Chemical-to-Mechanical Energy Conversion in Biomacromolecular

J. Phys. Chem. B 2008, 112, 8319–8329

8319

Chemical-to-Mechanical Energy Conversion in Biomacromolecular Machines: A Plasmon and Optimum Control Theory for Directional Work. 1. General Considerations Evgeni B. Starikov,†,‡ Itai Panas,§ and Bengt Norde´n*,§ Institute for Nanotechnology, Research Center Karlsruhe, Post Box 3640, D-76021 Karlsruhe, Germany, and Department of Physical Chemistry, Chalmers UniVersity of Technology, SE-412 96 Gothenburg, Sweden ReceiVed: February 22, 2008; ReVised Manuscript ReceiVed: April 10, 2008

To rationalize coherence and mechanochemical aspects of proteins acting as molecular machines, a plasmon concept for dealing with protein nonequilibrium dynamics is introduced and tested with respect to thermodynamic consistency. A stochastic optimum-control theory for protein conformational diffusion is developed and the corresponding stochastic Newton’s second law derived for optimum-controlled conformational diffusion in proteins. The plasmon concept is shown to be consistent with this theory, in that optical plasmons can pump entropy out of (or into) the protein, decreasing (or increasing) its conformational diffusion and, at the same time, help decrease intra- and intermolecular friction, as well as (potentially) break the symmetry of the latter. Instead, acoustic plasmons may break the spatial symmetry of a protein’s “potential of mean force”, thus converting it into an effective Brownian ratchet potential by applying quasistatic deformational corrections to the former. These concepts seem to be of rather general applicability and might also be useful when studying, for example, intercalation of cationic dyes into DNA duplexes, positively charged oligopeptide transduction through cell membranes, or even DNA translocation through nanopores. I. Introduction paper,1a

In a recent Ross proposes an ingeniously simple, direct molecular mechanism of gaining mechanical work from the chemical energy of enzymatic adenosine triphosphate (ATP) hydrolysis to explain the spectacular observation of ATP synthase acting as a biological motor that exhibits specific subunit rotations during ATP hydrolysis and synthesis.1b Indeed, the ATP4- + H2O f ADP3- + HPO42- + H+ reaction triggers a Coulombic repulsion between the reactant molecules ADP3and HPO42-, the kinetic energy of which is proposed to be converted by the enzyme into the desired mechanical work via collisions of the mutually receding anions with the neighboring amino acid residues. Meanwhile, such an approach does not disclose any mechanistic details of how the huge inertia of the enzyme as a whole is surmounted, causing the latter to carry out some directional mechanical work. Moreover, this approach does not take into account potentially important molecular details of the enzyme’s active site where the ATP hydrolysis takes place. For example, a friction-symmetry-breaking “Brownian ratchet” (as suggested recently after using a chiral-spring model of ATP synthase to describe the two mutually helically twisted γ peptide helices observed in the crystal structure1c–e) would exhibit one and the same rotational sense, irrespective of whether it is run in a motor mode (ATPase) or a chemical generator mode (ATP synthase). By contrast, if the enzyme would act close to equilibrium and the effect of increased ATP concentration is an incremental shift in conformational equilibria backward, the result should be that the sense of rotation of the motor is oppositite to that of the generator. Last, but not the least, the model1a treats molecular motors as inertial, conservative systems * Corresponding author. E-mail: [email protected]. † Institute for Nanotechnology. ‡ E-mail: [email protected]. § Chalmers University of Technology.

and does not properly take into account friction of any kind, which is, in fact, indispensable in the condensed state. Another interesting proposal has been advocated recently for the chemical-to-mechanical energy conversion in ATP-driven molecular motors.2a It is argued that the ATP hydrolysis ought to result in a significant decrease of the enzyme’s permanent dipole moment, which may generate an internal “electrical” torque via asymmetric dipole-dipole couplings between the active site of the enzyme and its hydration shell, and thus the main part of the desired mechanical work should be performed. Within such an approach, the presence of the enzyme’s active site during the ATP hydrolysis has been taken into accounts implicitly, at leastsbut the protein molecule itself plays only a passiVe role. On the other hand, a recent analysis2b indicates that most of the energy of the ATP hydrolysis is released in the dissociation into separate (ADP + Pi) fragments in water. This general electrostatic effect should then drive the protein conformational changes, according to the essential characteristics of the protein’s overall free energy landscape. Hence, qualitatively, the principal directionality of the enzymatic process involved can be thought of as being produced by the electrostatic free energy changes, so there is no need to introduce hypothetical protein/water dipoles, because the entire electrostatic free energy2b must be considered. In general, it is not yet completely clear for the present whether the ATP-driven protein motors work according to the “power stroke” or Brownian ratchet mechanism (see, for example, ref 2c and references therein). The models in refs 1a and 2a adhere to the former, whereas those of refs 1c–e adhere to the latter. We feel that these two standpoints deserve equally close attention, but then there is an urgent need for some unifying concept to bring the both of them under one and the same roof. Moreover, the main issue of the chemical-tomechanical energy conversion is not simply the onset of inertial motions (in the sense of ref 1a) but rather the conversion of potential energy into heat and entropy via some kind of

