A step-by-step explainer · protein thermodynamics
The two numbers that every stability measurement, every binding assay and every mutation-effect predictor in structural biology ultimately reports — and why the second one is the one you can actually trust.
The whole thing in one sentence
ΔG is nothing but a population ratio written on a logarithmic ruler — and ΔΔG is what survives when you compare two of those ratios and everything the change didn't touch cancels out.
Free energies are always differences between two states, so you must say which minus which. Throughout this page:
\(\Delta G_{\text{fold}} = G_{\text{folded}} - G_{\text{unfolded}}\)
→ negative means stable (the protein prefers to be folded).
\(\Delta G_{\text{bind}} = G_{\text{bound}} - G_{\text{free}}\)
→ negative means it binds.
\(\Delta\Delta G = \Delta G(\text{mutant}) - \Delta G(\text{wild type})\)
→ positive means the mutation made things worse
(less stable, or weaker binding). ("Wild type"
and "mutant" are defined in step 4.)
All three conventions are flipped somewhere in the literature — plenty of papers and databases quote \(\Delta G_{\text{unfold}}\) (positive = stable), and some define \(\Delta\Delta G\) the other way round so that positive means stabilizing. A ΔΔG value with no stated convention is half a number. When you pick up a dataset, always check its sign convention against a mutation whose effect you already know rather than assuming.
Step 1
Put a protein in water and you do not have a protein. You have \(10^{15}\) copies of it, each jiggling independently, each at any moment either folded up into its working shape or flopping around as a limp chain. Molecules cross back and forth constantly. What is stable is not any one molecule — it is the proportion.
So the honest measurement of "how stable is this protein" is a headcount: of every 120 molecules, how many are folded right now? That ratio is the physical fact. Everything else is bookkeeping.
The bookkeeping is this. Ratios are awkward: they run from 0 to ∞, they multiply rather than add, and a ratio of 5,000 doesn't look five times more impressive than 1,000 even though it is. So we take a logarithm and scale it by the thermal energy \(RT\):
\[ \Delta G_{\text{fold}} \;=\; -RT\,\ln\!\frac{[\text{folded}]}{[\text{unfolded}]} \]Read it right to left and nothing mysterious is happening. The fraction is the headcount. The \(\ln\) turns a multiplying ratio into an adding one. The \(RT\) puts it in energy units so it can be compared with, and added to, other energies. The minus sign is the convention that makes "more folded" come out negative. ΔG is a population ratio wearing energy clothing.
In the box below, each dot is one molecule sitting in one of two wells. The dots hop across on their own, accepting or rejecting each hop by the Metropolis rule — downhill hops always, uphill hops with probability \(e^{-\Delta G/RT}\). Nobody tells the crowd what ratio to settle at. Watch it find one anyway, and watch ΔG get read off the counts.
The simulation proposes a hop and accepts it with probability \(\min\!\left(1, e^{-(G_{\text{target}}-G_{\text{source}})/RT}\right)\). Writing \(p_{F\to U}\) and \(p_{U\to F}\) for the two acceptance probabilities, exactly one of them is 1 and the other is \(e^{-|\Delta G|/RT}\), so their ratio is \(p_{U\to F}/p_{F\to U} = e^{-\Delta G_{\text{fold}}/RT}\).
At steady state the traffic in each direction must balance, \(N_F\,p_{F\to U} = N_U\,p_{U\to F}\), which rearranges to \[ K = \frac{N_F}{N_U} = e^{-\Delta G_{\text{fold}}/RT}. \] Taking logs gives back the definition above. Note the direction of the logic: the equilibrium constant is the primitive object, and ΔG is defined from it. This is also why ΔG is only ever defined for a pair of states you have actually specified.
Where does \(RT\) itself come from, and why is free energy \(E - TS\) rather than just \(E\)? That is a story about counting microstates, told in full in the companion page Free Energy.
