calculating free energies using adaptive biasing force method
TRANSCRIPT
Calculating free energies using Adaptive biasing force method
Joanne ChiuDepartment of Mechanical Engineering,
Stanford2009/6/3
IntroductionFree Energy
◦Determination of the most stable conformation
◦Computing the reaction rate◦Determination of probabilities at equilibrium◦First step in building a course grained model
of proteinExamples
◦Fuel Cell reaction rate◦Protein-ligand binding and ion partitioning
across the cell membrane
2009/6/3
However…Free Energy Calculation
◦Needs first and second derivatives of the order parameter with respect to Cartesian coordinates
◦Expensive computation and long calculation time
◦Accuracy
2009/6/3
Thermodynamics Integration
2009/6/3
JkT
U
d
dA
dqqekTAqU
ln)(
))((ln)()(
Lecture 08◦Assume there is no momentum in
the Hamiltonian of the system.
J is already the first derivative of q, thus we need to get the second
derivative of q.
Adaptive Biasing ForceBased on computing the mean force
on◦ A free computation using the equation:
2009/6/3
2
1 1
)(
kk k qmm
mdt
d
d
dA
where
iii
qmdt
d
dq
dU
The governing equation of motion:
Adaptive Biasing Force (Cont.)
◦Applying a biasing force
2009/6/3
bF
Take a long time to go over the energy
barrier
Much easier to reach another state
Local minimum
Global minimum
Numerical Results
2009/6/3
Glycophorin A: key interactions between the two helical segments
Simulation SetupTwo molecules connected with a
spring and the potential is given by:
Because there is no long range force, free energy should be the same as potential energy.2009/6/3
2210 2xx
kU
1x 2x
Free energy calculationThermodynamics Integration
2009/6/3
12
2
122
2
12
212
0
212
0
212
0
0
)()(
ln)(
ln)(
xxx
dqee
dqxxxee
dx
xdA
dxeekTxA
dxxxxekTxA
xxxU
xxxU
xxxU
U
Replace impulse by Gaussian
distribution
Thermodynamics Integration
Calculated free energy (U) will be slightly different from the potential energy(U0).
2009/6/3
2120
2
2
2
ln)(
ln)(
2120
212
0
xxxUU
dxekTxA
dxeekTxA
xxxU
xxxU
Free energy calculationAdaptive Biasing Force
2009/6/3
1212
2
2
2
1
2
1
12
1212
2
1
2
1)(
21
)(
FFvvdt
d
d
xdA
xxqmm
vvmdt
dm
dt
d
d
xdA
vvxx
kk k
Assume the mass of two particles are 1:
0 1 2 3 4 5 60
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Reaction Coordinate
Ene
rgy
Potential
ABF
ABF simulation result
2009/6/3
2210 2
1xxU
21 x 52 x
Potential
Match
ConclusionABF is easier to implement and
only need to evaluate the first derivatives and the derivatives with time.
With applying biasing force, the phase space can be explored more completely.
The free energy converges faster than Thermodynamics Integrator.
2009/6/3
Future WorkComparing the simulation results of
TI and ABF.Changing the potential to create a
low/ relatively high energy barrier and comparing the results .
Adding biasing force to see if the accuracy is improved.
Adding long range force (more molecules) to calculate the new free energy by applying both methods.
2009/6/3