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PARAMETERIZATION OF SOLVENT MOLECULES AROUND A SOLUTE
Adnan Khan1 Peter Kramer1 & Rahul Godawat2
1Department of Mathematical Sciences
2Department of Chemical Engineering
Rensselaer Polytechnic institute
1.THE PROBLEM
6. THE DD-I MODEL
Two important lines of research in the behavior of proteins are
As a first model (following Garde et.al), we guess the correct form to be
- Given the sequence : predict the structure of the protein
- Given the structure : predict the function of the protein
dX  
D
dt  2D dW
kT
bulk
bulk
B
Want to study the functioning of a protein given the structure
We use Monte Carlo methods to simulate this SDE
The behavior depends on the surrounding molecules

To explicitly deal with sufficient number of solvent molecules is
difficult due to computational costs
We calculate the diffusivity and compare it with the diffusivity obtained
directly from the MD data
Initial configuration for the Buckyball Water system
2. OUR APPROACH
5. BENCHMARKING
We want to parameterize the state of water molecules near a protein
To benchmark our code we use two exactly solvable processes
As a basic model we will use
dX  A(X, ) B(X, )dW
The Ornstein Uhlenbeck process governed by
dX  Xdt  2dW
A is the drift vector and B is the diffusion tensor; these need to be determined, dW is
white noise
The Langevin process with quadratic potential, which is governed
by the equation


There are two approaches we use
- We guess the correct form of the coefficients
- We use MD data to determine the coefficients
7. THE DD-II MODEL
dX  Vdt
dV  aVdt  aXdt  adW
We next use a more general drift-diffusion parameterization
We are interested in the overdamped case which corresponds to
a>>1 

dX  U dt  D


long
The OU process can be considered a coarse-grained version of
the Langevin process
3. GENERAL PROGRAM
A measure of diffusivity which is used in the literature is
We start by considering a simple solute: specifically, we consider a Buckyball
(C60), as this has approximate spherical symmetry
Once we have a good parameterization for water around this simple solute, we
shall consider more complex solutes that mimic the geometric and chemical
properties of a protein
We shall then use the appropriate parameterization when simulating the
dynamics of an actual protein
We use inputs obtained from MD simulations to develop our parameterization
X(t  2 )  X(t) X(t)  x 
2
D(X, ) 

X(t   )  X(t) X(t)  x
2
6
2
The parameters Ulong Dlong & Dlat are calculated directly from the MD
data

We then calculate the diffusivity, and compare it to the value obtained
directly from the MD data
We are interested in the regime where the V dynamics are coarse-grained,
i.e. we want a value of  such that v<<  <<X where is the momentum
relaxation time and is the characteristic time for the X dynamics
We also use the two aformentioned exact systems to understand how to
calculate the best value of  for our models
8. CONCLUSIONS & FUTURE WORK
We note that the DD-II model does better than DD-I model in capturing the
features of the diffusivity
The MD simulation was performed with a Buckyball (C60) hydrated by 1380 water molecules
A 3.5 x 3.5 x 3.5 cubic box was used, which was periodic in all dimensions
The simulation parameters were
–Integrator time step 2fs
–Temperature 300K (using Nose Hoover thermostat)
lat
 X  X 
I 
 dW
X 

We calculate this for our test models and compare the results to the exact
solutions
4. MD SIMULATION
The interactions were
–LJ (Lennard Jones) cutoff 1nm
–Particle Mesh Ewald (PME)
long

X  X
dW

 D
X  X

We shall now use the DD-II model with input from MD simulation of water
around more complex solutes
Numerical vs Exact Diffusivity for OU
and Langevin systems
We hope to construct effective models for the dynamics of water around
different types of proteins, using these simulations