Step 2
Step 1 said ΔG is the ratio. It didn't say what sets the ratio. That comes from a contest with two sides:
Energy favours the folded shape. Folding buries greasy
side chains away from water and packs atoms close enough to attract each
other. Every such contact lowers the energy.
Counting favours the unfolded shape. There is essentially
one way to be correctly folded and an astronomical number of ways to be a
floppy chain. Even if each of those ways is individually unfavourable,
there are so many of them that collectively they are hard to beat.
That contest is exactly \( \Delta G = \Delta H - T\Delta S \): an energy term and a counting term, weighed against each other with temperature as the exchange rate.
We can do the contest honestly, with no hand-waving, on a toy protein: a 12-bead chain living on a square grid. The widget below enumerates every single one of its 120,292 possible shapes — that is the complete conformational ensemble, computed in your browser, not sampled or estimated. Each shape gets an energy \(E = -\varepsilon H\), where \(H\) is the number of bead pairs that touch on the grid without being chain neighbours. Then:
\[ \Delta G_{\text{fold}} \;=\; -RT \ln \frac{Z_{\text{folded}}}{Z_{\text{unfolded}}}, \qquad Z = \sum_{\text{shapes}} e^{-E/RT} \]Here \(Z\) is just a weighted headcount: every shape counted once, but discounted by \(e^{-E/RT}\) according to how energetically expensive it is. The folded state is the set of maximally compact shapes (a 3×4 block, 6 contacts); the unfolded state is everything else. Build the picture up:
The widget computes \(\Delta G(T)\) exactly from the density of states \(g(H)\) (the number of shapes with \(H\) contacts), which for this chain is \(g = [29940, 36976, 27600, 15552, 6424, 3552, 248]\) for \(H = 0 \ldots 6\); these sum to 120,292, the total number of 11-step self-avoiding walks on the square lattice. Then \(Z_{\text{folded}} = 248\,e^{6\varepsilon/RT}\) and \(Z_{\text{unfolded}} = \sum_{H<6} g(H)\,e^{H\varepsilon/RT}\).
The entropy is obtained as \(\Delta S = -\,\partial \Delta G/\partial T\) by numerical differentiation, and then \(\Delta H = \Delta G + T\Delta S\). Note that \(\Delta H\) and \(\Delta S\) both drift with temperature — that drift is the heat capacity change of unfolding, which is one of the reasons real proteins can be denatured by cooling as well as heating.
The transition here is broad, and honestly so: with only 12 beads, \(\ln(g_{\text{unfolded}}/g_{\text{folded}}) = 6.2\), so \(\Delta S\) is small and the tug-of-war tips gently. A real 150-residue protein has an entropy difference two orders of magnitude larger, which is why real melting curves are sharp enough to look like a switch.
Step 3
Before using these numbers, calibrate your intuition for the ruler. At room temperature (298 K):
\[ RT = 0.592\ \text{kcal/mol}, \qquad RT\ln 10 = 1.364\ \text{kcal/mol} \]So every 1.36 kcal/mol multiplies the folded-to-unfolded ratio by ten. A typical small protein sits at \(\Delta G_{\text{fold}}\) between −3 and −10 kcal/mol — which sounds like a lot until you notice that it is the residue of hundreds of kcal/mol of favourable contacts almost exactly cancelling hundreds of kcal/mol of lost chain entropy. Protein stability is a small difference between two enormous numbers. Hold on to that; it is the whole reason step 4 exists.
Now, how is ΔG actually obtained in a lab? You cannot count molecules. What you can do is titrate in a denaturant like urea, which selectively stabilises the unfolded state, and watch an optical signal (circular dichroism, tryptophan fluorescence) track the shifting population. Over the accessible range the effect of urea is empirically linear:
\[ \Delta G_{\text{fold}}([\text{urea}]) = \Delta G_{\text{fold}}^{\,\text{H}_2\text{O}} + m\,[\text{urea}] \]The number everybody quotes, \(\Delta G^{\text{H}_2\text{O}}\), is therefore not measured at all — it is an extrapolation back to zero denaturant from the narrow window where both states are visible at once. Run the experiment below a few times and watch how much that extrapolated number moves between repeats.
