Physical Biology: From Gene Regulation to Living Matter

# Can statistical mechanics predict gene expression? Watch the video first. Use this companion to revisit the reasoning and its evidence limits. Start with one gene in E. coli. Upstream sits the promoter, where RNA polymerase binds to begin transcription. While polymerase is bound, messenger RNA can be made; otherwise the gene is silent. Now add a repressor, a transcription factor that binds a short DNA site called an operator. In simple repression, a single operator overlaps the promoter, so a bound repressor keeps polymerase out. The lac genes are the classic example. At any instant the promoter is in one of three states: empty, bound by polymerase, or bound by repressor. Molecules keep binding and falling off, so instead we ask what fraction of time it spends in each. Statistical mechanics answers with one rule. At equilibrium, a state's probability is proportional to e to the minus E over k B T. E is the state's energy, and k B T, Boltzmann's constant times temperature, is the typical thermal kick. A state lower by two k B T is about seven times more likely; lower by ten, twenty thousand times. Assume expression tracks how often polymerase is bound. Fold-change compares expression with repressor to expression without. If polymerase binds weakly, the model collapses to: one over one plus R over N N S, times e to the minus delta epsilon over k B T. Two knobs: how many repressors, and how tightly they bind. It roughly halves. With strong repression the one in the denominator is negligible, so fold-change is inversely proportional to R. Cells also tune repressors with signals. LacI is allosteric: it switches between an active shape that grips DNA tightly and an inactive shape that binds weakly. An inducer such as IPTG binds it and tips the balance toward inactive, lifting repression. In 2018, Razo-Mejia and colleagues added this two-state, M W C picture to the same statistical mechanics, measuring a yellow fluorescent reporter cell by cell with flow cytometry. With inducer-binding constants fit to one strain, the theory predicted induction curves for eighteen strains, three operators by six repressor copy numbers, and the measurements largely matched. That is the central idea of physical biology: molecular energy, to probability, to cellular phenotype. One caution: the model assumes the promoter is at equilibrium. That held up for simple repression in E. coli, but it isn't guaranteed for every gene. ## Evidence guide ESTABLISHED BIOLOGY: transcription depends on molecular interactions. PHILLIPS-LAB PRIMARY RESULT: simple repression and allosteric induction provide scoped tests. MODEL PREDICTION: state weights assume an appropriate equilibrium model. EXPERIMENTAL MEASUREMENT: expression tests those predictions; driven genetic circuits need additional kinetics. Sources: [garcia2011], [razomejia2018], [yang2026]. See the course bibliography and claim audit.