Intracellular stochasticity
Biochemical reactions inside a cell are carried out by individual molecules. Molecules collide, bind, unbind, are produced and are degraded at unpredictable times. This creates intrinsic randomness in cellular dynamics.
Start with one molecular reaction
Consider
\[A\xrightarrow{k}B.\]If there are \(n_A\) molecules of \(A\), a simple stochastic reaction model assigns the propensity
\[\boxed{a(n_A)=k n_A.}\]The propensity has units of events per unit time. It measures how rapidly this reaction is expected to occur in the current state.
Probability during a short interval
For a sufficiently small time interval \(\Delta t\),
\[\boxed{P(\text{reaction in }[t,t+\Delta t))\approx a(n_A)\Delta t.}\]If there are more \(A\) molecules, there are more opportunities for the reaction, so its propensity is larger.
The state changes discretely
When the reaction occurs, the molecule numbers change by whole numbers:
\[(n_A,n_B)\longrightarrow(n_A-1,n_B+1).\]There is no such thing as 0.37 of a molecular reaction event in this description.
Random waiting time
If the total propensity is \(a_0\), the waiting time \(\tau\) until the next reaction is exponentially distributed:
\[\boxed{\tau\sim\operatorname{Exp}(a_0).}\]Therefore
\[P(\tau>t)=e^{-a_0t},\qquad E[\tau]=\frac1{a_0}.\]A larger total propensity means that the next reaction tends to occur sooner.
Several possible reactions
Suppose a system has reactions with propensities
\[a_1,a_2,\ldots,a_M.\]The total propensity is
\[\boxed{a_0=\sum_{j=1}^{M}a_j.}\]The waiting time to the next event uses \(a_0\). Conditional on an event occurring next, reaction \(j\) is selected with probability
\[\boxed{P(\text{next reaction}=j)=\frac{a_j}{a_0}.}\]Birth–death example
A minimal model of protein production and degradation is
If the current protein count is \(n\), the propensities are
\[a_{birth}=s,\qquad a_{death}=\delta n.\]A production event changes \(n\to n+1\), while a degradation event changes \(n\to n-1\).
The deterministic version
The corresponding deterministic mean-field equation is
\[\boxed{\frac{dP}{dt}=s-\delta P.}\]Its equilibrium is
\[P^*=\frac{s}{\delta}.\]This gives a smooth trajectory. The stochastic model instead gives a sequence of random integer-valued jumps.
One stochastic trajectory versus the deterministic curve
The graph below uses an exact event-driven simulation of the birth–death process. A fixed pseudo-random seed is used so that the example is reproducible whenever the page loads.
Why one trajectory is not the average
A single stochastic simulation answers: what is one possible history of this cell?
The deterministic solution usually answers a different question: what is the average or macroscopic tendency predicted by the model?
We should therefore not expect one stochastic trajectory to lie exactly on the deterministic curve.
Repeat the experiment many times
Starting identical cells from the same initial condition produces different stochastic trajectories because the reaction times and event choices differ.
The distribution across many simulated cells tells us about variability, not only the mean.
Mean and variance
For a random molecule count \(N(t)\), two basic summaries are
\[E[N(t)]\]and
\[\operatorname{Var}(N(t))=E[N(t)^2]-E[N(t)]^2.\]The mean describes the central tendency; the variance measures the spread between possible cellular outcomes.
Stationary distribution of the birth–death model
For constant production \(s\) and linear degradation \(\delta n\), the long-run molecule count has a Poisson distribution with parameter
\[\lambda=\frac{s}{\delta}.\]Thus
\[\boxed{P(N=n)=e^{-\lambda}\frac{\lambda^n}{n!}.}\]For a Poisson distribution,
\[E[N]=\lambda,\qquad \operatorname{Var}(N)=\lambda.\]The full distribution matters
Take \(s=4\) and \(\delta=0.5\). Then \(\lambda=8\). The deterministic equilibrium is therefore 8 molecules, but the stochastic equilibrium is not a fixed value of 8. It is a distribution of possible integer counts centred around 8.