Step 4
The wild type is the naturally occurring version of the protein: the sequence the organism actually makes. In practice it is whatever construct the lab or the database has designated as its baseline, which is often not quite the organismal sequence — it may carry a purification tag, a truncated terminus, or a stabilising background mutation added to make the protein behave in a cuvette.
A mutant is the same protein with one or a few
residues swapped. The notation you will meet everywhere is
V54A (or Val54Ala): the wild-type residue, its
position in the sequence, then what it was changed to. Read left to right
as "valine 54 became alanine". V54A/L98F is a double mutant.
The wild-type letter is written down precisely because the mutant is only
defined relative to a stated baseline.
And that is the distinction worth being careful about. Thermodynamics also uses a reference — the standard state, \(c^{\circ} = 1\,\text{M}\), 1 bar, a fixed temperature — but that is a universal convention: everyone on earth uses the same one, so two labs' absolute \(\Delta G_{\text{bind}}\) values are directly comparable (more on this at the end of step 5). Wild type is a local reference, chosen per protein and per paper. Two groups can report honest, correct, and different ΔΔG values for the same mutation if their wild-type constructs were not the same molecule.
The good news is that re-referencing is just subtraction. If you have ΔΔG for constructs B and C both measured against baseline A, then C measured against B is \(\Delta G_C - \Delta G_B = (\Delta G_C - \Delta G_A) - (\Delta G_B - \Delta G_A)\). Which sequence you call "wild type" is a gauge choice: arbitrary, harmless, and only dangerous when it goes unstated.
Here is the question people actually ask. Not "how stable is this protein" but "what did that mutation do to it?" Take the wild-type protein, mutate one residue, and compare:
\[ \Delta\Delta G \;=\; \Delta G_{\text{fold}}(\text{mutant}) - \Delta G_{\text{fold}}(\text{wild type}) \]With our convention, positive ΔΔG = destabilising. That is the definition, and it looks almost too trivial to name. Its importance is not in the arithmetic — it is in what the arithmetic deletes.
Look at what these energies are made of. The absolute free energy of a folded protein in water is a colossal number: every bond, every water molecule, every ion in the box. Nobody can compute it and nobody can measure it. When you subtract the unfolded state from the folded state (a vertical step in the cycle below), almost all of that colossus cancels — same molecule, same solvent, same everything except the shape. You are left with a few kcal/mol.
Then you subtract wild type from mutant, and a second layer cancels: every residue the mutation didn't touch, contributing identically to both proteins, drops out. ΔΔG is what remains after two rounds of cancellation — a number that depends only on the thing that changed.
The widget below makes this literal. It starts from four absolute free energies, each a meaningless 5-digit number. Press the button to close the thermodynamic cycle around them, and use Shift the reference to add an arbitrary constant to all four — the absolute numbers scramble, ΔΔG does not move.
Free energy is a state function: the difference between two states does not depend on the path taken between them. Going around the square and returning to the start must therefore accumulate zero:
\[ \Delta G_{\text{fold}}^{\text{WT}} + \Delta G_{\text{mut}}^{\text{folded}} - \Delta G_{\text{fold}}^{\text{mut}} - \Delta G_{\text{mut}}^{\text{unfolded}} = 0 \]Rearranged, this says the two definitions of ΔΔG agree:
\[ \underbrace{\Delta G_{\text{fold}}^{\text{mut}} - \Delta G_{\text{fold}}^{\text{WT}}}_{\text{what the experiment measures}} = \underbrace{\Delta G_{\text{mut}}^{\text{folded}} - \Delta G_{\text{mut}}^{\text{unfolded}}}_{\text{what the simulation computes}} \]The right-hand side is what alchemical free-energy calculations evaluate: you never simulate folding, you gradually morph one side chain into another in each state and take the difference. How that morphing is done, and how it quietly fails when the two states are too dissimilar, is the subject of Free energy perturbation.
