How molecular histories became biological evidence

From chemical event clocks to engineered oscillators, single-molecule expression, noise-control limits, and the growth of a cell that remembers.

CCBS · EST 5053 · Fall 2026 · lecture 5 · lecturer-written lineage · core · exposition

Lecture 4 lets us ask whether a reaction system restores a disturbed state. Lecture 5 asks a different question: which molecular history does one cell follow while restoration is continually interrupted by events? The research history makes this change of question concrete. It runs from mathematical representations to experiments that could distinguish them, then to interventions that changed the circuit or the cell.

The engineered toggle and repressilator were not the beginning of stochastic gene-expression theory. They were unusually clear constructions in which dynamical behavior became a design objective. Later measurements revealed individual transcription episodes, separated shared from reporter-specific fluctuations, and followed oscillators through many generations. Each measurement exposed an object that a previous average had concealed.

The exposition develops the calculations needed to use these ideas. This page instead asks why each calculation became useful, what evidence supported it, and which interpretation remained conditional. Dates organize intersecting lines of work; they do not claim a single inventor for a broad field.

1A stable state is not a history

Reproducible bulk chemistry and variable individual outcomes can be descriptions of the same system.

Consider constant addition and individual first-order removal. The concentration equation approaches a stable value. At that value, addition has not stopped and removal has not stopped; their mean fluxes agree. A single small compartment therefore keeps changing even when a bulk assay reports no change. Stability determines the response to a perturbation, not the absence of molecular events.

Three questions now separate. What is the mean count? What fraction of compartments are empty? How long does one compartment remain above a threshold? The first can close as an ordinary differential equation for this simple model. The second needs a distribution on integer states. The third needs a probability law for histories. Neither of the latter questions is answered by drawing a noisy-looking line around the first answer.

This distinction preserves continuity with older kinetics. The stoichiometric matrix $\Gamma$ still states what each event changes. The new object is a propensity $a_r(n)$, the instantaneous event rate conditional on the present count state. The same reaction specification can generate a concentration limit, a master equation and a sample path. Gillespie's later review explicitly develops these representations together.¹

The historical question changes when an experiment ceases to report only an aggregate trajectory. The simple addition–removal calculation in the exposition establishes this compatibility before any complicated biological example is introduced.

Identify whether a claim concerns an average, a count distribution or a history before deciding that two descriptions disagree.

2Counts replace unresolved encounters

The randomness of a count model is conditional on what microscopic information the model does not retain.

Gillespie's 1977 article starts with a physical modelling question. Classical mechanics could, in principle, track molecular positions and velocities, but concentration kinetics discards that information. A stochastic chemical description instead retains molecule counts and assigns probabilities to reaction events. The probabilistic representation is attached to this resolution of the state.²

We distinguish the species from their random counts. Chemical species are $X_1,\ldots,X_s$. Their random count vector is $N_X(t)=(N_{X_1}(t),\ldots,N_{X_s}(t))^T$, and $n$ is one realized count state. Concentrations are denoted by $c$. This keeps the same reaction bookkeeping as Lecture 3 while changing the object whose evolution we describe.

A well-mixed count state makes the next-event clocks depend on the present counts. For a resolved elementary encounter $A+B\xrightarrow{k}C_{AB}$, the number of eligible pairs is $n_An_B$. If the macroscopic event flux is $kc_Ac_B$, the count propensity is $kn_An_B/(N_AV)$. Here $N_A$ is Avogadro's constant and $V$ compartment volume. Pair counting and units connect the descriptions.

Identical reactants expose what the rate constant means. For $2A\xrightarrow{k}C$ with concentration event flux $kc_A^2$, the eligible unordered pairs number $\binom{n_A}{2}$. Their individual hazard is $2k/(N_AV)$, giving $a(n)=kn_A(n_A-1)/(N_AV)$. The factor two converts the concentration constant to a per-pair constant. It does not change the chemistry. Gillespie's 2007 review states both pair conversions explicitly. The exposition derives their normalization.¹

Reaction occurrence and species disappearance remain different rates. Each dimerization event removes two $A$ molecules. Its instantaneous count drift is therefore $-2a(n)$. This stoichiometric factor is separate from counting unordered pairs. A constant imported from a paper must be read with its defining flux or disappearance equation.