Why randomness matters more at small copy number
For a Poisson variable, the standard deviation is
\[\sigma=\sqrt{\lambda}.\]The relative size of the fluctuations is therefore
\[\boxed{\frac{\sigma}{E[N]}=\frac1{\sqrt{\lambda}}.}\]At mean 4, the relative standard deviation is \(1/2\). At mean 100, it is only \(1/10\). Thus fluctuations are relatively much more important when molecule numbers are small.
Intrinsic noise
Intrinsic noise comes from the randomness of the biochemical events themselves. Even two cells with identical parameters and environments can experience different reaction histories.
Extrinsic noise
Extrinsic noise comes from differences in the wider cellular environment: cell size, ribosome abundance, metabolic state, upstream regulators or other slowly varying factors.
Observed cell-to-cell variability can contain both intrinsic and extrinsic contributions.
Gene-expression bursts
Gene expression can be especially noisy because promoters may switch randomly between inactive and active states:
\[G_{off}\;\underset{k_{off}}{\overset{k_{on}}{\rightleftharpoons}}\;G_{on}.\]When the promoter is active, several mRNA molecules may be produced before it switches off again. This can create bursts of transcription and, subsequently, bursts of protein production.
A stochastic gene-expression model
A simple reaction network is
\[G_{off}\rightleftharpoons G_{on},\]\[G_{on}\xrightarrow{s_m}G_{on}+mRNA,\]\[mRNA\xrightarrow{\delta_m}\varnothing,\]\[mRNA\xrightarrow{k_p}mRNA+P,\]\[P\xrightarrow{\delta_p}\varnothing.\]Each reaction has its own propensity. Together they produce random promoter switching, transcription, translation and degradation.
Noise can change biological outcomes
If a regulatory system contains a threshold, fluctuations can push an individual cell across it even when the average state remains below the threshold.
Noise can therefore alter switching times, gene-expression states, cell-cycle transitions and other cellular decisions.
Noise and bistability
In a deterministic bistable system, two stable states are separated by a threshold. In a stochastic system, sufficiently large fluctuations can occasionally move the state across that threshold.
The result can be random switching between phenotypic states.
The chemical master equation
Instead of following one trajectory, we can describe the probability of every possible molecular state.
Let \(P(n,t)\) denote the probability that the system has state \(n\) at time \(t\). The chemical master equation describes how these probabilities change because probability flows into and out of each state.
For the birth–death process,
\[\frac{dP_n}{dt}=sP_{n-1}+\delta(n+1)P_{n+1}-(s+\delta n)P_n.\]The first two terms are probability entering state \(n\); the final terms represent probability leaving it.
Gillespie's stochastic simulation algorithm
The Gillespie algorithm generates exact trajectories of a well-mixed continuous-time Markov reaction model under its assumptions.
At each step it calculates all propensities, sums them to obtain \(a_0\), samples the waiting time to the next reaction, chooses which reaction occurs according to \(a_j/a_0\), updates the molecular counts, and repeats.
Where the random numbers enter
If \(U_1\) is uniform on \((0,1)\), the waiting time can be generated by
\[\boxed{\tau=-\frac{\ln U_1}{a_0}.}\]A second independent uniform random number can select the reaction according to the cumulative propensity fractions.
The random numbers do not replace the biological rates. The rates determine the probability law from which the random event is generated.
CTMC interpretation
The stochastic reaction network is a continuous-time Markov chain whose state is the vector of molecular counts.
The current state determines the propensities, and those propensities determine the random waiting time and next state transition.
From CTMC to SDE
When molecule numbers are sufficiently large, many reaction events may occur over short intervals. Under suitable approximations, a discrete reaction model can be approximated by a stochastic differential equation.
The SDE retains random fluctuations but represents state variables continuously rather than as individual molecular counts.
When is an ODE enough?
A deterministic ODE can be appropriate when molecule numbers are large, fluctuations are small relative to the mean, and the biological question concerns average behaviour.
A stochastic model becomes particularly useful when copy numbers are small, rare events matter, thresholds or switches are present, or cell-to-cell variability is itself important.
A practical modelling workflow
List the molecular species and reactions. Specify how each reaction changes molecule counts. Write a propensity for each reaction. Check units and conservation laws. Simulate individual trajectories when needed. Repeat simulations to estimate distributions, means, variances and event probabilities. Compare stochastic predictions with deterministic behaviour. Use the chemical master equation, moment methods or approximations when the full probability distribution is required.