Biochemical reaction networks
A biochemical reaction network is a collection of molecular reactions that interact because the product of one reaction can become the reactant of another. The mathematics tells us how all molecular concentrations change together through time.
Start with one reaction
\[A\xrightarrow{k}B.\]One molecule of \(A\) is converted into one molecule of \(B\). Under mass-action kinetics, the reaction rate is
\[v=k[A].\]Every occurrence removes \(A\) and creates \(B\), so
\[\frac{d[A]}{dt}=-k[A],\qquad \frac{d[B]}{dt}=k[A].\]The same reaction rate appears with a minus sign for the consumed species and a plus sign for the produced species.
Why mass action depends on reactants
For a reaction
\[A+B\xrightarrow{k}C,\]an \(A\) molecule must encounter a \(B\) molecule. Under the mass-action assumption, the rate is
\[\boxed{v=k[A][B].}\]If either reactant concentration is zero, the reaction cannot occur. Increasing either concentration increases the encounter rate.
Stoichiometry controls how much changes
Consider
\[2A+B\xrightarrow{k}C.\]Each reaction event consumes two units of \(A\), one of \(B\), and produces one of \(C\). If its rate is \(v\), its contributions are
\[\frac{d[A]}{dt}=-2v,\qquad \frac{d[B]}{dt}=-v,\qquad \frac{d[C]}{dt}=v.\]The coefficients in the reaction therefore become coefficients in the differential equations.
Now connect several reactions
Take the network
There are three reaction rates:
\[v_1=k_1[A],\qquad v_2=k_2[B],\qquad v_3=k_3[B].\]Now follow each species.
Species \(A\) is consumed only by reaction 1:
\[\frac{d[A]}{dt}=-v_1=-k_1[A].\]Species \(B\) is produced by reaction 1 and consumed by reactions 2 and 3:
\[\boxed{\frac{d[B]}{dt}=k_1[A]-k_2[B]-k_3[B].}\]Species \(C\) is produced by reaction 2:
\[\frac{d[C]}{dt}=k_2[B].\]Species \(D\) is produced by reaction 3:
\[\frac{d[D]}{dt}=k_3[B].\]The important bookkeeping rule
| If a reaction... | Contribution to the species equation |
|---|---|
| produces the species | add its rate |
| consumes the species | subtract its rate |
| uses or produces multiple copies | multiply by the stoichiometric coefficient |
| does not involve the species | no contribution |
A model-generated network trajectory
For the network above, take \(k_1=0.7\), \(k_2=0.35\), \(k_3=0.15\), with \([A](0)=1\) and all other concentrations initially zero.
Initially \(A\) is abundant, so it falls rapidly. \(B\) first rises because it is being supplied from \(A\). Later it falls because its input weakens while conversion to \(C\) and \(D\) continues. The products accumulate.
Why an intermediate can rise and then fall
For \(B\),
\[\frac{d[B]}{dt}=\underbrace{k_1[A]}_{\text{input}}-\underbrace{(k_2+k_3)[B]}_{\text{output}}.\]At first input exceeds output, so \(B\) rises. At its maximum,
\[\frac{d[B]}{dt}=0,\]so input and output are momentarily equal. Afterwards output exceeds input and \(B\) falls.
This simple balance idea is extremely useful when interpreting larger biochemical networks.
Conservation laws
In this example, material is only redistributed among \(A,B,C,D\). Adding the four equations gives
\[\frac{d}{dt}([A]+[B]+[C]+[D])=0.\]Therefore
\[\boxed{[A]+[B]+[C]+[D]=\text{constant}.}\]Such conservation laws can reduce the effective dimension of a model and provide useful checks on numerical simulations.
Reversible reactions
A reversible reaction
\[A\;\underset{k_-}{\overset{k_+}{\rightleftharpoons}}\;B\]is treated as two reactions:
\[v_+=k_+[A],\qquad v_-=k_-[B].\]Hence
\[\frac{d[A]}{dt}=-v_++v_-,\qquad \frac{d[B]}{dt}=v_+-v_-.\]At equilibrium the forward and reverse fluxes may balance even though molecular reactions continue microscopically.
Reaction flux
A reaction rate is often called a flux. In a network, flux describes how quickly material moves through each reaction pathway.
Two networks can have similar concentrations but different internal fluxes, so concentrations and reaction rates answer different biological questions.
Stoichiometric matrix
For a larger network, bookkeeping can be organised into a matrix. Let \(x\) be the vector of species concentrations, \(v(x)\) the vector of reaction rates, and \(N\) the stoichiometric matrix. Then
\[\boxed{\frac{dx}{dt}=Nv(x).}\]Each column of \(N\) corresponds to a reaction. Negative entries represent consumption and positive entries represent production.
Example stoichiometric matrix
For
\[A\xrightarrow{v_1}B,\qquad B\xrightarrow{v_2}C,\qquad B\xrightarrow{v_3}D,\]with species ordered \((A,B,C,D)\),
\[N=\begin{pmatrix}-1&0&0\\1&-1&-1\\0&1&0\\0&0&1\end{pmatrix},\qquad v=\begin{pmatrix}k_1[A]\\k_2[B]\\k_3[B]\end{pmatrix}.\]Multiplying \(Nv\) reproduces all four ODEs at once.
Why matrix notation matters
Writing every equation manually is manageable for three reactions but difficult for hundreds. The matrix form separates two ingredients:
\[\text{network structure }N\quad+\quad\text{kinetic laws }v(x).\]This makes large biochemical systems easier to construct, analyse and simulate computationally.
Not every reaction follows simple mass action
Elementary molecular reactions are naturally represented by mass-action kinetics, but reduced biochemical models may use Michaelis–Menten, Hill, inhibition or other effective rate laws.
For example, an enzyme-mediated conversion may be represented as
\[v=\frac{V_{\max}[S]}{K_m+[S]}.\]The same stoichiometric bookkeeping still applies; only the formula for the reaction rate changes.
Feedback
A downstream molecule may affect an earlier reaction. For example, if product \(P\) inhibits an upstream enzyme, the corresponding rate may decrease as \([P]\) increases.
Feedback can stabilise concentrations, create switches, generate oscillations or produce more complex dynamics.
Steady states
A steady state satisfies
\[\frac{dx}{dt}=0,\]or equivalently
\[Nv(x^*)=0.\]This does not necessarily mean every reaction has stopped. Non-zero reaction fluxes can balance so that concentrations remain constant.
Deterministic versus stochastic reaction networks
ODE models treat concentrations as continuous quantities and describe average macroscopic behaviour. Inside a cell, however, some molecules may occur in very small copy numbers.
Then individual reaction events are random. A stochastic reaction network represents discrete molecular counts and probabilistic reaction events, often using continuous-time Markov chains or the Gillespie stochastic simulation algorithm.
Propensity in a stochastic network
For a stochastic reaction such as
\[A\xrightarrow{k}B,\]the propensity is related to the number of available \(A\) molecules. It determines the probability that the reaction occurs during a very short time interval.
This connects biochemical reaction networks directly to stochastic-process modelling.
What can reaction-network models study?
They can represent metabolic pathways, signalling cascades, gene regulation, enzyme systems, phosphorylation cycles, protein production and degradation, cell-cycle control and many other intracellular processes.
A practical modelling workflow
List the molecular species. Write every biologically relevant reaction. Assign a kinetic law to each reaction. Build the stoichiometric changes. Combine them into ODEs or the matrix equation \(dx/dt=Nv(x)\). Check conservation laws and units. Estimate parameters. Then analyse trajectories, steady states, stability, sensitivity and, when molecule numbers are small, stochastic effects.