Refine Structures

Restating question

Before using the potential energy functions to determine the best structure for two candidate structures given the sequence, one has to refine the structures:

• If one of the structures is the correct one:
  • Need to determine side chain conformations before calculating potential
• If better structure is only approximate:
  • Need to refine backbone and side chains first

Methods for Refining Structures

• Energy minization
  • Move towards the local conformation
  • Fast compared the other two
  • restricted to local changes
• Molecular dynamics
  • tries to simulate the biological process
  • Computationally very intensive
• Simulated annealing
  • tries to shortcut the route to some of those global free energy minima by raising the temperature and sample all the space, and then cooling down so we trap a high probility conformation

Energy Minization

Gradient Descent

one dimension:

\[\begin{aligned} x_{i}=x_{i-1}-\varepsilon f'(x_{i-1}) \end{aligned}\]

N dimensions:

\[\begin{aligned} \vec{x}_{1}=\vec{x}_{0}-\varepsilon\nabla U_{0}(\vec{x}_{0}) \end{aligned}\] \[\begin{aligned} \nabla U=\left(\cfrac{\partial U}{\partial x_{1}},\ldots,\cfrac{\partial U}{\partial x_{n}}\right) \end{aligned}\]

Force:

\[\begin{aligned} F=-\nabla U \end{aligned}\] \[\begin{aligned} \vec{x}_{1}=\vec{x}_{0}+\varepsilon F_{0}(\vec{x}_{0}) \end{aligned}\]

each step is moving in the direction of the force

Not guaranteed to get to the correct energetic structure

Molecular Dynamics

• Seeks to simulate the motion of molecules
• Can escape local minima

\[x(t_i)=x(t_{i-1})+v(t_{i-1})\times(t_i-t_{i-1})\] \[v(t_i)=v(t_{i-1})+\cfrac{F(t_{i-1})}{m}\times(t_i-t_{i-1})\] \[v(t_i)=v(t_{i-1})-\cfrac{\nabla U(t_{i-1})}{m}\times(t_i-t_{i-1})\]

Short simulations take tremendous computing resources

Length of simulation and protocol determine radius of convergence

Simulated Annealing

• At low temperature we cannot escape local minima
• At high temperature the kinetic energy exceeds the potential energy barrier
• This approach allows us to balance the need for speed and the need to be at high temperature

Step

• Start at high temperature
• Find most probable states
• Reduce temperature to trap these states

Metropolis Algorithm

• Goal: efficiently search a large conformation space
• Can be understood in terms of physical processes, but much more general
• Note the difference from molecular dtnamics:
  • Molecules move under physical forces but temperatures are far outside of normal range
  • A sampling method(a search strategy) not a simulation

Acceptance Criteria:

• Randomly choose neighboring state
  • Always accept moves that reduce potential
  • Go uphill (higher potential) based on odds ratio

\[\cfrac{P(S_{\text{test}})}{P(S_{n})}=\cfrac{e^{-E_{\text{test}}/\text{kT}}}{Z(T)}/\cfrac{e^{-E_{n}/\text{kT}}}{Z(T)}=e^{-(E_{\text{test}}-E_{n})/\text{kT}}\]

Boltzmann equation

Full Algorithm

Iterate for a fixed number of cycles or until convergence:

1. Start with a system in state $S_n$ with energy $E_n$
2. Choose a neighboring state at random; we will call it the proposed state: $S_{\text{test}}$ with energy $E_{\text{test}}$
3. If $E_{\text{test}}\lt E_{n}$: $S_{n+1}=S_{\text{test}}$
4. Else set $S_{n+1}=S_{test}$ with probalility $P=e^{-(E_{\text{test}}-E_{n})/\text{kT}}$
   • otherwise $S_{n+1}=S_{n}$

Noted that if $kT$ is very high, we almost always take the new state (higher potential). And this is what allows us to climb those energetic hills.

Full Algorithm (prob. version)

To identify minima given a probability function: $P(S)$

1. Start with a system in state $S_n$
2. Choose a neighboring state at random: $S_{\text{test}}$
3. Compute acceptance ratio $a=\cfrac{P(S_{\text{test}})}{P(S_n)}$
4. If $a\gt 1$: $S_{n+1}=S_{\text{test}}$
5. Else set $S_{n+1}=S_{\text{test}}$ with probability $a$ and $S_{n+1}=S_{n}$ with probability $1-a$

Not specific to protein structure. Used to sample diverse probability distributions.