Energy landscapes and folding dynamics

Lorenzo Bongini

Dipartimento di Fisica Universita’di Firenze

Bongini@fi.infn.it

Introduction

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

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

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

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: T, Tf, Tg

Analyzed sequences

• S0 hydrophobic omopolymer

• S1 good folder

• S2 bad folder

Native Configurations

Spontaneous folding

In contact with a thermal bath (Langevin or Nosè-Hoover dynamics) the system folds spontaneusly in a temperature range

Good and bad folders

Also a good folder can reach its NC but spends there just a small fraction of its time

Dynamical Stability

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)

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.

Searching for saddles

• 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

1. We build a database of local minima of the potential

2. For every pair of minima A and B we build and intermediate configuration

3. We apply to C a steepest descent untill we reach a minimum

• 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 isn’t already there.

4. If the distance between C and C’ is lower than a threshold we stop. C and C’ are on the “RIDGE”, the stable manifold that divides the attraction basins of A and B

5. We start two steepest descent form C and C’ monitoring their distance while they “colano” along the ridge. If their distance passes the threshold we go back to 3.

6. When the gradient gets 0 we are in the saddle. We refine it with Newton.

Transition rates

Comparison of the numerically determined transition rates and the Langer estimate:

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)

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

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•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)

Metric properties of the EL

Directly connected minima are generally “near” to each other

Connections between minima near to the

NC are in general shorter.

near in this case has a very general meaning,

both metric and topologic

Let’s call the N-th shell the set of all minima separated by the NC by at least N saddles

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 doesn’t show any connection with the folding propensity of a sequence?

Because we didn’t take in to account the dynamical weights of the connections.

homopolymer

good folding eteropolymer

Long jumps are much more probable for the good folding sequence

HENCE

The mobility over the landscape is higer

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.

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?

The connectivity graph is scale free

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

Inserting dynamical weights

Circles = homoplymer

Squares = bad folder

Crosses = good folder

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

Circles = homoplymer 7.1

Squares = bad folder 2.5

Crosses = good folder 2.9

Inserting dynamical weights

Squares = bad folder

Crosses = good folder

The weighted laplacian matrix governs the master equation. Therefore the smaller its eigenvalues the slower the dynamics

Tentative conclusions

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

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

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

The reduced modelBy switching off Lennard-Jones interactions one gets a

model characterized by a discrete energy spectrum

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

The landscape steepness also depends on the

number of hydropobic interactions

The deformation of the energy landscape depend on a parameter independent of the sequence details the relative strenght of hydrophobic interactions

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