10.1021/jp801580d CCC: $40.75  2008 American Chemical Society Published on Web 06/21/2008

8320 J. Phys. Chem. B, Vol. 112, No. 28, 2008 collective excitations as a result of a charge generation process; this is just the phenomenon that has not yet been satisfactorily described in microscopic terms. Our present paper attempts to tackle the above posers. Since the enzyme molecule is very likely not only a passive source of inertia, but may actively participate in the chemical-tomechanical energy conversion process, we shall adopt a plasmaphysical viewpoint in order to deal with protein relaxation dynamics and statistical thermodynamics in response to electrostatic interactions within (or outside) the biomacromolecule. A similar approach was earlier used in the context of DNA polyanions, colloids, and dense polymer blends (see, for example, refs 3–5a). Here we also discuss possible implications of the proposed approach on molecular modeling techniques. II. Plasmon Representation of Protein Collective Dynamics With a “protein plasma” we do not mean a conventional plasma consisting of free electrons and ions/holes coupled by electrostatic forces. The nuclei and electrons in a protein, of course, are not completely free to move but are an integral part of the whole protein’s dynamical elastic framework. However, let us for a moment assume that the macromolecule may be considered composed of ionic elements, which are amino acid side chains bearing electrostatic charges (immersed into the polarizable medium consisting of other side chains, peptide groups, and solvent), and apply the laws of plasma physics. Of course, the amino acid side chains are molecular fragments whose masses are much greater than those of electrons and holes in semiconductors, as well as those of elementary atomic cations. If we consider that the singly positively charged amino acid side chains are lysine (molecular mass 128 amu) and arginine (molecular mass 156 amu), whereas those singly negatively charged are glutamine (molecular mass 129 amu) and asparagine (molecular mass 115 amu), we may neglect the differences in their masses in the zero approximation and use an average mass m of a charged protein residue, 132 amu, in what follows. We may ask, what is a typical plasma density, that is, concentration of charged residues in a protein molecule? From a statistical analysis on a large number of high-resolution protein structures deposited in the Protein Data Bank (the details of this statistical study will be reported elsewhere), we note a linear regression of the number of charged side chains in the protein crystal unit cells onto the pertinent unit cell volumes, corresponding to a rather high charged-residue concentration on the order of 0.001 Å-3 for all the proteins involved. Interestingly, the high density of charged residues without clear amino acid sequence conservation had earlier been pointed out for many different proteins (with enzymes among them, see, for example, refs 5b–f), but those works performed no quantitative analysis. With such a charged residues density value in mind, we may proceed with the physical characterization of the protein plasma including only formal side chain charges. Next, we must also specify the relative dielectric permittivity ε of the protein; let it be equal to 20, for the sake of consistency with previous work.1a We will now ask how the Coulomb interaction energy of the protein plasma relates to the “thermal quantum” kBT at room temperature. This relation is usually cast in form of the “plasma interaction parameter”, Γ ) e2/kBT (3/4πn)1/3, where kB is the Boltzmann constant, T the absolute temperature, e the electron charge, and n the concentration of the charged particles in the system. For our protein plasma, Γ ≈ 90, which not unexpectedly shows a huge prevalence of the potential energy (Coulomb interaction) over the kinetic energy (thermal motion). On the