Our course distinguishes resolved elementary steps from composite processes. Transcription or effective removal uses a bare squiggly chemical arrow, with its rate law stated separately. A constant or first-order propensity does not by itself resolve a molecular mechanism. Stochastic gene-expression models can combine elementary binding with effective descriptions of larger cellular operations.

A compartment boundary needs physical evidence

A small counting volume is not automatically an isolated chemical system. A dendritic spine is a synaptic head joined to a neuron's dendrite through a narrow neck. Its geometry fixes a useful local volume. Whether calcium can be treated as a local signal depends on how neck exchange competes with binding and clearance, not just on the head's molecule count.

Sabatini, Oertner and Svoboda used fluctuations to test that boundary. Their 2002 study compared repeated calcium responses in identified spines and neighboring dendrites. Similar average traces could conceal different trial-to-trial fluctuations. Those fluctuations supplied evidence about diffusional coupling that a mean trace alone could miss. The indicator also bound and transported calcium, so native exchange had to be inferred with that perturbation in mind.²⁸

The result supports a condition-specific calcium compartment, not a universal Poisson law for a small volume. The exposition separates the geometric calculation, measured clearance and indicator-dependent exchange inference. This is an early example of the lineage's recurring theme: choosing the observation also changes what can be inferred about the mechanism.

The umbrella example in the core illustrates hidden information, not a proof that chemical noise is fake. Conditional on rain and both locations, the carrying rule is deterministic. Once rain is modelled as a random input, the same rule defines a stochastic state process. The biological task is to justify which state and clocks match the observation.

Explain what information a count-state model retains, what it averages over, and what physical assumptions justify its event clocks.

3The rate equation returns as a limit

Stochastic and deterministic chemistry were connected mathematically before the canonical synthetic circuits were built.

Kurtz's 1972 chemical-kinetics paper asks how a family of count processes behaves as system size increases. Counts grow while concentrations are held at the same scale. Each reaction changes concentration less, while the number of firings grows. Under density-dependent scaling and regularity assumptions, stochastic concentration approaches the deterministic solution over a fixed finite time interval.³

The scaling is part of the statement. A bimolecular count propensity contains an inverse-volume factor. Doubling all counts without changing that coefficient does not simply make the same chemistry larger. It changes the concentration kinetics. The exposition therefore converts constants as well as variables before comparing simulations with Lecture 3's $\dot c=\Gamma v(c)$.

At finite size the exact mean identity is $d\mathrm E[N_X]/dt=\Gamma\mathrm E[a(N_X)]$. For a nonlinear propensity, averaging the rate is not the same as evaluating it at the average state. A bilinear clock depends on a covariance. Identical-reactant clocks also retain the finite-count exclusion of a molecule as its own partner. A large-system theorem supplies one route to neglecting relative fluctuations. An affine reaction model supplies a different route to exact mean closure.

The time qualification matters for biological memory. Convergence over a finite window does not imply that a noisy toggle never escapes a basin when observed for arbitrarily long times. Increasing system size may make escape exceptionally rare while leaving it possible at every finite size. A deterministic attractor and a finite stochastic residence time are compatible conclusions.

Read a deterministic limit together with its volume scaling, time horizon and closure assumptions.

4Event clocks make the theory usable

An enormous probability state space need not be enumerated to sample one exact history of its model.

The CME is linear in probabilities, but a network of several count variables can have an enormous or infinite set of states. Gillespie's direct method bypasses that enumeration. At the current state it needs only the enabled channels and their propensities. The total hazard determines the waiting time; each channel's share of that hazard determines its probability of firing.²

The 1977 paper derives the joint next-event law, not a small-timestep approximation. With hazards fixed while waiting, no-event survival is exponential. Two uniform random numbers select the time and channel. After the jump, the hazards are recalculated. The exposition reproduces this derivation and supplies executable code because this representation makes otherwise intractable models usable.

“Exact” belongs to the specified Markov jump model. It does not certify an effective promoter law, spatial mixing or a growth model. Gillespie, Hellander and Petzold's later perspective places direct simulation beside tau-leaping and Langevin descriptions, each with its own conditions.⁴ Faster computation changes which questions can be explored, but cannot supply biology absent from its state.

