Lecture 2

Question 1

• What is an order parameter?

Answer 1

• An order parameter or collective variable, allows the different configurations/states of a system to be distinguished
• Allows system to be driven from A to B (or v.v) through application of enhanced sampling method of choice.

Question 2

Give an example of an order parameter to characterise the adsorption of a potassium atom on to a silver surface.

Answer 2

• State A: atom not interacting with surface
• State B: K fully adsorbed on to surface
• OP = min | |r_Ar – r_Au­^i,Surf |

Question 3

• What is the problem with the surface adsorption OP?

Answer 3

• Only in terms of state A (adsorbed) and state B (free).
• To drive a free energy pathway, OP must recognise intermediate states as well

Question 4

• What order parameter could be used to describe a protein folding event?

Answer 4

• Radius of gyration of a protein describes how much an average chain deviate from geometric centre
• Many atoms stay at this mean distance and calculation of them is costly
• May ned to remap values to positive numbers that can be made in to mathematical objects more easily

Question 5

• What may be a more suitable OP for a protein fold? What is the result of this better choice?

Answer 5

• Instead coarse grain the protein atoms in to a sub set of chains to save computational time
• Average distance between two chains could be used to give similar indication of fold
• FES remapped in to simpler CG representation (6D –> 2D)

Question 6

• What are the requirements of an OP? what is the difficulty in this requirement?

Answer 6

• Must be differentiable with respect to atomic position
• This is simple in the cases illustrated but can be very challenging for complex systems that is simple enough to compute but accurately represents dynamics.
• Can also be difficult when no prior knowledge of configurational space is known.

Question 7

• How many OP’s or collective variables would we need to simulate the nucleation of crystals from molecules in solution.

Answer 7

• Need two collective variables to describe 2 step process.
• One describing density difference of un/aggregated particles.
• One describing order of clumped particles.

Question 8

• Choice of order parameter determines the … and … of enhanced sampling results
• For this reason, it may be necessary to use …

Answer 8

• Choice of order parameter determines the nature and reliability of enhanced sampling results
• For this reason, it may be necessary to use multiple

Question 9

• What is affected by our OP choice in free energy-based methods.

Answer 9

• Our choice of OP determines the resulting free energy surface (FES)
• This multidimensional hyper surface is re-mapped on to a simple coarse-grained FES (1D/2D)

Question 10

• What is problem with this new FES?

Answer 10

• Can often be an oversimplification as only representative of the chosen order parameter.
• The true FES, which represents all configurational space does not equal the FES we are sampling, constrained to our OP of choice

Question 11

• (IMP) How does statistical mechanics relate to our FES assumption?

Answer 11

• Stat mech confirms true FES ≠ our FES according to this OP, as uses configurational partition function, Z which does not contain info about kinetics/ dynamics of the system and depends only on particles position in the system only.

Question 12

(PPQ) How can we use commitor analysis to probe the accuracy of our OP?

Answer 12

• Select a value, OP* of OP corresponding to a putative transition state (maximum)
• Sample an ensemble of configurations characterized by OP*
• Run several MD simulations for each, varying velocity (making them statistically independent)
• Find probability of OP* configuration making to it to B
• Plot histogram of these probabilities.

Question 13

• (PPQ) Which of these histograms indicates a more accurate OP?

Answer 13

• (a) indicates no matter where you start you always end up with same probability of ending up in A or B, NO indication this is a TS
• (b) indicates probability of ending up in B increasing as you move further along coordinate, 0.5 at middle. Representative of the putative TS
  
  • Therefore, b is better.

Question 14

• Why can the probability of going from state A to B not always be used?

Answer 14

• In ice nucleation the rate is extremely low
• Describing the path in terms of an order parameter would be completely unrealistic time wise

Question 15

(IMP)

• … … … is a path based enhanced sampling method where instead of computing an overall (v.low) … , path is divided into a series of …
• Each … has an increasing value of our …, … each corresponding to a possible … with said value of …
• As λ increases likelihood of going from … to … , rather than back to A, …

Answer 15

(IMP)

• Forward flux sampling is a path based enhanced sampling method where instead of computing an overall (v.low) probability, path is divided into a series of interfaces
• Each interface has an increasing value of our OP, λ, each corresponding to a possible configuration with said value of λ
• As λ increases likelihood of going from A to B, rather than back to A, increases

Question 16

• How is the probability of the process overall related to interfaces that make it up?

Answer 16

• Sum or product?? of computable windows

Question 17

(IMP) How are trajectories generating in FFS?

Answer 17

• Begin by looking at fluctuations of a long unbiased MD run
• At each interface λ_i large # of trial molecular dynamics runs done
• The few that reach the next interface λ_i+1 are used as a starting point to reach the following interface.

Question 18

• What is the problem with this trajectory generation? How can it be avoided?

Answer 18

• It is technically a bias result as each final trajectory is a function of its starting configuration
• To try and avoid this we must make sure initial simulation is run for the maximum amount of time, to ensure probabilities are correct.

Question 19

• Why is our overall nucleation mechanism not biast?

Answer 19

• Information is not biast from A to B as we are not changing system parameters with forces as you do in conventional MD, resulting in a fairly true mechanism.
• Other free E based methods will often add forces which will modify the Hamiltonian giving the correct thermodynamic properties of the system, but not the kinetics or mechanism.