Step 5
Nothing in steps 1–4 was about folding specifically. All the machinery needed was two states and a ratio. Swap in a different pair of states — a drug free in solution versus bound in its target's pocket — and every formula carries over:
\[ \Delta G_{\text{bind}} = RT \ln \frac{K_d}{c^{\circ}}, \qquad \Delta\Delta G_{\text{bind}} = RT \ln \frac{K_d^{\text{mut}}}{K_d^{\text{WT}}} \]\(K_d\), the dissociation constant, is the ligand concentration at which half the target sites are occupied — a concentration you can read straight off a binding curve. Because it has units, it needs a reference concentration \(c^{\circ} = 1\,\text{M}\) inside the logarithm to make sense; more on that below. A \(K_d\) of 20 nM gives \(\Delta G_{\text{bind}} = -10.5\) kcal/mol.
In the widget, 40 target sites bind and release ligand stochastically. No binding curve is drawn for you — run the titration and it gets built out of measured occupancies, one concentration at a time. Then mutate the interface and watch the whole curve slide right.
Folding is unimolecular: one molecule in, one molecule out, so the equilibrium constant \(K = [F]/[U]\) is a pure number and \(\Delta G = -RT\ln K\) is unambiguous.
Binding is bimolecular — \( \text{P} + \text{L} \rightleftharpoons \text{PL}\) — so \(K_d = [\text{P}][\text{L}]/[\text{PL}]\) carries units of concentration. You cannot take the logarithm of something with units, so a reference concentration must be chosen. Convention picks \(c^{\circ} = 1\ \text{M}\), giving \(\Delta G^{\circ}_{\text{bind}} = RT\ln(K_d/c^{\circ})\). Choose \(c^{\circ} = 1\ \text{mM}\) instead and every binding free energy on earth shifts by \(RT\ln 1000 = 4.09\) kcal/mol.
And here is the payoff, one more instance of the theme of this page: \(\Delta\Delta G_{\text{bind}} = RT\ln(K_d^{\text{mut}}/K_d^{\text{WT}})\) contains a ratio of two concentrations, so \(c^{\circ}\) cancels. The absolute number depends on a convention; the difference does not. This is a large part of why predicted ΔΔG values are taken far more seriously than predicted absolute binding affinities.
Start with a crowd of molecules splitting between two states. The split is a ratio; taking \(-RT\ln\) of it gives ΔG, which is the same information on a ruler where "ten times more of it" is a fixed step of 1.36 kcal/mol (step 1). What sets the ratio is a contest between energy, which likes the one compact shape, and counting, which likes the hundred thousand floppy ones — that contest, written out, is \(\Delta H - T\Delta S\) (step 2). Measuring ΔG in practice means extrapolating from the narrow window where both states are visible, which is why it carries a real error bar (step 3).
Then the move that makes the whole field work: don't ask for absolute free energies at all. Ask for the difference between two of them. Every contribution the mutation didn't touch appears identically in both terms and vanishes — so ΔΔG is a small, robust, computable number where ΔG was a huge, fragile, incomputable one (step 4). And because none of the reasoning ever mentioned folding, the same two symbols describe binding just as well, with the standard-state ambiguity cancelling in exactly the same way (step 5).
Which returns us to the kernel: ΔG is a population ratio on a logarithmic ruler, and ΔΔG is what is left when everything unchanged cancels out.
ddg_monomer — the engine behind tools like Levitate Bench's
DDG — estimates ΔΔG by scoring two relaxed structures and subtracting.
Inside a ΔΔG prediction takes that
pipeline apart: repacking, the ramped repulsion, fifty repeats averaged
three at a time, and what r = 0.73 does and does not license.