A history becomes evidence through its sampling rule

The simulator and the estimator solve different problems. An exact path tells us which events occurred and how long each state persisted. A stationary histogram requires a further choice: how do we sample that path? Erban, Chapman and Maini's 2007 tutorial makes this choice explicit. Its addition-removal example records counts at regular physical times before comparing the histogram with the analytical stationary law.²⁷

Counting event records instead measures an activity-weighted law. Let $\pi(n)$ be the stationary time distribution and $a_0(n)=\sum_r a_r(n)$ the total outgoing event rate. On an ergodic class with finite positive mean event rate, a pre-event histogram weights states in proportion to $a_0(n)\pi(n)$. Faster-firing states appear more often, even if they occupy little time. Residence-time weights or predetermined sampling times answer the time-occupancy question.

More elapsed time and more independent paths buy different evidence. Independent paths sampled at one fixed time estimate that ensemble without an ergodicity assumption. A long time average estimates a stationary expectation only if the path samples the relevant stationary class. Adjacent frames remain correlated, so duration measured in correlation times matters more than frame count. The exposition derives the estimators and their finite-window errors.

Extinction makes the distinction unavoidable. In the core's critical branching model, the finite-time ensemble mean stays at one while almost every path eventually reaches zero. Rare surviving paths carry that mean. This is not stationary nonergodicity: the ensemble started from one molecule is not stationary. The separate finite two-absorber example shows literal stationary nonergodicity. A metastable but ergodic switch observed too briefly is a third case. These are different limits, not disagreements between an exact simulator and its CME.

Separate the model, its exact path sampler and the estimator used to turn paths into evidence. Check the sampling measure and time limit before interpreting a histogram.

5Before synthetic circuits: a fate decision

Stochastic regulatory modelling already addressed a concrete biological decision before the year-2000 engineering landmarks.

Arkin, Ross and McAdams studied the choice between lysis and lysogeny after bacteriophage lambda infects E. coli. The alternatives are not slightly different fluorescence levels. One route ultimately produces phage and lyses the host; the other establishes a lysogen. A model must predict a fraction taking each route under stated infection conditions, not just an average regulator concentration.⁵

Their 1998 kinetic model resolved a regulatory network including competing CI and Cro controls and upstream processes influencing the decision. Early expression histories determine which regulatory state develops. The study compared predicted lysogenization fractions across phage-to-cell conditions with observations. Molecular events were connected to a population of alternative outcomes through an explicit mechanism.

This supplies a reason to compute more than stationary variance. An early excursion can select a fate whose consequences remain after the initiating molecules disappear. The final fraction of lysogens is a history-dependent observable, and expression timing can matter even when eventual protein abundance is large.

The larger theory has earlier roots. Paulsson's 2005 review traces stochastic gene-expression calculations to work in the 1970s and distinguishes later developments in promoter states, propagation and feedback.⁶ That retrospective supports a limited historical point: the engineered circuits did not create the question from nothing. It is not a claim here to have established the first use of every expression model.

Recognize a cell-fate fraction as an observable of competing molecular histories, not simply a property of the mean trajectory.

6Two constructions in 2000

The toggle and repressilator made a dynamical behavior itself the target of genetic construction.

In the genomics-era expansion of systems and synthetic biology, two papers published together in Nature provided particularly legible demonstrations. Gardner, Cantor and Collins constructed a toggle switch from mutual repression. Elowitz and Leibler constructed the repressilator from three repressive links arranged in a cycle.⁷⁸ Could a small engineered network realize memory or oscillation inside a living cell?

The toggle's objective is persistence after an input is removed. Mutual repression can create two restoring states separated by a threshold. The repressilator's objective is repeated change: each repressor suppresses the next, whose decline eventually relieves repression farther around the loop. Cyclic negative regulation matters, but kinetic response, nonlinearity and delays determine whether the particular system oscillates.

These are effective regulatory descriptions. Gene-expression laws commonly lump binding, transcription and translation into a Hill-shaped response. Such a law can be useful without describing a literal cooperative elementary collision. The elementary mechanisms, reduced rate equation and reporter are different resolutions that must not be conflated.

The constructions expose distinct noise questions. For a toggle: how often is memory lost, and does a perturbation cross a basin boundary? For an oscillator: how much do amplitude and phase vary, and for how long is timing predictable? The core playground makes these questions computationally accessible. The experiments explain why they matter.