Starikov et al. other hand, there should also be a powerful Debye screening of the electrostatics in proteins, with the Debye radius rD ) (kBTε ε0/e2n)1/2 ≈ 1.69 Å, where ε0 is the dielectric permittivity of vacuum. To this end, it should be noted that the mean distance between the charged particles in the protein plasma is jr ) (3/ 4πn)1/3 ≈ 6.2 Å, whereas at r > rD the Coulomb interaction can be safely neglected. To conclusively demonstrate the latter finding, we can also estimate the mean free path between two successive collisions, which is huge for the protein plasma, l ) (πn rD2)-1 ≈ 111 Å. Thus, proteins can safely be thought of as a collisionless nondegenerate plasma. An important feature of any plasma is its collective dynamical properties, which can be described in terms of acoustic (cooperative motion of both positive and negative charges) and optical (cooperative motion of all the negative charges with respect to all the positive ones) plasmons. For the protein plasma we may estimate the optical plasmon frequency as νp ) (e2n/ mε ε0)1/2 ∝ 1012 s-1. We note that the characteric time of the optical plasmons (picoseconds) is the same as that of the conventional elastic vibrations in proteins. Furthermore, νp is on the same order of magnitude as the universal frequency factor of the chemical transition state theory, kBT/h (h is Planck constant), at room temperature. Both of these results can be anticipated, since charged amino acid side chains are actually an integral part of the whole protein elastic network. It is thus tempting to assume that the communion of the charged residues in an enzyme plays a role of the “keyhole” that responds to the proper electrostatic changes in the enzymes’ active sites (or outside the enzyme molecule) by switching on nonequilibrium collective dynamics of the protein as a whole and herewith helps efficiently surmount the inertia of the whole huge enzyme molecule. The above assumption allows us to suggest a general physical procedure to deal with an enzymatic reaction, where any charge redistribution due to the chemical reaction in the enzyme’s active site ought to result in a generation of nonequilibrium optical plasmons in the whole enzyme molecule. Such a mechanism seems to be a general one, since charge redistributions resulting from charge/proton transferssfollowed by structural reorganizations of enzyme molecules as a wholesare ubiquitous in biochemical reactions.6–9 Further, collective vibrational modes in plasmas, either collisional or collisionless, are always damped. However, in collisionless plasmas, the dominating relaxation mechanism for their collective excitations to obey is Landau damping, whereby the plasmon loses its energy in accelerating separate charged particles, whose velocity is less than the phase velocity of the plasmon wave. Since every charged amino acid side chain is anyway involved into the whole protein elastic network, this is how the pertinent vibrational modes of the latter (which play direct catalytical/allosteric roles) might get excited. Indeed, the vibrational mode coupling/vibrational energy transfer is known to play an important role in enzyme molecular dynamics.10 In fact, the enzyme vibrations relevant to enzymatic action have long ago been quantified and analyzed by the dispersed polaron (spin boson) model (in a way inherently similar to our present approach), providing a lot of crucial information on the basic trends in the phonon spectra of enzymes.11 The importance of the acoustic plasmons in enzymatic reactions (although their energies are much less than those of optical plasmons) should not be underestimated as well. Such plasmons can be generated at virtually no additional energy cost, as they should correspond to the lowest-frequency vibrations, and their population above the Debye temperature will not be

Energy Conversion in Biomacromolecular Machines significantly dependent on temperature. Therefore, it is of interest to estimate Debye temperature of a typical protein, and we need to know the value range for sound velocity in it. The sound velocity is noticeably dependent on the protein structure, on the concentration of protein solutions, or on the relative humidity of protein crystals12–15 and may usually range from 1500 to 2000 m s-1. With this in mind, and taking into account the conventional density value for a protein molecule,16 the Debye temperature of a typical protein should range from 140 to 180 K, just in accordance with the experimental observations.17 This means that at room temperature the population of acoustic plasmons in a protein should practically be temperatureindependent. Moreover, as the charged residues are an integral part of the general elastic framework of a protein, acoustic plasmons should be intertwined with the acoustic vibrations of the protein as a whole, in that some nonequilibrium acoustic plasmons in a protein would trigger some amount of acoustic phonons beyond their equilibrium population at the given temperature. To this end, it is important to note that the understanding of enzymatic reactions draws a clear borderline between the socalled protein Vibrations, which occur on a (sub)picosecond time scale and promote lowering of the activation barrier, and protein motions, which are the lowest-frequency vibrations promoting an increase in the enzymatic reaction rate via dynamical recrossing of the activation barrier.18a However, here we have also to mention that the latter standpoint is heavily debated in the literature. Specifically, all the possible “promoting” modes were carefully analyzed and characterized already long ago6a and were shown just to help enzymes in reaching the corresponding reaction transition states but should neither lower potential barriers nor increase enzymatic reaction rates by barrier recrossing, since the transmission factor is basically the same in enzymes and solutions.18b,c Thus, we may hope that our combined acoustic + optical plasmons picture could in principle be potentially helpful in resolving these basic conceptual issues. Is there then some interconnection between the optical and acoustic plasmons in proteins? In plasma physics, whenever two motions with very different time scales are present, we can treat the high-frequency vibrations as a “gas of quasiparticles” propagating in a background of low-frequency density modulations. Within the latter representation, it is possible to consider the higher-frequency optical plasmons as exerting a gross ponderomotiVe force on the lower-frequency acoustic plasmons (or, consequently, generic acoustic waves).19,20 The above consideration shows, in principle, how the free energy liberated in the enzymatic ATP hydrolysis, -50 kJ/mol (under the physiological conditions; see ref 1a and references therein), can be efficiently harnessed to drive versatile biochemical processes. Specifically, this free energy can be spent to generate a sufficient number of optical plasmons in protein (up to several hundreds of optical plasmon quanta) which, in turn, exert a ponderomotive-like force on the lowest-frequency protein vibrations (hydration water hull relaxations,2 etc.) and thus perform some biologically useful work to trigger directional processes, like, for example, molecular motor motions. It is noteworthy that the ponderomotive effects in plasma are intimately connected with nonlinear dynamics and dynamical instabilities.21 This is a very important aspect for working out correct mathematical models of enzymes, with the special reference to molecular motors. Reference 2 contains clear instructions concerning how the specific protein structure can be used to formulate a correct “engineering-mechanical” model of the molecular motor. Recasting the corresponding equations

