Semenov Institute of Chemical Physics, RAS

New results in applications of p-adic pseudo-differential equations to the protein dynamics

Vladik Avetisov Albert BikulovSergey KozyrevVladimir Osipov

Alexander Zubarev in cooperation with

Viktor Ivanov and Alexander Chertovich

What is a protein?

“Protein? It is very simple” - biologist will say. - Protein is a well folded polymeric chain of a few hundreds amino aside residues.

Proteins looks like “nono-pocket devices” constructed from helixes and sheets. They are fabricated in a cell to provide all biochemical reactions including the protein fabrication too.”

“Nothing of the kind!” – physicist will rejoin.- “Protein looks like an amorphous “nono-drop” consisting of a few thousands closely interacting atoms. There is no symmetry here. O-o-o…! Protein is too complex to be described by a simple way.”

- C carbon atoms, - O oxygen atoms, - N nitrogen

atoms, - hydrogen atoms

are not shown

Figure 5. Hierarchical arrangement of the conformational substates in myoglobin. (a) Schematized energy landscape. (b) Tree diagram. G is Gibbs’ energy of the protein, CC(1-4) are conformational coordinates.

Hans Frauenfelder, in Protein Structure (N-Y.:Springer Verlag, 1987)

p.258.

“The results sketched so far suggest two significant properties of substates and motions of proteins, nonergodicity and ultrametricity.”

Hans Frauenfelder, in Protein Structure (N-Y.:Springer Verlag, 1987)

p.258.

Hans Frauenfelder was first who drown ultrametric tree for the protein states to underline the protein complexity.

«In <…> proteins, for example, where individual

states are usually clustered in “basins”, the

interesting kinetics involves basin-to-basin

transitions. The internal distribution within a basin

is expected to approach equilibrium on a relatively

short time scale, while the slower basin-to-basin

kinetics, which involves the crossing of higher

barriers, governs the intermediate and long time

behavior of the system.”

O.M.Becker and M.Karplus.

J.Chem.Phys. 106, 1495 (1997)

“What does ultrametricity mean physically in protein dynamics?”

This means that the protein dynamics is characterized by a hierarchy of time scales.

Given such picture, we will take an interest to Frauenfelder’s

question,

“Are proteins ultrametric?”

p-Adic mathematics gives us natural tools to try to find an answer.

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Historically, at the beginning, we have suggested that the protein dynamics can be

modeled by a random walk over a hierarchy of embedded basins of states…,

and so, the protein dynamics can be described by the р-adic pseudo-differential equation of ultrametric

diffusion

p

ydtxftyfyxAt

txfp

Q

)(),(),(||),( 1

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activation of masuringfor unit scale a is e,temperatur-( /~

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states;protein theof space cultrametrian is

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p

p

In protein dynamic applications, this equation is interpreted as the muster

equation for the transitions between the protein states.

Assuming all this, we have tried to show that our approach is relevant to

observable features of the protein dynamics.

<If you said “A”, do not be “B”>.

Vitalii Goldanskii

h

experiment: kinetics of CO rebinding to myoglobine

(H. Frauenfelder group, since the 1970s)

Mb-COCO

Mb

Measured quantity :.The total concentration of the Mb unbounded to CO.

Mb-CO

rebinding CO to Mb

CObreaking

of chemical

bound Mb-CO

Mb*

strained confor-

mational state

Conformational rearrangements of the Mb

Mb1

active confor-

mational state

laser pulse

p-adic model of CO rebinding kinetics

xdtxftSrB

,

measured quantity:

Zp

reaction sink

Initial distribution

Br

Protein “diffuses” over unbounded

states

ultrametric space of the protein

undounded states

ultrametric diffusion

reaction sink due to CO binding to Mb

protein leaves unbounded

states

experiment and theory

T1>T2>T3

O O

Good agreement between ultrametric model and experiment certainly supports an idea

that protein dynamics possesses ultrametricity (!)

This was a pioneer experience in applications of p-adic pseudo-differential equation to the protein dynamics: it was presented on the First Conference on p-

Adic Mathematical Physic (Moscow, 2003)

Now, I will present new experience in applications of the same p-adic equation to drastically different phenomenon related to the protein dynamics.

This is the spectral diffusion in proteins.

Spectral diffusion in proteins

a chromophore marker

protein

1. A chromophore marker is injected into a protein, then the protein is frozen up to a few degrees of Kelvin, and the adsorption spectrum is measured.

At low temperature this spectrum is very wide, due to different arrangements of the protein atoms around a chromophore in individual protein molecules.

2. Then, a set of chromophore markers are burned using short light impulse at a particular absorption frequency, and a narrow hole is arisen in the absorption spectrum.

3. Then, the hole wide is monitored during the time. Because proteins with unburned markers “diffuse” over the protein state space, the hole is broadening and covering with time.

Thus, spectral diffusion phenomenon is directly coupled with the protein dynamics.

Spectral diffusion features

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“Weighting time” experiments:

The “weighting time”, tw , starts immediately after the burning of a hole, i.e. it is the current time for spectral diffusion:

“Aging time” experiments:

The “aging time”, tag , is the time interval between protein freezing and hole burning.

07.04 ~min10, agwag ttt

Spectral diffusion broadening obeys the power law with an exponent drastically smaller then in familiar diffusion.

The aging time, tag, grows, spectral

diffusion broadening becomes slower .

How protein dynamics is coupled with spectral diffusion

(tw )

tw

A marker absorption frequency changes randomly only at those instants when protein gets very peculiar area of the protein state space. This area relates to a few degrees of freedom of the closest neighbors of the chromophore marker.

Description of spectral diffusion in proteins

1. Calculation of the distribution of the first passage (first reaching) time P() for ultrametric diffusion

1,~,121

agag ttP

ydyx

txftyf

t

txf

pQ p

1||

,,,

2. Construction of the random walk on a “frequency line”, which has random delays with the distribution P()

1,, 2

1122

1

wagwag tttt

note, that the first reaching time depends on the aging time tag

p-adic equation for the protein dynamics

Spectral diffusion broadening:(analytical description and computer simulation)

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2,2

ydyx

txftyf

t

txf

pQ p

1||

,,,

0,2

w

Spectral diffusion aging:(analytical description and computer simulation)

2,2

07,0 agag tt

The exponents in the power laws of spectral diffusion broadening and aging are determined only by a degree of Vladimirov’s pseudo-differential operators (2) (!)

ydyx

txftyf

t

txf

pQ p

1||

,,,

Protein is a complex object.

However, the protein dynamics is described by simple p-adic equation.

Conclusion:

Probably, p-adic pseudo-differential equations will be as much important to keep paradoxical union of order and randomness in the “biological mechanics”, as Newton’s equations are in classical mechanics.