Translate an engineered network's design objective into the relevant stochastic observable: memory lifetime or timing precision.

7Read the original oscillator evidence

The original modelling and experimental panels must be compared at the level of their actual observables.

Elowitz and Leibler's model explores how repressor and RNA dynamics can sustain oscillations. Its experimental fluorescence follows a GFP reporter controlled by the circuit while cells grow and divide.⁸ The reporter is an observation of the network, not an instantaneous, perfectly calibrated measurement of every repressor molecule.

The first comparison is qualitative dynamical behavior: does an oscillatory circuit operate in cells? Precision then requires sharper observables such as peak intervals, amplitude variation and lineage-to-lineage phase. A mismatch between a simulated protein trajectory and a GFP trace cannot be attributed wholly to intrinsic expression noise without accounting for reporter dynamics, growth and sampling.

Population averaging adds another distinction. Cells with regular individual oscillations but different phases can yield a nearly flat population signal. Conversely, a population wave does not establish that every cell follows the same trajectory. The original circuit motivates single-cell measurements and stochastic models; it does not by itself partition every source of irregularity.

Compare model and experiment through their observation maps before assigning a mismatch to a particular noise source.

8Expression becomes a stochastic filter

RNA lifetime and protein lifetime determine how a noisy input is amplified and averaged.

Thattai and van Oudenaarden's 2001 treatment put intrinsic expression fluctuations into a network framework. A transcript is not simply a deterministic intermediate: while alive it repeatedly produces protein, transmitting its random history downstream.⁹ Translation supplies amplification and protein persistence supplies temporal averaging.

For constant transcription and linear translation/removal, the stationary protein Fano factor is $1+k_p/(\gamma_m+\gamma_p)$. Here $k_p$ is translation propensity per RNA, and $\gamma_m,\gamma_p$ are individual removal constants. The exposition derives the covariance producing this expression. In the short-RNA limit, the extra term approaches the mean number of translations during an RNA lifetime.

The same result describes filtering. Protein integrates RNA fluctuations over a memory time; fast fluctuations average more efficiently than slow ones. Relative noise depends on how many independent transcripts contribute during that memory, not only on how many RNAs are visible at an instant. Expression biology thereby connects to the restoring and averaging times of the earlier lectures.

A broad protein distribution need not look like isolated bursts in a movie. Many overlapping transcripts can produce a nearly continuous translation stream while retaining a large Fano factor. The burst interpretation needs a molecular parent and a temporal resolution. Paulsson's review makes this distinction explicit.⁶

Relate expression noise to a source, propagation step and averaging lifetime without mistaking a Fano factor for a visible burst.

9Two reporters separate variation

A second, matched reporter adds information that a single expression histogram cannot contain.

Elowitz, Levine, Siggia and Swain's 2002 experiment placed two distinguishable reporters in the same cell under matched regulatory control. Both experience much of the same environment, but their independent reaction histories can differ.¹⁰ The joint distribution contains two directions of variation: together and apart.

Swain, Elowitz and Siggia formalized the decomposition by conditioning on shared cellular variables or histories.¹¹ Under matched conditional means and independent conditional fluctuations, covariance measures the shared component, while half the variance of the reporter difference measures the reporter-specific component. The exposition derives this from total variance.

“Intrinsic” identifies variability internal to a specified subsystem; “extrinsic” describes variation transmitted through its environment. If two reporters share one fluctuating transcript, RNA fluctuations are shared for that comparison. Independent transcripts contribute differently. These labels are relative to the construction and conditioning, not permanent molecular properties.

The design also supplies a warning. Different maturation kinetics or correlated detection errors can alter the inferred components. More reporters are useful because they change identifiability, not because every added color automatically names a biochemical source.

State the reporter-matching and conditional-independence assumptions that turn a joint measurement into a noise-source decomposition.

10A common language for noise

The same variance can combine event strength, sensitivity and memory in different proportions.

By 2004, expression experiments had reported different apparent contributions to cellular variation. Paulsson's Summing up the noise in gene networks organized such results using common source and response calculations.¹² The question was which architecture and timescales made each measurement possible.

The covariance balance connects this work to Lecture 4. Linearized restoration is encoded by a Jacobian $A$; event increments supply a covariance-injection matrix $D$. Stationary local covariance $\Sigma$ satisfies $A\Sigma+\Sigma A^T+D=0$. Systems can differ because they inject different disturbances, restore them differently or pass them through different downstream filters.

