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

1
New results in applications of p-adic
pseudo-differential equations to the protein
dynamics Vladik Avetisov Albert Bikulov Sergey
Kozyrev Vladimir Osipov Alexander Zubarev in
cooperation with Viktor Ivanov and Alexander
Chertovich
2
What is a protein?
3
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.
4
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

5
Hans Frauenfelder was first who drown ultrametric
tree for the protein states to underline the
protein complexity.
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.
6
What does ultrametricity mean physically in
protein dynamics?
In ltgt 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)
This means that the protein dynamics is
characterized by a hierarchy of time scales.
7
Given such picture, we will take an interest to
Frauenfelders question, Are proteins
ultrametric? p-Adic mathematics gives us
natural tools to try to find an answer.
8
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,
9
and so, the protein dynamics can be described by
the ?-adic pseudo-differential equation of
ultrametric diffusion
In protein dynamic applications, this equation is
interpreted as the muster equation for the
transitions between the protein states.
10
Assuming all this, we have tried to show that our
approach is relevant to observable features of
the protein dynamics.
ltIf you said A, do not be Bgt. Vitalii
Goldanskii
11
experiment kinetics of CO rebinding to
myoglobine (H. Frauenfelder group, since the
1970s)
Measured quantity . The total concentration of
the Mb unbounded to CO.
laser pulse
12
p-adic model of CO rebinding kinetics
Protein diffuses over unbounded states
ultrametric diffusion
Zp
Br
reaction sink
reaction sink due to CO binding to Mb
ultrametric space of the protein undounded states
measured quantity
protein leaves unbounded states
13
experiment and theory
T1gtT2gtT3
Good agreement between ultrametric model and
experiment certainly supports an idea that
protein dynamics possesses ultrametricity (!)
14
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)
15
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.
16
Spectral diffusion in proteins
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.
a chromophore marker
protein
17
Spectral diffusion features
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
Spectral diffusion broadening obeys the power law
with an exponent drastically smaller then in
familiar diffusion.
Aging time experiments The aging time, tag
, is the time interval between protein freezing
and hole burning.
The aging time, tag, grows, spectral diffusion
broadening becomes slower .
18
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.
19
Description of spectral diffusion in proteins
p-adic equation for the protein dynamics
1. Calculation of the distribution of the first
passage (first reaching) time P(?) for
ultrametric diffusion
note, that the first reaching time ? depends on
the aging time tag
2. Construction of the random walk on a
frequency line, which has random delays ? with
the distribution P(?)
20
Spectral diffusion broadening (analytical
description and computer simulation)
21
Spectral diffusion aging (analytical description
and computer simulation)
w
The exponents in the power laws of spectral
diffusion broadening and aging are determined
only by a degree ? of Vladimirovs
pseudo-differential operators (??2) (!)
22
Conclusion
Protein is a complex object. However, the
protein dynamics is described by simple p-adic
equation.
23
Probably, p-adic pseudo-differential equations
will be as much important to keep paradoxical
union of order and randomness in the biological
mechanics, as Newtons equations are in
classical mechanics.