Energy landscapes and folding dynamics

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Energy landscapes and folding dynamics

• Lorenzo Bongini
• Dipartimento di Fisica Universitadi Firenze
• Bongini_at_fi.infn.it

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Introduction

• In several models folding times have shown to
  correlate with equilibrium quantities such as the
  difference between the temperatures of the
  folding and q transitions. How comes? How does
  the shape and topology of the energy landscape
  influence protein folding?

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Summary

• Folding dynamics and thermal activation
• A metric description of the energy landscape
• Topological properties of the connectivity graph
• Short range versus long range interactions

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A simple model

• 2-dimensional off-lattice model
• 2 kinds of amminoacyds hydrophobic and polar
• Harmonic potential between consecutive
  amminoacyds
• Effective potentials of Lennard-Jones kind to
  mimic the solvent effect.
• Angular potential to introduce a bending cost

F. H. Stillinger, T. H. Gordon and C. L.
Hirshfeld, Phys. Rev. E 48 (1993) 1469
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Achievements of the model

• Reproduces spontaneus folding
• Allows to distinguish between sequences with a NC
  more stable and less stable (good and bad
  folders)
• Reproduces the 3 transition temperatures Tq, Tf,
  Tg

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Analyzed sequences
Native Configurations

• S0 hydrophobic omopolymer
• S1 good folder
• S2 bad folder

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Spontaneous folding
In contact with a thermal bath (Langevin or
Nosè-Hoover dynamics) the system folds
spontaneusly in a temperature range
d
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Good and bad folders
Dynamical Stability
Also a good folder can reach its NC but spends
there just a small fraction of its time
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Verifying the funnel hypothesis
The energy funnel is steeper for good folders,
but this just speaks of the equilibrium
L. Bongini, Biophys. Chem. 115/2-3, 145-152
(2004)
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What is the dynamic while changing of minimum?
Both energy and configurational distance
undergo abrupt changes upon jumps between basins
of attraction of different minima. This suggests
a thermally activated barrier jump.
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Searching for saddles

• We build a database of local minima of the
  potential
• For every pair of minima A and B we build and
  intermediate configuration
• We apply to C a steepest descent untill we reach
  a minimum

• If the new minimum is A we build a new
  configuration C intermediate between C and B and
  we go back to 3
• If the new minimum is B we build a new
  configuration C intermediate between C and A
  andwe go back to 3

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• If the new minimum is neither A nor B the two
  minima are not directly connected and we stop
  investigating their connection. We add the new
  minimum to the database if it isnt already
  there.

• If the distance between C and C is lower than a
  threshold d we stop. C and C are on the RIDGE,
  the stable manifold that divides the attraction
  basins of A and B
• We start two steepest descent form C and C
  monitoring their distance while they colano
  along the ridge. If their distance passes the
  threshold d we go back to 3.
• When the gradient gets 0 we are in the saddle. We
  refine it with Newton.

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Transition rates
Comparison of the numerically determined
transition rates and the Langer estimate
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What causes the discrepancies?

• Discrepancies increase with temperature
• As temperature increase they get correlated with
  the inverse of the Hessian determinant in the
  saddle (the smaller the coefficents of the second
  order term in the potential expansion the higher
  the discrepancy)

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Higher Order Estimates
Langer estimate is second order in the potential
O. Edholm and O. Leimar, Physica 98A (1979) 313
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Conclusions

• Jumping dynamics between different inherent
  minima is a thermal activation process both above
  and below the folding temperature
• then
• Folding towards the NC is not due to a change in
  dynamics but seems to be related to the different
  accessibility of the NC at different temperatures.

L. Bongini, R. Livi, A. Politi, A. Torcini, Phys.
Rev. E 68, 61111 (2003)
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Metric properties of the EL

• Directly connected minima are generally near to
  each other

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• Connections between minima near to the
• NC are in general shorter.

near in this case has a very general meaning,
both metric and topologic
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• Lets call the N-th shell the set of all minima
• separated by the NC by at least N saddles

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• It seems then that there exists a sort of
  entropic funnel, in the sense that the nearer to
  the NC the easiest it is to reach it
• How comes that this property doesnt show any
  connection with the folding propensity of a
  sequence?
• Because we didnt take in to account the
  dynamical weights of the connections.

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Long jumps are much more probable for the good
folding sequence

• homopolymer

HENCE The mobility over the
landscape is higer
good folding eteropolymer
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Conclusions

• The metric properties of the landscape seem
  insufficient to explain the folding propensity of
  a sequence
• Taking in to account the dynamical properties
  shows instead that good folding sequences are
  characterized by an higher mobility in the EL.

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Topological properties of the connectivity graph

• If folding dynamics can be summarized as non
  linear oscillations around minima interlaced with
  thermally activated jumps between their basins of
  attraction, then the folding problem can be
  rephrased in terms of a diffusion over the
  connectivity graph a graph whose nodes are minima
  and whose edge is a dynamical connection (i.e. a
  first order saddle). Edges are weighted by the
  jumping rates.
• How do connectivity graphs of sequences with
  different folding propensities differ?

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The connectivity graph is scale free

• The connectivity of all sequences decays with
• the same exponent ( from 3.02 to 2.76)

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Inserting dynamical weights

• Circles homoplymer
• Squares bad folder
• Crosses good folder

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The spectral dimension

• For graphs defined over a regular lattices it
  is well known that dynamics properties as mean
  first passage and return times depend on the
  lattice dimension
• The spectral dimension allows to extend these
  results to irregular graphs
• spectral dimension twice the exponent of the
  scaling of the low eigenvalues of the laplacian
  matrix

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Circles homoplymer 7.1
Squares bad folder
2.5 Crosses good folder 2.9
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Inserting dynamical weights
The weighted laplacian matrix governs the
master equation. Therefore the smaller its
eigenvalues the slower the dynamics

• Squares bad folder
• Crosses good folder

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Tentative conclusions

• Topological properties do not seem to provide
  tools to distinguish usefull tools to distinguish
  between good and bad folding connectivity graphs.

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Short versus long range interactions

• In the framework of a more realistic model we
  investigate the interplay between short range
  interaction (typically hydrogen bonds,
  responsible of the secondary structure) and long
  range ones (hydrophobic interactions) in order to
  understand how they contribute to the shape of
  the energy landscape of a protein

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The Model

• it is a 3 letter off lattice coarse grained model
  (Veitshans et al. 1997)
• the angular contribution is set so to force an
  average value of 105 degrees between consecutive
  beads
• there is a dihedral contribution

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The reduced model

• By switching off Lennard-Jones interactions one
  gets a model characterized by a discrete energy
  spectrum

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Switching on hydrophobicitycauses a spread of
the energy levels

• The spread is higher for higher for energies
• The spread depends on the number of
  hydrophobic interactions

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The landscape steepness also depends on the
number of hydropobic interactions
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• The deformation of the energy landscape depend
  on a parameter independent of the sequence
  details the relative strenght of hydrophobic
  interactions

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Conclusions

• The flattening of the landscape and the creation
  of very low minima (kinetic traps) explain why
  folding is slower than the formation of secondary
  structure motifs
• Relevant (as far as the folding propensity is
  concerned) point like mutations are those
  strongly altering the molecule hydrophobicity