Paulsson's 2005 review extends this language and examines its assumptions.⁶ A two-stage model does not include every cellular process, and measured fluctuations need not select one source uniquely. An effective first-order description of growth removal does not make discrete division identical to independent molecular degradation.

The methodological gain is a set of separable questions. What injects variability? How strongly does the output respond? For how long does it remember? These questions remain useful even when a Gaussian approximation is inadequate for the entire distribution.

Interpret a noise measurement by separating disturbance strength, sensitivity and temporal averaging.

11RNA episodes become visible

Counting transcripts and following their appearance made promoter activity a measurable temporal process.

Golding and colleagues' 2005 bacterial experiment used RNA carrying many MS2 binding sites and fluorescent MS2 protein to reveal labelled transcripts in living cells. Intensity calibration connected fluorescent spots to RNA number. Time records showed episodes of appearance rather than a homogeneous initiation stream.¹³

The assay also changed its object. MS2 binding stabilized the labelled transcripts, making dilution and partition important to their disappearance from a lineage. These traces cannot be interpreted with an ordinary short bacterial RNA lifetime silently imported from another experiment. The authors also examined approximately binomial RNA partitioning at division.

Raj and colleagues' 2006 mammalian-cell study used single-molecule RNA detection to characterize variable transcript numbers and transcriptional episodes.¹⁴ A promoter-state picture connects episodes to a count distribution, but that distribution is not a direct movie of each chromatin conformation or initiation complex.

Both assays separate a model state from an observed molecule. An “on” state means a condition in which initiation is enabled at an effective rate. It may lump many molecular events. A binary state can be useful without asserting that a locus literally has only two biochemical configurations.

Trace an RNA-burst claim through labelling, calibration and lifetime assumptions before using it as a promoter model.

12Catalysis reveals protein packets

Single-molecule sensitivity can come from amplified catalytic activity rather than directly imaging each molecule.

Cai, Friedman and Xie used β-galactosidase activity to infer low protein expression in individual bacteria. Fluorescent product accumulated in a small chamber. Its accumulation rate reported enzyme activity, so a change of slope could reveal added active enzyme even when direct protein imaging was difficult.¹⁵

In the reported weak-expression regime, the inferred packet averaged $5\pm2$ active tetramers, corresponding to $20\pm8$ monomers. That stoichiometry matters when comparing translated polypeptides with enzymatic activity. Burst frequency was inferred as $0.11\pm0.03$ per cell cycle. These parameters belong to a particular reporter and condition.

A hidden count was connected to an observable slope through biochemistry. Lecture 2's catalytic amplification and Lecture 3's composite rate law form part of the measurement chain. The result is a calibrated inference with an observation model, not simply a count read off a fluorescence trace.

A translation/lifetime estimate now has a meaningful comparison. Translation once per ten seconds for a mean three-minute transcript life gives about eighteen monomers. Agreement in scale is informative, but neither establishes a universal packet size nor equates a transcriptional episode with a translational packet.

Convert an activity-based measurement into a molecular inference while keeping calibration and oligomeric state explicit.

13From bursts to a fitted law

Eliminating a fast transcript should preserve its integrated output, not erase its fluctuations.

Friedman, Cai and Xie's 2006 framework connects random expression bursts to stationary protein abundance. It treats abundance continuously, uses exponentially distributed packets and represents removal by continuous drift. Its gamma distribution describes asymmetric variation without a Gaussian-noise assumption.¹⁶

The paper explicitly distinguishes its Fano-like abundance ratio from the discrete result, which contains an additional one. The continuum treatment omits individual counting variance. A continuous density can also diverge near zero without assigning a point probability to exactly zero.

Shahrezaei and Swain's 2008 treatment derives analytical count distributions from expression models. In the fast-RNA limit, competing translation and RNA-removal clocks leave a geometric protein packet, and the stationary count is negative binomial.¹⁷ Its gamma limit requires an abundance rescaling. The exposition derives both results and their relationship.

Within a simple burst model, mean and Fano factor estimate effective frequency and size. Other promoter mechanisms, extrinsic changes and observation noise can give similar marginal distributions. A parameter interpretation conditional on one model is stronger than a descriptive fit, but weaker than identifying a unique molecular mechanism.

