Calculating Free Energies Using Adaptive Biasing Force Method

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

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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

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Thermodynamics Integration

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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:

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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

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bF

Take a long time to go over the energy

barrier

Much easier to reach another state

Local minimum

Global minimum

Numerical Results

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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

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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).

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2120

2

2

2

ln)(

ln)(

2120

212

0

xxxUU

dxekTxA

dxeekTxA

xxxU

xxxU

Free energy calculationAdaptive Biasing Force

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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

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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.

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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.

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