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TitleInnovations in Biomolecular Modeling and Simulations. Vol. 1
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Table of Contents
Innovations in Biomolecular Modeling and SimulationsVolume 1
CHAPTER 1:Personal Perspective
CHAPTER 2:Fashioning NAMD, a Historyof Risk and Reward: KlausSchulten Reminisces
CHAPTER 3:Towards BiomolecularSimulations with ExplicitInclusion of Polarizability:Development of a CHARMMPolarizable Force Field basedon the Classical DrudeOscillator Model
CHAPTER 4:Integral Equation Theory ofBiomolecules and Electrolytes
CHAPTER 5:Molecular Simulation in theEnergy Biosciences
CHAPTER 6:Enhancing the Capacity ofMolecular Dynamics Simulationswith Trajectory Fragments
CHAPTER 7:Computing Reaction Rates inBio-molecular Systems UsingDiscrete Macro-states
CHAPTER 8:Challenges in ApplyingMonte Carlo Sampling toBiomolecular Systems
CHAPTER 9:Coarse-grain Protein Models
CHAPTER 10:Generalised Multi-levelCoarse-grained MolecularSimulation and its Applicationto Myosin-V Movement
CHAPTER 11:Top-down Mesoscale Modelsand Free Energy Calculations ofMultivalent Protein-Protein andProtein-Membrane Interactionsin Nanocarrier Adhesion andReceptor Trafficking
CHAPTER 12:Studying Proteins and Peptidesat Material Surfaces
CHAPTER 13:Multiscale Design: From Theoryto Practice
Subject Index
Document Text Contents
Page 1

RSC Biomolecular Sciences

Innovations in Biomolecular
Modeling and Simulations
Volume 1

Edited by Tamar Schlick


s in




g an



e 1


Page 2

Innovations in Biomolecular Modeling and Simulations
Volume 1

Page 190

To pick a re-entry point, one of the boundaries a of cell i is randomly
picked using a probability law obtained from the cell flux and steady-state

pboundary a of cell i ¼

k;k 6¼iWkNki=Tk

where boundary a is the boundary between cell i and j. Although the
implementation is different, this is similar to Warmflash et al. (2007).


Based on this, it appears that using two lattices is not necessary and that the
scheme correctly works with a single lattice. Also the idea of lattice is no longer
discussed in a more recent publication.

See also Dickson et al. (2011)


an application of this method to unfolding and refolding of RNA. In this paper
as well, a single lattice is used. Dickson and Dinner (2010)

present some

theoretical results regarding non-equilibrium umbrella sampling (an analysis of
the convergence of the weights using the local scheme), a comparison with and
discussion of forward flux sampling, and recent applications of these methods.

7.4.2 Reactive Trajectory Sampling

The second method, which is related in some fashion to the previous class of
techniques, can be attributed to Huber and Kim (1996).

In this reference, the

method is developed assuming that an approximate reaction coordinate has
been chosen. However, it is not difficult to extend this approach to a general
decomposition of the conformational space O in a manner similar to, for
example, Vanden-Eijnden and Venturoli (2009a)

with Voronoi cells. This

method will be discussed in more details below. It consists in running a large
number of simulations (or ‘‘walkers’’) in parallel in such a way that a given
number of walkers are maintained in each cell or macro-state. Macro-states
that are near an energy barrier will tend to be depleted and therefore a strategy
is applied to duplicate walkers in this macro-state, in a statistically correct way.
This is done by assigning statistical weights to each walker. For example a
walker with weight w can be split into two walkers, starting from the same
location in O, with weights w/2. Conversely, macro-states that are at low energy
will tend to become overcrowded and walkers are then removed. If for example
we have two walkers with weights w1 and w2, we randomly select one with
probabilities (w1/(w1þw2), w2/(w1þw2)) and assign to it the weight w1þw2.
This approach ensures an efficient sampling of phase space.
In order to calculate a reaction rate, the macro-state corresponding to region

B is transformed into a cemetery state, that is any walker that enters this macro-
state is removed from the simulation and re-inserted in region A. In this
fashion, although the simulation is effectively out of equilibrium, the popula-
tion of walkers is kept constant. This method allows computing all the relevant
quantities of interest, such as reaction rates, free energy, metastable states, etc.
We note that contrary to Markov state models, this approach does not suffer

165Computing Reaction Rates in Bio-molecular Systems Using Discrete Macro-states

Page 191

from non-Markovity errors and that in the limit of infinite sampling it provides
an exact answer.
In Zhang et al. (2007),

this technique was applied to explore the transition

paths ensemble in a united-residue model of calmodulin.