Interpret a burst distribution with its discrete or continuous state, elimination limit and model-dependent parameters.

14From one gene to a proteome

A broad survey tests how far a simple noise mechanism travels across genes and expression levels.

Taniguchi and colleagues built a chromosomal fluorescent-fusion library and measured bacterial expression with single-molecule sensitivity. Their calibrated assay corrected autofluorescence and normalized protein measurements by cell size, addressing abundance and noise much more broadly than one reporter.¹⁸

Gamma-shaped fits were useful across many protein distributions, while the interpretation of fit parameters depended on expression level. At low expression, stochastic expression events supplied a simple kinetic interpretation. At high expression, an extrinsic contribution remained important instead of disappearing according to independent-event inverse-mean scaling. A similar shape did not preserve one interpretation across regimes.

The survey also found weak correspondence between a cell's instantaneous RNA and protein counts for a given gene. This need not contradict translation from RNA. Protein integrates transcripts from an earlier interval, whereas current RNA is a snapshot of a faster process. Causally connected variables can have weak same-time correlation.

Furusawa and colleagues studied another route to broad distributions: multiplicative variation in intracellular reaction dynamics coupled to growth.¹⁹ Log-normal behavior means logarithmic abundance is approximately Gaussian. It does not assign log-normal laws to proteins as a category while assigning gamma laws to RNAs.

To compare studies, name the measured quantity and normalization. Count, concentration, fluorescence and logarithmic abundance have different distributions under a change of variable. A cross-gene regularity becomes a mechanism only after the observation map and contributing timescales are tested.

Use distributional regularities across genes without assigning a universal distribution to RNA or protein.

15Noise suppression has a sensing cost

A controller can react only to information supplied by its molecular signalling events.

A local negative-feedback calculation seems to offer a simple route to precision: steepen the restoring slope. But an effective law can hide its sensor. Lestas, Vinnicombe and Paulsson ask how much suppression is possible when information about a fluctuating species reaches a controller through a specified stochastic signalling channel.²⁰

The controller can use a general causal history of its signal. Its limitation is not an arbitrarily chosen linear feedback formula: signalling events lose information about a changing controlled count. Integrating longer helps only until old events become uninformative about the present.

For linear signalling, the high-budget lower bound on standard deviation improves only as a quartic root of signal budget at fixed controlled mean. The original bound contains $\sqrt{1+4N_2/N_1}$, where $N_2/N_1$ is the signal-to-controlled event budget. The exposition derives the algebra from stated information-theoretic inequalities.

The controlled species is treated in a continuous fluctuation approximation; signalling and control histories can remain discrete. The specified information route is essential. A new architecture, including correlated reactions absent from independent sensing and control events, needs a new analysis.

The design question is consequently which disturbances can be suppressed, with what information and event expenditure. The coupled-reaction example below will show why a deterministic feedback diagram cannot determine stochastic performance on its own.

Read a noise-control bound with its information channel, resource convention and approximation assumptions.

16Observe the oscillator for longer

A later repressilator study changed the measurement before attributing improvements to a redesigned circuit.

Potvin-Trottier, Lord, Vinnicombe and Paulsson revisited the repressilator in 2016 using a microfluidic mother-machine device. A retained lineage could be followed for many generations while newborn cells were washed away. Long records exposed persistent single-cell cycling more clearly than short observations could.²¹

Clear cycling across the tracked cells supported the basic cyclic design and showed that some apparent irregularity reflected the earlier imaging platform. It did not establish the absence of molecular noise or explain every earlier observation solely by population averaging.

The next interventions targeted observation machinery that was also part of the biology. The separate reporter plasmid and protein-degradation interactions could transmit fluctuations to the repressors. A reporter sharing finite cellular machinery can perturb the circuit it measures. Relocating it or changing degradation interactions alters a coupling, not just camera precision.

An observable can improve because it is followed longer, because the reporter interferes less, or because the oscillator becomes more precise. Those explanations require different comparisons. A single before/after statement about “less noise” would lose the experiment's central logic.

Separate improved observation, reduced reporter perturbation and genuine redesign in a claimed improvement of single-cell dynamics.

