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Semenov Institute of Chemical Physics, RAS New results in applications of p-adic pseudodifferential equations to the protein dynamics Vladik Avetisov Albert Bikulov Sergey Kozyrev Vladimir 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 “nonopocket 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 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. 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. “What does ultrametricity mean physically in protein dynamics?” «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) 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. 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…, w3 w2 w1 w3 w2 w4 w3 w1 w2 w1 w1 w2 w3 w4 ... are thetransitions rates and so, the protein dynamics can be described by the рadic pseudo-differential equation of ultrametric diffusion f ( x, t ) A | x y |p 1 f ( y, t ) f ( x, t )d ( y) t Qp Q p is an ultrametric space of theproteinstates; f x, t is a populationdensityin a statex at instantt ; A | x y |p 1 is a transition ratebetween statesx and y; ~ T0 / T (T - temperature, T0 is a scale unit for masuringof activation barrierson theenargylandscape). 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 experiment: kinetics of CO rebinding to myoglobine (H. Frauenfelder group, since the 1970s) Measured quantity :. strained conformational state The total concentration of the Mb unbounded to CO. Mb* Mb1 CO breaking of chemical bound Mb-CO Mb Mb-CO h CO laser pulse active conformational state rebinding CO to Mb Mb-CO p-adic model of CO rebinding kinetics Protein “diffuses” over unbounded states ultrametric diffusion Zp Br reaction sink Initial distribution reaction sink due to CO binding to Mb ultrametric space of the protein undounded states measured quantity: S t f x, t d x Br protein leaves unbounded states experiment and theory O O T1>T2>T3 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 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. protein Thus, spectral diffusion phenomenon is directly coupled with the protein dynamics. 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: tw t w 2 2 1/ 2 ~ t w0, 27 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. tag , tw 104 min ~ tag0.07 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 f x, t f y , t f x, t d y 1 t | x y |p Qp 1. Calculation of the distribution of the first passage (first reaching) time P() for ultrametric diffusion 2. Construction of the random walk on a “frequency line”, which has random delays with the distribution P() p-adic equation for the protein dynamics Ptag , ~ tag 1 2 1 , 1 note, that the first reaching time depends on the aging time tag 1 2 2 1 t ag , t w t ag 1 2 tw , 1 Spectral diffusion broadening: (analytical description and computer simulation) 2,2 tw ~ tw0,27 f x, t f y , t f x, t d y 1 t | x y |p Qp Spectral diffusion aging: (analytical description and computer simulation) 2,2 tag tag0,07 2,0 w 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) (!) Conclusion: Protein is a complex object. However, the protein dynamics is described by simple p-adic equation. f x, t f y , t f x, t d y 1 t | x y |p Qp 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.