Biology · Systems & Quantitative Biology

Stochastic processes, biological noise and individual variation

An invented two-state process begins in an off state. After one step it has probability one quarter of being on. That number is neither a quarter of an individual nor a promise that every four observed individuals contain exactly one on state. Probability and a realised observation describe different things.

Research Design, Data & Scientific Practice introduces observations, comparisons and uncertainty. Gene Regulation & Cell Signalling develops biological feedback, while Population Ecology & Life-History Patterns develops rates and population boundaries. Their English chapters are available in this working manuscript; publication remains pending. Start with arithmetic, averages and ordinary graph reading. A derivative is introduced as an instantaneous rate of change. Exponentials and the circular sine–cosine example are explained where used; a first reading can follow their values and arrows, then return to their formulas with further mathematical preparation.

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By the end, you can

Distinguish a probability distribution from an individual trajectory and separate observed variation from claims about its biological sources.

Keep possible histories and their probabilities distinct

A stochastic process specifies probabilities for change. The original off–on model assumes that the probability of the next state depends only on the current state, with the same transition rule at each step. From off, the probabilities are three quarters to remain off and one quarter to become on. From on, they are one half for each next state. Each row sums to one.

Starting off, there are four possible two-step histories. Off–off–off has probability 0.75 × 0.75 = 0.5625. Off–off–on has probability 0.75 × 0.25 = 0.1875. Off–on–off and off–on–on each have probability 0.25 × 0.5 = 0.125. The histories are not equally likely simply because four are listed.

Two histories end on. Their probabilities add to 0.3125. One realised individual follows one history; the model assigns probabilities across possible histories before observing which occurs. Many independent runs would provide evidence about those probabilities, but a small realised count need not equal its expectation exactly.

Four histories have different probabilities

All histories retain their ordered states and unequal probabilities. Probabilities are model assignments, not observed fractions or partly on individual states.

Read the diagram in words

Starting off, the four two-step histories have probabilities 0.5625, 0.1875, 0.125 and 0.125, summing to one. Two end on, with combined probability 0.3125. The successive on probabilities are 0, 0.25, 0.3125 and 0.328125; their stationary value is one third.

Current state
Current state	Next state off	Next state on
Currently off	0.75	0.25
Currently on	0.5	0.5

Complete two-step history
Complete two-step history	Probability of the history	Next state on
Off, off, off	0.5625	×
Off, off, on	0.1875	✓
Off, on, off	0.125	×
Off, on, on	0.125	✓

Model probability of being on
Discrete step	Model probability of being on
0	0
1	0.25
2	0.3125
3	0.328125

q(t+1)=0.25+0.25q(t)

Stationary on probability, not an individual schedule: 0.3333333333333333

Original discrete-time off–on process with constant transition rows and initial state off. The current state determines the next-state probability within this assumed model. No actual gene, molecule, reporter, biological manipulation or measured transition rate is specified.

A distribution can change while an individual state stays discrete

Let qₜ be the model probability of being on at step t. An individual is either off or on. The probability qₜ is a continuous number between zero and one describing uncertainty or an ensemble, not a third intermediate individual state. The next-step probability is 0.25(1 - qₜ) + 0.5qₜ = 0.25 + 0.25qₜ.

Starting with q₀ = 0 gives q₁ = 0.25, q₂ = 0.3125 and q₃ = 0.328125. A stationary on probability solves q = 0.25 + 0.25q, giving q = 1/3. Stationary means that this probability distribution is unchanged by another application of the rule. Individual transitions can continue even under that stationary distribution.

The assumption that only the present state matters is a modelling choice. A process with age, previous exposure or an unobserved state might need a richer description. Nor does a discrete transition model supply a particular molecular mechanism. Its usefulness must be assessed against the biological question and observations it is intended to explain.

Paired records reveal a relationship, not its cause alone

Cells can vary even when they share an inherited background. Variation may involve stochastic molecular events, different cellular conditions, persistent individual differences and measurement error. What appears random also depends on what the model includes and what the observer knows. Calling variation noise does not mean that it has no biological consequences.

The original paired snapshots give four cells two assigned signals each. In one case the pairs are (1, 1), (1, 1), (3, 3), (3, 3); in another they are (1, 1), (1, 3), (3, 1), (3, 3). Each signal has the same four values, mean two and population variance one in both cases. Pairing changes the relationship between signals while leaving their separate distributions unchanged.

The population covariance averages the products of the two signals' deviations from their respective means. It is one in the first case and zero in the second. A common influence is compatible with joint variation, but covariance alone does not identify an actual intrinsic or extrinsic noise source. Such an interpretation needs justified observation and mechanistic assumptions; these invented snapshots provide no reporter design or biological diagnosis.

Pairing can change while separate distributions stay the same

Both snapshots use the same axis scales and retain four distinct cell identities, including records with coincident coordinates. Covariance summarises the assigned pairing; it does not identify a causal mechanism.

Read the diagram in words

Each signal has two values of one and two values of three in either snapshot. Both means are two and both population variances one. Common pairing gives covariance one; the alternative pairing gives zero. Cells with identical coordinates remain separate named records.

Together in this snapshot
Cell identity within each alternative snapshot	First assigned signal	Second assigned signal
c1	1	1
c2	1	1
c3	3	3
c4	3	3

Together in this snapshot
Population mean X	Population mean Y	Population variance X	Population variance Y	Population covariance
2	2	1	1	1

Different pairings in an alternative snapshot
Cell identity within each alternative snapshot	First assigned signal	Second assigned signal
c1	1	1
c2	1	3
c3	3	1
c4	3	3

Different pairings in an alternative snapshot
Population mean X	Population mean Y	Population variance X	Population variance Y	Population covariance
2	2	1	1	0

cov(X,Y)=sum((X-meanX)(Y-meanY))/4

Two original alternative snapshots of four equally weighted abstract cells, with arbitrary signal units. They are not sequential measurements or empirical reporter data. Population moments divide by four; no sampling estimate, molecular construction or causal source of variation is asserted.

Worked example

Add histories and compare pairings

At step two, the on probability is 0.1875 + 0.125 = 0.3125, obtained by adding the two histories that end on. The recursive rule gives the same result: 0.25 + 0.25 × 0.25 = 0.3125.

For the common pairing, each signal's deviations from mean two are -1, -1, +1, +1. Their paired products are 1, 1, 1, 1, with average covariance one. For the separate pairing, the products are 1, -1, -1, 1, with average zero.

All marginal values and both means remain the same. The comparison changes how values are paired across the four cells. It is a comparison of alternative snapshots, not a claim that the same cells changed over time or that a particular molecular source was identified.

Check your understanding

Why are the four two-step histories not assigned probability one quarter each?
Show reasoning
Their transition probabilities differ. A history's probability is the product of its specified successive transition probabilities; the number of listed histories alone does not make them equiprobable.
If the stationary on probability is one third, does one individual spend every third step on?
Show reasoning
No. The stationary distribution describes probabilities, while individual histories remain stochastic. It does not impose a fixed three-step schedule or exact realised proportions in a short record.
Why does zero covariance in the separate snapshot not prove that the signals are biologically independent?
Show reasoning
It describes this assigned relationship only. Zero covariance is not generally sufficient for independence, and causal interpretation would additionally require appropriate measurement and mechanistic evidence.

What to carry forward

Distinguish possible events, their probabilities and realised records, then ask what assumptions allow a variation pattern to support a biological explanation.

References

Resources supporting this lesson’s explanations and examples.