17Redesign what the clock counts

Timing can depend on decay to a molecular threshold rather than the precision of peak amplitude.

Once a repressor is abundant and its synthesis is suppressed, dilution can lower its concentration toward a threshold releasing the next gene. At constant growth rate $\lambda$, concentration starting at $x_0$ follows $x(t)=x_0e^{-\lambda t}$. If $x_c$ is the release threshold,

The logarithm converts a multiplicative amplitude difference into a modest absolute timing difference. At one volume doubling per generation, a twofold difference in peak adds one generation to the crossing time. This isolates one dilution segment, not a fitted model of every experimental phase.

Threshold fluctuations still matter. Potvin-Trottier and colleagues used additional TetR binding sites as a titration load and modified active-degradation features.²¹ The sites alter how abundance translates into regulatory release. Binding, growth depletion and stochastic timing therefore belong to the mechanism, rather than an added noise term around an unchanged ODE.

Growth-normalized and laboratory-clock timing are different observables. A period robust in generations can change in hours with doubling time. Individual phase precision and population coherence are also distinct: independent phase fluctuations eventually disperse a synchronized population. Population coherence is evidence about timing, not a substitute for measuring single-cell phase.

Relate timing precision to dilution, threshold variability and the time units in which robustness is claimed.

18Identical drift, different event coupling

Reaction stoichiometry carries covariance structure invisible in a deterministic rate equation.

Xiao, Fang, Yan and Doyle make this distinction explicit in their coupled-reaction analysis. A single event can change two species together, while independent events can produce the same average rates. Covariance input depends on each jump's outer product, not only on mean fluxes.²²

The exposition works a feedforward example where X and Y have independent additions and Y promotes X removal. Replacing two additions with one simultaneous X/Y addition preserves the deterministic equations and stable fixed point. It changes the off-diagonal covariance injected by events. A declared parameter set isolates precisely that difference.

Fluctuating Y generates later X removal, tending to produce negative X/Y covariance. Simultaneous addition can offset that effect. The outcome depends on event-coupling sign and restoring dynamics. Coupling is not universally beneficial: disturbances must be propagated through the specific network.

This clarifies noise-limit comparisons. A result for a controller receiving information through a specified independent channel need not apply unchanged when events jointly alter signal and controlled species. Compare the model contracts before interpreting an apparent exception to a bound.

Check event coincidence when circuits share an ODE but differ in observed noise.

19Fast states and fitted states

A state may be safely omitted for one prediction while remaining essential for another.

Two questions meet here. Given a larger model, what survives eliminating a state? Given data, can we determine which hidden states exist? The first is a reduction question; the second is an inference question.

Holehouse and Grima's 2019 study revisits stochastic feedback with fast promoter switching. A deterministic reduction can suggest a Hill-shaped expression law, yet copying that shape into a reduced CME need not reproduce the full stochastic system.²³ Binding changes the relation between free and bound regulator counts, and conditional distributions matter under feedback.

This does not invalidate all Hill propensities. “Fast” must identify what is held fixed during averaging. If binding redistributes free protein while conserving a total, fixing free protein can be the wrong operation. The stochastic question thereby connects to Lecture 6's slow variables and binding-time conservation.

Braichenko, Holehouse and Grima's 2021 comparison asks what mechanism expression data can identify. Different promoter descriptions can have similar count distributions but different initiation-time statistics.²⁴ A fitted two-state model may be effective without identifying the literal number of molecular states.

Reduction and inference ask different questions. Reduction starts with a specified larger model and asks what survives elimination. Inference starts with observations and asks which models they distinguish. A successful reduction need not make the original mechanism identifiable, and a good fit need not validate a proposed reduction.

Distinguish preserving an observable under reduction from identifying a mechanism, and use timing to test omitted states.

20Growth perturbs cellular memory

Circuit topology affects recovery from growth perturbation, even when the disturbance has a deterministic component.

Zhang and colleagues in Xiaojun Tian's group compared growth feedback in self-activation and toggle circuits. Their 2020 study found self-activation memory more vulnerable to growth-mediated dilution under the tested conditions, while mutual repression retained memory more robustly.²⁵ This compares specific constructions and perturbations, not every possible toggle and self-activator.

