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Stochastic roadmap
simulation for the study of
ligand-protein interactions
Mehmet Serkan Apaydin, Carlos E. Guestrin,
Chris Varma, Douglas L. Brutlag and JeanClaude Latombe (Stanford Departments of
Computer Science and Biochemistry)
Presented by Max Shneider
Definitions
 Stochastic - Random, probabilistic
 Roadmap - Compact graph structure
 Torsional – twisting or turning
 Putative binding sites – different cavities on a
protein where a ligand could potentially bind
 Funnel of Attraction – all ligand conformations
within 10 Å RMSD of a binding site
conformation
Monte Carlo Simulation (MC)

Generate paths corresponding to potential motions
of the ligand and protein:
1.
2.
3.

Select initial conformation of interest
Sample new conformation according to move set
Accept or reject new conformation based on energy
difference with original
Drawbacks:


Only generates one simulation path at a time
Can get stuck in local minima of the energy function
(repeatedly sampling many similar conformations)
Stochastic Roadmap Simulation (SRS)
 Example conformation representation - ligand and
protein parameters specified as vector (1, 2, …, d)


Ligand parameters – 3D coordinates of one atom,
torsional angles of remaining atoms
Protein parameters – backbone torsional angles
 Conformational parameters determine interaction
between atoms of molecule and between molecules
and the medium (ie. van der Waals, electrostatic)
 Assumes that interactions are described by an
energy function that depends only on the
conformation of the molecules
SRS Roadmap
 Encode many pathways as a directed graph

Node = conformation with each i sampled randomly from
allowed range according to some distribution


Find nearest neighbors using some metric (ie. RMS,
Euclidean Distance)
Edge = probability of the molecules transitioning from a
node i to one of its neighbors j:


ij  


P
1
Ni
e
Eij / k Bt
1
Ni
if Eij  0
otherwise
Pii  1   Pij
i j
 Roadmap contains many simultaneous MC paths

Can get individual MC path by starting with top node,
choosing successor node at random according to edge
probabilities (note: you never need to do this with SRS)
SRS Roadmap (cont.)
A
E
B
F
C
PAA
A
A
PAB
B
G
PCC
PBB
C
PAC
B
PBD
C
PBE
PCF
PCG
D
D
E
F
G
D
E
F
G
SRS - Properties
 Implicitly defines a Markov chain that
captures stochastic nature of molecular
motion

Markov property: probability of where the
system will go next depends only on its current
states, not where it has been
 Doesn’t suffer from MC’s drawbacks
 No local minimization problems
 Orders of magnitude speed-up (can process
paths simultaneously in closed form using
linear algebra methods)
Escape Time
 Measure of binding affinity (expected number of MC
simulation steps for ligand to escape the “funnel of
attraction” of the protein’s binding site)

Longer escape time = high energy barriers around
catalytic site
 Averaged over many molecular motion pathways
 Naïve approach – perform many simulation runs on
roadmap (start from potential bound conformation, end
when ligand escapes from funnel), average number of
steps taken in each run
 Slow, and only provides estimate of escape time!
Escape Time (cont.)
 Better solution – use first-step analysis (from Markov
chain theory)

Each of the neighboring nodes is either:
 In the funnel (expected number of further steps =
that node’s escape time)

Not in the funnel (stop)
 i  1


 Pij  0 
jF  i 
 Pij  j
jF  i 
F
I
PIJ1 PIJ2
J1
J2
Can define a linear system with one equation for each
roadmap node, solve all escape times simultaneously
Very fast, and computes escape times exactly!
Ligand-Protein Modeling
 Proteins rigid, ligand flexible
 One atom in ligand designated as the base with 5
DOF, each additional atom had 1 torsional DOF

Bonds in a ring were rigid, no DOF
 Bond angles and lengths assumed constant
 Potential function used to calculate free energy
incorporated electrostatic, van der Walls, and
solvation free energies (resolution of 1 Å or 0.5 Å)
 Modeled solvent with dielectric of 80, solute with
dielectric of 1
 Repeated experiments with 6 Å and 8 Å funnels,
obtained similar results
Study 1 - Effects of Mutations




Site-directed mutagenesis – biological method in
which a few amino acids are deleted, replaced by
other amino acids, or have their side chains altered
Computational mutagenesis – same as above, but
using computers (faster/easier, but less accurate)
Lactate dehydrogenase
(LDH) – catalyzes reduction
of pyruvate to lactate when
bound to NADH
Mutated residues near LDH’s
catalytic site (computational
mutagenesis), observed
effects on binding affinity (via
escape time)
Study 1 – Mutations
 His193Ala, Arg106Ala, both of these together
 Cause large reduction in energetic structure of active
site
 Show sensitivity of SRS to coarse changes in system
 Asp195Asn, Gln101Arg, Thr245Gly
 Cause small or no reduction in energetic structure of
active site
 Show sensitivity of SRS to fine changes in system
 Generated roadmaps contained 4,000 nodes
sampled over whole conformation space, 100 extra
nodes sampled around bound conformation
 Other sampling schemes gave similar results
Study 1 – Results
=

Study 2 – Distinguishing Catalytic Site
 Shape and electrostatic complementarity between
catalytic site and ligand  tight bond
 Singh et al. (1999) – Studied 3 different ligand-protein
complexes:


Bound state energy not good at discriminating catalytic
site from other putative sites
Instead compared average path weight of most
energetically stable paths entering and leaving the
sites  energy barrier around catalytic site
 Study expands on this idea:
 SRS/first-step analysis measures whole energy barrier,
not just small part corresponding to most feasible paths
 Escape time more precise than average path weight
Study 2 - Methods
 Each test conducted with true bound
conformation and 4 other putative
conformations with:


Lowest energies, close to protein surface
(< 5 Å), and distant from each other (> 10 Å)
20 roadmaps/complex, each with set of
random conformations and 100 extra
conformations around each putative binding
site conformation
 Took < 4 mins. to generate roadmaps, and
< 4.5 mins to compute escape times on
desktop computer
Study 2 - Results
Summary


Can compute escape time efficiently with SRS to
analyze ligand-protein interactions
Study 1 – showed high sensitivity of SRS by
performing computational mutagenesis on catalytic
site of protein


In all 6 cases, SRS simulation results agreed with
expected biological interpretation of mutation
Study 2 – used escape time as metric to distinguish
catalytic site from 4 other putative sites

In 5/7 cases, escape time distinguished the catalytic
site by over 2 orders of magnitude difference from
other putative binding sites
Future
 SRS is a general tool, and could be used to
efficiently compute other interesting metrics in
addition to escape time (binding time, total
energy difference along binding paths, etc.)
 Combine SRS with other techniques to model
simultaneously interactions of many
molecules (current representation only
models one ligand-protein complex)
Discussion Questions
 What are some explanations for why the
escape times of the putative sites were higher
than the catalytic site in study 2’s failed
cases?
 The paper showed that escape time could be
useful in distinguishing the catalytic site.
What are other possible applications of
escape time?
 Did the way in which they modeled ligands
and proteins affect the results of the studies?