Protein Folding and Phenotypic Variability — Epoche B2
Beyond the Central Dogma: Protein Folding and Phenotypic Variability The central dogma is usually drawn as a chain of arrows, DNA to RNA to protein, and the last arrow is silently read as a function: one sequence, one structure, one function, one phenotype. Thermodynamics does not permit that reading. A polypeptide in water does not have a structure; it has a Boltzmann-weighted ensemble of structures, and the native fold is simply the member of that ensemble that free energy favours — by a margin that turns out to be astonishingly small. This essay develops that margin quantitatively for a single-domain protein of about a hundred residues at body temperature in dilute buffer, and shows that it is the arithmetic of a few kilojoules per mole, rather than anything written in the genetic code, that decides whether a substitution is invisible or lethal. The two states and the quantity that separates them Take the simplest description that is still honest for a small protein: a two-state equilibrium between the native fold $N$ and the unfolded ensemble $U$, with no populated intermediate. Fix one convention and keep it throughout — all free energies below are of unfolding , $$ \Delta G_U \;=\; G_U - G_N \;=\; \Delta H_U - T\,\Delta S_U , $$ so that $\Delta G_U \gt 0$ means the folded state is the stable one, and a destabilising change lowers $\Delta G_U$. Here $\Delta H_U$ is the enthalpy of unfolding, $T$ the absolute temperature and $\Delta S_U$ the entropy of unfolding, which includes both the chain and the surrounding water. The chain term alone can be estimated. If each residue of a chain of length $L$ has roughly $\nu \approx 3$ accessible backbone conformations, the unfolded ensemble contains $$ \Omega \approx \nu^{L} = 3^{100} \approx 5\times 10^{47}, \qquad S_{\text{conf}} = R\ln\Omega \approx 0.91\ \text{kJ mol}^{-1}\text{K}^{-1} \;\Rightarrow\; T\,S_{\text{conf}} \approx 2.8\times 10^{2}\ \text{kJ mol}^{-1} $$ at $T = 310\ \text{K}$, with $R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$. Two warnings attach to that number. It is a floor, since $\nu = 3$ counts backbone states only and ignores side-chain rotamers; and it is only one of the two entropy terms, because unfolding exposes hydrophobic surface and orders the water around it, so the solvent contribution to $\Delta S_U$ is negative at this temperature and works against the chain term. The reliable route to the magnitudes is therefore calorimetric rather than combinatorial. Exposing that surface also raises the heat capacity, by $\Delta C_p \approx 6\ \text{kJ mol}^{-1}\text{K}^{-1}$ for a chain of this length, which makes both large terms strongly temperature-dependent: $$ \Delta H_U(T) = \Delta H_U(T_m) + \Delta C_p\,(T - T_m), \qquad \Delta S_U(T) = \frac{\Delta H_U(T_m)}{T_m} + \Delta C_p \ln\frac{T}{T_m} $$ where $T_m$ is the melting temperature, at which $\Delta G_U = 0$ and hence $\Delta S_U(T_m) = \Delta H_U(T_m)/T_m$. Calorimetry on a mesophilic protein of this size gives roughly $\Delta H_U(T_m) \approx 400\ \text{kJ mol}^{-1}$ at $T_m \approx 335\ \text{K}$; those large published values belong to the melting point, not to body temperature. Substituting $T = 310\ \text{K}$ into both relations, $$ \Delta H_U \approx 250\ \text{kJ mol}^{-1}, \qquad T\Delta S_U \approx 226\ \text{kJ mol}^{-1}, \qquad \Delta G_U = \Delta H_U - T\Delta S_U \approx 24\ \text{kJ mol}^{-1} $$ Three things follow. The stability is the small residue of two quantities an order of magnitude larger, about a tenth of the terms that produce it. It falls inside the $20$ to $60\ \text{kJ mol}^{-1}$ band — five to fifteen kilocalories per mole — observed for natural mesophilic proteins, without having been fitted to it. And the gap between the combinatorial chain estimate, $283\ \text{kJ mol}^{-1}$, and the calorimetric $T\Delta S_U \approx 226\ \text{kJ mol}^{-1}$ is the opposing solvent term, about $-57\ \text{kJ mol}^{-1}$ at $310\ \text{K}$, which is the sign the hydrophobic effect requires. Nothing in the sequence guarantees the sign of such a near-cancellation, and that is why the last arrow of the dogma cannot be a function. From free energy to the fraction that works The margin matters because the cell does not see $\Delta G_U$; it sees how many copies of the protein are folded at a given instant. Equilibrium fixes that directly. Writing $K$ for the unfolding equilibrium constant and $f_U$ for the fraction of molecules unfolded, $$ K = \frac{[U]}{[N]} = \exp\!\left(-\frac{\Delta G_U}{RT}\right), \qquad f_U = \frac{K}{1+K} = \frac{1}{1+e^{\Delta G_U/RT}} . $$ At $310\ \text{K}$ the natural unit is $RT = 2.58\ \text{kJ mol}^{-1}$, and every multiple of it changes $f_U$ by a factor of $e$. That single fact carries the argument, as the table shows. $\Delta G_U$ at $37\,^{\circ}$C Unfolded fraction $f_U$ What the cell experiences $60\ \text{kJ mol}^{-1}$ $8\times 10^{-11}$ hyperstable, typical of thermophilic proteins; far more stability than function requires $30\ \text{kJ mol}^{-1}$ $9\times 10^{-6}$ mid-range for a mesophilic enzyme: about one molecule in $10^{5}$ unfolded at any instant $20\ \text{kJ mol}^{-1}$ $4\times 10^{-4}$ still fully functional, but one molecule in $2300$ is exposed to aggregation and degradation $10\ \text{kJ mol}^{-1}$ $2\times 10^{-2}$ two per cent unfolded; misfolding-linked variants sit here $0\ \text{kJ mol}^{-1}$ $0.5$ half the population unfolded: the phenotype is lost although the gene is intact The entire range from a robust protein to a non-functional one spans about $30\ \text{kJ mol}^{-1}$, which is roughly twelve times $RT$ and less than a tenth of the enthalpy of unfolding. One genotype, held at a fixed sequence, therefore specifies not a structure but a position on this scale — and the position moves with temperature, pH and ionic strength, none of which the gene encodes. The funnel is wide at the top, where the unfolded ensemble contains about 5 times 10 to the 47 conformations, and narrows downwards to the n