In a self-activator, lowering the regulator can cross the threshold below which production no longer maintains the high state. In a toggle, a shared decrease can leave the imbalance between repressors largely intact, allowing recovery of the same identity. The perturbation's direction relative to the separatrix matters. A scalar noise amplitude cannot encode that geometry.

Growth dilution and division partition are separate operations. Volume increase supplies $-\lambda c$ in a concentration balance without removing molecules. At symmetric division, independent partition gives a binomial daughter count while volume halves. Conditional mean concentration is preserved. Proportional partition would create no concentration jump.

This connects Lectures 3, 4 and 5 without reducing every disturbance to randomness. Lecture 3 supplies concentration accounting, Lecture 4 basin geometry, and Lecture 5 event and reset statistics. A prescribed growth pulse can be deterministic while its interaction with molecular histories yields variable outcomes.

Analyze the direction and timing of a growth disturbance before attributing lost memory to generic noise.

21Growth can move the landscape

A toggle can withstand a shared concentration perturbation yet lose bistability when its arms respond unequally to growth.

Zhu, Chu and Fu studied how unbalanced growth responses reshape cell-fate decisions. Their 2023 Nature Chemical Biology paper compares an engineered bacterial toggle across growth conditions. The two repressive arms respond unequally, changing their synthesis-to-dilution relationships. The published article supplies the experimental and model evidence used here.²⁶

The phenotype comparison separates this mechanism from burden-mediated feedback. The authors found no significant difference in growth rates between the two phenotypes in that comparison. They then measured each arm's expression capacity separately across growth conditions. Unequal responses can reshape the circuit even when a high-expression phenotype is not detectably slowing its own growth. This is a specific control, not a claim that growth feedback is absent from every circuit.²⁶

A transient state change moves a point in an existing phase portrait. A parameter change can move the nullclines themselves. Intersections and separatrices shift; a range with two stable states can become one with only a single stable state. Even a noiseless trajectory may then lose a previously stored identity.

The measured and predicted bifurcation ranges must stay separate. Decreasing growth in the original construct took the measured circuit from bistability to a single red state. The model also predicted another transition at higher growth, outside that construct's accessible experimental range. Modified repressor-binding sites shifted the decision boundary in further experiments. The paper therefore tests both physiological dependence and a molecular intervention, without measuring every branch of the model's diagram.²⁶

Here $f_X,f_Y$ are effective concentration addition laws, $\delta_X,\delta_Y$ true degradation constants, and $\lambda$ growth rate. Unequal growth dependence in addition cannot be represented solely by the same extra removal term in both equations. Host physiology changes the circuit's deterministic dynamics.

The Tian- and Fu-group results are compatible. A topology can be resilient to a specified common perturbation while sensitive to asymmetric parameter changes. Neither result makes division equivalent to concentration halving, and neither says every switch is caused by an independent random kick.

A useful intervention separates growth-dependent drift, random partition and unequal expression responses. First make partition deterministic while changing growth. Then add partition randomness on a fixed growth schedule. Finally vary the two arms independently. The exposition supplies the count–volume equations and event-clock correction for implementing these comparisons.

Distinguish a perturbed state from a perturbed phase portrait in growth-coupled switching.

22Choose the next discriminating experiment

The lineage ends with a design rule: measure what competing mechanisms predict differently.

Each advance adds an observable or intervention. Count kinetics makes discrete events explicit. Simulation supplies histories without enumerating every state. Sampling rules determine which probability those histories estimate. Paired reporters separate conditional sources. Molecular assays reveal packets. Long tracking tests phase memory. Growth perturbations distinguish a disturbed state from a changed attractor structure.

A student project can begin with the core's toggle or self-activator, but its contribution should be a comparison. Define a target region, retain censored non-switching paths and separate fixed from growth-dependent parameters. Or compare independent and coupled additions with identical drift, predicting covariance before simulation. State the competing explanation so that the result can distinguish it.

Lecture 6 inherits a specific debt. Fast binding, short-lived RNA and rapid promoter switching are candidates for elimination, but need not leave the same reduced object. A conditional mean propensity, a packet distribution and dynamics in conserved totals preserve different information. The next lecture develops systematic reductions instead of simply trusting that fast events disappear.

Propose a measurement or intervention that separates mechanisms left indistinguishable by the current dataset.

References and reading locators

The cited claims use inspected full-text passages. These locators identify the argument, assay or limitation to read.