In Zhang et al.

it is shown that the method initially developed in Huber and Kim

is really applicable to a much wider class of problems and proposes

some generalizations of this procedure.
We mention that a similar technique has been applied to simulated annealing

to find minima of rough (or even fractal) functions (see Huber and McCammon


Detailed discussion of reactive trajectory sampling. We now discuss in more
details the method of Huber and Kim (1996),

Zhang et al. (2007, 2010)


which we rename reactive trajectory sampling method (RTS), in the broader
context of macro-state models (e.g. Voronoi cell partitioning). In this
approach, systematic errors arising from non-Markovian effects are avoided by
directly calculating reactive trajectories from A to B and obtaining the prob-
ability flux entering B (or A for the backward rate), Metzner et al. (2006).


When the energy barrier is high, this can be very inefficient since very few
trajectories (if any) will make it to B when started from A. However, a simple
trick allows improving the efficiency of the calculation to the extent that the
decay of the statistical errors becomes essentially independent of the energy
barrier height.
As before we split the space of possible configurations into cells. Then a large

number of random ‘‘walkers’’ are initialized and advanced forward in time. The
basic idea is to use a strategy whereby, in cells that get overcrowded (too many
walkers), we merge walkers, thereby reducing their numbers, while in cells that
are depleted (near transition regions), we split walkers to increase their number.
The end goal is to maintain a given target number of walkers in each cell. With
such an approach we are able to observe a constant stream of walkers going
from A to B (and vice versa) irrespective of the height of the energy barrier. We
now explain the details of the method.
Assume we have nw walkers whose position gets updated at each time step. It

is possible to resample from these walkers without introducing any bias in the
calculation using the following procedure. Eachwalker, whose position is denoted
xi, is assigned a probabilistic weight wi, for example initially equal to 1/nw.
A walker can be split into p walkers with weight wi/p. After the split, each walker
can be advanced independently. Averages can then be computed using:

hf i ¼ lim



wif ðxiÞ ð7:46Þ

This equation is always true, irrespective of how many times the splitting
procedure is applied, or how many steps are performed, as long as the initial
position of the walkers is drawn from the equilibrium distribution. This is
proved from the fact that the equilibrium distribution is by definition invariant
under the dynamics under consideration for x(t).

166 Chapter 7

Page 380

trajectory fragments 120–3
background to 117–20
computing rates

forward flux sampling 123,
126–9, 135–6

milestoning 121, 122, 123–6,

illustrative 2-D model
system 132–5, 136

kinetics and equilibrium
applications 129–32

transition interface sampling 143,
148–9, 150, 193–4

transition matrix 175–82
transition path sampling 118, 140,

143, 146–8
transition state theory 118, 145
tree code algorithm 93–5
Tsukuba convention 11, 13
tyrosine, UV spectrum of 4

U1 snRNA 169
umbrella sampling 45–6, 140, 144,

163–5, 194

unit cell, computational 323–6
united-residue force field

(UNRES) 6, 231

virtual screening 293–5
viruses 18, 104

HCV IRES 162–3, 165,

HDV ribozyme 143, 143–4, 146,

viral capsids 237–8
see also antiviral drugs; HIV

Voronoi cells 152, 156,
161, 164

Watson-Crick base pairs,
electrostatics and 56–9

weighted ensemble Brownian
dynamics see Brownian

XRISM 53, 57

Z-DNA 61–2, 63

355Subject Index

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