Lecture 3.2

Energy landscapes

Learning objectives

After this lecture, you should be able to do the following.

Account for how a molecular mechanics energy function is a sum of separate bonded and nonbonded terms, and attribute a structure’s high energy to the term responsible.

  1. Name the bonded and nonbonded terms in an additive energy function, and state which geometric feature of a structure each one scores.
  2. Explain what an additive fixed-charge energy function leaves out of a molecule’s physics, and name one claim it therefore cannot support.
  3. Explain how an energy function’s functional form differs from the parameter values assigned to its terms, and classify a stated change to a term, such as a new value of \(k\) or a new kind of equation, as a change of form or of parameters.
  4. Interpret the harmonic bond term \(\tfrac{1}{2}k(r - r_0)^2\) as a penalty that grows with the square of the departure from a reference length, and state what \(k\) and \(r_0\) each control.
  5. Explain why only differences in energy between structures of the same system carry meaning, and state what a single value measured against an arbitrary reference state reports on its own.
  6. Given two bonds whose force constants differ, displaced by the same amount from their reference lengths, predict which contributes more energy and how that energy changes when the force constant or the stretch doubles.
  7. Predict which term a steric clash shows up in when two atoms are brought closer, and compare how the van der Waals and electrostatic terms each change with distance.
  8. Given a per-term breakdown of one structure’s energy together with its geometry, attribute the high total to the bond lengths, the angles, the torsions, or a nonbonded contact.

Interpret a potential energy surface as the object a structural model sits on, and relate its minima and barriers to what the molecule does.

  1. Define a potential energy surface and state what its horizontal axes and its height each report.
  2. Define local minimum, global minimum, barrier, and basin, and identify each on a labeled one-dimensional energy curve.
  3. Compare what the relative depth of two minima reports about which state is favored, and when that reading fails, with what the barrier between them decides about where a downhill optimization ends.

Trace a gradient-based optimization across an energy surface, and predict how the starting point, the step size, and the stopping rule change the structure it returns.

  1. State what a geometry optimization returns and which question about a starting structure it answers.
  2. Explain what a convergence criterion placed on the gradient tests, and why a converged optimization is not evidence that the structure it returned is the lowest one available.
  3. Explain why an optimization that only moves downhill cannot leave the basin it started in, and what that costs when the starting structure sits in the wrong basin.
  4. Explain why the update \(x \leftarrow x - \eta\, dE/dx\) moves a coordinate toward lower energy, and state what the sign of the slope says about which way to step.
  5. Given a starting point on an energy curve and the sign of the slope there, determine which way the next step moves the coordinate and whether the energy falls.
  6. Determine what happens to the atom positions of a deposited structure when it is minimized, and name what to check before using the result.
  7. Predict what a step size too large and a step size too small each do to the path an optimization takes and to where it stops.
  8. Given a double-well energy curve and two starting points on opposite sides of the barrier, determine which minimum each optimization run ends in.
  9. Given an optimization that returned a geometry no molecule would adopt, together with the structure it started from and the settings it ran under, attribute the outcome to the starting point, the step size, or a term missing from the energy function.

Account for the decisions a deposited coordinate file forces before its energy can be evaluated, and determine how pH and a group’s \(pK_a\) set the protonation state assigned to it.

  1. State what a one-unit difference between the pH and a group’s \(pK_a\) does to the ratio of its protonated to its deprotonated form.
  2. Name the decisions made when preparing a deposited structure for energy evaluation: where hydrogens go, which protonation and tautomer state each ionizable group takes, which waters and ions to keep, and which absent residues to build in.
  3. Explain why hydrogen atoms are absent from most diffraction-derived coordinate files, and state why an energy function cannot be applied until they are placed.
  4. Explain why an ionizable group inside a protein can take a \(pK_a\) far from its value for the free amino acid, and name two features of its surroundings that shift it.
  5. Explain why the data behind a routine diffraction structure cannot settle which group in a pair of equivalent ionizable residues carries the proton, and name a measurement that can.
  6. Predict how changing the protonation state of one group in a binding site changes the charge it presents and the hydrogen bonds it can make.
  7. Given whether the pH sits above or below a group’s \(pK_a\), predict whether the group is mostly protonated or mostly deprotonated, and which way the protonated fraction moves as the pH rises.
  8. Given two preparations of one structure differing in a single protonation assignment, differentiate the downstream results that would change from those that would not.

Relate a free-energy difference to the equilibrium it implies through \(\Delta G = -RT\ln K\), and predict how that energy responds when the fold change it corresponds to grows, shrinks, or inverts.

  1. State what \(\Delta G = -RT\ln K\) relates to what, and which sign of \(\Delta G\) goes with an equilibrium lying toward products.
  2. Identify \(RT\) as the thermal energy scale, and state what it sets the size of when an energy difference is converted into a ratio such as an equilibrium constant.
  3. Explain why the number of kcal/mol corresponding to one factor of ten in an equilibrium constant is tied to a stated temperature, and state what it means that every further factor of ten costs that same amount.
  4. Explain why the energy an energy function reports for one arrangement of atoms is not a free-energy difference between two states, and name what the second accounts for that the first does not.
  5. Explain why a difference of one or two kcal/mol, small beside the total energy a force field reports for a protein, moves an equilibrium by orders of magnitude.
  6. Given the free-energy difference between two states, predict which state is more populated, and how the population ratio changes when that difference doubles or changes sign.
  7. Predict how the free-energy difference corresponding to a fold change in an equilibrium constant changes when the fold change is multiplied by ten, squared, or inverted.
  8. Predict which way the energy corresponding to a fixed fold change moves when the temperature rises, and state why reporting the temperature alongside it matters.
  9. Relate one free-energy difference to the three ratios it can be read as: an equilibrium constant, the relative populations of two states, and a fold change in a measured potency.

Judge whether a reported number supports the claim drawn from it, given how the number was produced and how large the uncertainty of that method is.

  1. Compare the typical error in an energy difference from a fixed-charge energy function with the energy that corresponds to a tenfold change in an equilibrium constant.
  2. Given two structures ranked by a reported energy difference and a stated error for the method, determine whether the ranking is supported.
  3. Given a predicted \(pK_a\) with a stated error and a pH that lies either within that error of it or well outside it, distinguish an assignment the prediction settles from one it leaves open, and identify a measurement that could settle the one it leaves open.
  4. Given a methods section describing an energy result, distinguish the preparation decisions it reports from the ones it leaves unstated, and identify which omission most limits what the result can be used for.
  5. Given a claim about how a molecule behaves and one minimized structure offered in support, judge whether the structure supports the claim and state what further evidence would settle it.
  6. Given an unfamiliar quantity reported in energy units for a system you have not studied, identify what it treats as the reference state, and find the flaw in a claim drawn from it that ignores its zero, the object it scores, or its error.

Readings

Back to top