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Dynamics of Protein Molecules:
Modeling and Applications
Huan-Xiang Zhou
Physics/CSIT/IMB
7/28/2012
Model I: General
• Each atom is modeled as a mass point, with
mass mi, position ri, and velocity vi.
• The atoms interaction with each other, with
a potential energy U({ri}).
• Motion of each atom is governed by
Newton’s equation:
midvi/dt = Fi = -iU({ri})
Model II: Harmonic Oscillator
• Molecule with two atoms, at r1 and r2.
• Interact with a potential U(r) = ½kr2, where
r = |r1 - r2|.
• Only non-trivial motion is along r,
mdv/dt = F = -kr
1.5
U 1.5 r
1
1
0.5
0
0 0.2 0.4 0.6 0.8 1 1.2
0.5 t
-0.5
-1
0
-1.5 -1 -0.5 0 0.5 1
r 1.5
-1.5
r = Asin(wt)
w2 = k/m
Energy and Temperature
• Potential energy: U = ½kr2 = ½kA2sin2(wt).
• Kinetic energy: K = ½mv2 =
½mw2A2cos2(wt) = ½kA2cos2(wt).
• Total energy E = K + U = ½kA2.
• <K> = ¼kA2 = ½kBT, E = ½ kA2 = kBT. The
amplitude of oscillation is determined by the
temperature.
Model III: Ethane
• Construct an energy function:
• Bonded terms: f
½∑kb(b- b0)2 + ½∑kq(q- q0)2
• Torsion terms:
½∑Vf[1+cos(nf-d)]
• Nonbonded interactions:
∑(A/r12-B/r6) + ∑C/r
½Vf[1+cos(3f)]
Model III: Ethane
• Molecular dynamics: if position is x0 and
velocity is v0 at time t, what will position
and velocity be a short internal Dt later?
• Position at time t+dt can be expanded:
x(t+Dt) = x0 + (dx/dt)Dt + ½(d2x/dt2)Dt2
= x0 + v0Dt + ½(F/m)Dt2
• Solvent is an integral part of the system.
Applications
• Functional dynamics – AchE
• Zhou et al. (1998), Proc. Natl. Acad. Sci. USA 95, 9280-9283.
• Protein folding – Trp cage
• Simmerling et al. (2002), J. Am. Chem. Soc. 124, 11258-
11259.
• Snow et al. (2002), J. Am. Chem. Soc. 124, 14548-14549.
• Chowdhury et al. (2003), J. Mol. Biol. 327, 711-717.
• Protein-protein interactions – Src kinases
• Young et al. (2001), Cell 105, 115-126.
Applications
• Functional dynamics – AchE
• Zhou et al. (1998), Proc. Natl. Acad. Sci. USA 95, 9280-9283.
• Protein folding – Trp cage
• Simmerling et al. (2002), J. Am. Chem. Soc. 124, 11258-
11259.
• Snow et al. (2002), J. Am. Chem. Soc. 124, 14548-14549.
• Chowdhury et al. (2003), J. Mol. Biol. 327, 711-717.
• Protein-protein interactions – Src kinases
• Young et al. (2001), Cell 105, 115-126.
Acetylcholinesterase
• breaks acetylcholine
and allows for rapid
transmission of
neural signals
Transimission of Neural Signals
Ach as a neural
transmitter was
first discovered
by Otto Loewi,
who won the
Nobel Prize in
1936.
Neural Signals
Rapid breakdown
of Ach is essential!
Burst of Ach Breakdown of Ach
Apparent Poor Choice of Structure
• Active site is buried
deeply in the center,
with a ring of
aromatic groups as a
gate.
Fundamental Questions
• How could rapid breakdown be achieved?
• Is there any advantage in the gated channel?
Zhou et al., PNAS (1998)
Dynamic Gate
2.6
2.4
2.2
2.0
s (A)
o
1.8
1.6
1.4
1.2
0 200 400 600 800
t (ps)
Fraction of Open State
1.0
0.8
0.6
po
0.4
2.4%
0.2
0.0
1.0 1.5 2.0 2.5 3.0
o
s (A)
Gate Appears Open to Intended Substrate
~ ps ~ ns
• but is effectively closed for slightly larger
ligand. Dynamic gate provides mechanism for
enzyme specificity!
Implications
• Dynamics is essential for the proper
functioning of AchE and many other
proteins.
• Dynamics can be exploited to achieve
selectivity.
Applications
• Functional dynamics – AchE
• Zhou et al. (1998), Proc. Natl. Acad. Sci. USA 95, 9280-9283.
• Protein folding – Trp cage
• Simmerling et al. (2002), J. Am. Chem. Soc. 124, 11258-
11259.
• Snow et al. (2002), J. Am. Chem. Soc. 124, 14548-14549.
• Chowdhury et al. (2003), J. Mol. Biol. 327, 711-717.
• Protein-protein interactions – Src kinases
• Young et al. (2001), Cell 105, 115-126.
Gellman & Woolfson, NSB (2002)
Experimental Study
Hagen group, JACS (2002)
Simulation Studies
• Simmerling et al. (2002), J. Am. Chem. Soc. 124,
11258-11259.
• Snow et al. (2002), J. Am. Chem. Soc. 124, 14548-
14549.
• Chowdhury et al. (2003), J. Mol. Biol. 327, 711-717.
Simulation Details: Common
• AMBER force field, with solvent implicitly treated by
the Generalized Born model.
• MD simulations up to 100 ns.
Simulation Details: Specific
• Simmerling et al
– Simulations were carried out before NMR structure
determination.
– Done at 325 K on a Linux/Intel PC cluster.
• Pande and co-workers
– Done through Folding@Home distributed computing
– Aggregated simulation time of 100 ms, consisting of 1000’s of
simulations up to 1 ns and 100’s of simulations up to 80 ns.
• Duan and co-workers
– 100-ns simulation on a Pentium III PC.
Results: a Critique
• Simulations show RMSDs decrease with simulation
time.
• However, little insight is gained on protein folding.
Contact Formation Model
• Folding occurs through
accumulation of native contacts.
• Different segments of the protein
chain come into contact through
diffusion.
• Contact forms if both side chains
are in the “correct”
conformations.
Zhou, JCP (2003)
Why Is Folding Fast?
• The fraction of time the two side chains are
simultaneously in their respective “correct”
conformations is small.
• However, inter-conversion between correct and
incorrect occurs on a ps time scale, whereas chain
diffusion occurs on a ns time scale.
• Rapid local dynamics contributes to fast folding.
Applications
• Functional dynamics – AchE
• Zhou et al. (1998) Proc. Natl. Acad. Sci. USA 95, 9280-9283.
• Protein folding – Trp cage
• Simmerling et al. (2002) JACS 124, 11258-11259.
• Snow et al. (2002) JACS 124, 14548-14549.
• Chowdhury et al. (2003) JMB 327, 711-717.
• Protein-protein interactions – Src kinases
• Young et al. (2001) Cell 105, 115-126.
Critique
• Detailed simulations provide hints and insights to the
type of motion required for kinase activation.
• However, quantitative link between simulation and
experiment is not yet possible.
• Room for simpler models!
Balls Connected by Tether
Potassium Channel
inactivation ball
Tether Is Essential
• Inactivation has to be rapid (~ 1 ms).
• Without tether, inactivation domain must be present at
mM level.
• Dynamics of tether is fully exploited in channel
opening and closing.
Zhou JPC (2002)
