three species to occur, typically two of the species would form a precomplex, which would then react with the third species.

We can also write down the differential equations describing the above chemical reaction network. We will use the assumption of mass-action kinetics: that the reaction rate is proportional to the reaction rate constant and the amount of the reactant species. Furthermore, we will assume that the reaction volume is well-stirred, meaning that there is no spatial dependency of the species’ concentrations and that changes in the concentrations are instantaneous. The differential

equa-tions for the time-dependent species’ concentraequa-tions then read:

d

dtA = −k1·A·B +k0 (2.3)

d

dtB = −k1·A·B +k5·E d

dtC = k1·A·B −k2·C d

dtD = k2·C−k3f·D+k3b·F d

dtE = −k4·E−k5·E d

dtF = k3f ·D−k3b·F.

The equations directly follow from the reaction equations in equation (2.1) com-bined with mass-action kinetics, as we can see e.g. in the example of the third reaction in equation (2.3): The change of the concentration C, _dt^dC, is character-ized by an influx and an outflux. The influx comes from the reaction A+B →C and is proportional to the reaction rate k1 as well as the reactant concentrations A andB. The outflux comes from the reactionC →D+E. It is negative because it consumes C, and it is proportional to the reaction rate k2 as well as the reac-tant concentration C. Please note that, in mass-action kinetics, the concentration of the product itself does not influence the reaction rate, unless the product is a reactant at the same time.

The other differential equations are constructed in the same manner. We now have asystem of differential equations, which arecoupled: the change of one species de-pends on the other species. These equations contain all the information we have about the model and are equivalent to figure 2.1.

If we are interested in determining how the system would evolve in a given situa-tion, we need to further specify the initial conditions for all species,A(0) to F(0).

When we then solve the differential equations with the given initial conditions , we arrive at the time seriesA(t) toF(t) for all the species. In principle, we now know everything there is to know about the model with the given initial conditions. We can input any desired time point into the time series A(t) to F(t) and determine the individual species concentrations in that moment, or look at the entire time series. In subsection 2.1.3, we will discuss how to solve the differential equations.