J. Phys. Chem. B, Vol. 112, No. 28, 2008 8321 of motion to include the correct driving forces and damping terms will bring them into the well-known and well-studied form.22,23 The main problem here is how the above-mentioned “slow plasmon modes” should enter these equations of motion. III. Physics of Plasmon Contributions to Protein Conformational Diffusion The equations of motion for a molecular machine are taken in the Fokker-Planck and Langevin sense, where the concept of “slow conformational diffusion” allows for the treatment of the very process of ATP hydrolysisstogether with the Brownian motion directed by itson equal footing, by writing up the relevant Langevin equations containing the inertial term. This is discussed in detail in Appendix I. Indeed, such equations allow one to look for something like Brownian ratchet created and controlled by power stroke. The role of the power stroke should then be played by the very event of the ATP hydrolysis, whereas the Brownian ratchet should result from the controlled conformational diffusion. If we agree with the previous works26–29 that internal protein dynamics is so intimately intertwined with the enzymatic reaction kinetics, we must recognize the obvious fact that not only conformational diffusion may guide the enzymatic reaction but, vice versa, also the onset of the latter may appreciably modify the former. Bearing this in mind, we propose that the physical carrier and director of the feedback between the very enzymatic reaction and the protein conformational diffusion can be the protein plasmons introduced in the previous section. An elegant formalization of the mechanistic principle advocated in Appendix I is in principle possible through the optimum control theory, and since we deal with diffusion processes, this must be stochastic optimum control theory. The latter is introduced and its consequences for protein dynamics are analyzed in Appendix II. The main objective there is to develop an optimum strategy (or, in the case of a controlled stochastic process, a definite family of optimum strategies) to produce a Brownian ratchet out of the ordinary conformational diffusion, that is, to perform some directional work using the stochastic, noisy system. Interestingly, the stochastic optimum control theory has recently been successfully used to formulate a physically clear-cut efficiency criterion for Brownian motors.30 That work30 is based upon the overdamped Langevin equation for diffusion-mediated transport processes, while a stochastic optimum control theory for Langevin equations with the inertial term is presented in Appendix II. Further, Appendix III discusses implications on our plasma approach in terms of the well-known Onsager-Machlup functional (being mathematically just the Lagrangian that delivers the family of the most probable pathways of a diffusion process44a) given by eq A10 and, in particular, eq A12 (with both being deduced from eq A2, where the latter describes the stochastic dynamics within this pathway family). The impact of plasmons on conformational diffusion (CD) is formulated by introduction of distribution functions for the coupled plasmon-CD dynamics. Using the Bayesian principle, we are able to demonstrate that the total nonequilibrium thermodynamic force in eq A12 in the presence of plasmons should read (cf. eq A15 in Appendix III):

TB∇vS(x, v, t) ) -kBTB(∇v log f(plasmon|CD) + ∇v log f(CD) - ∇v log f(plasmon)) (1) The three terms on the right-hand side of eq 1 represent the “plasmon load”, as well as the conventional Onsager thermo-

8322 J. Phys. Chem. B, Vol. 112, No. 28, 2008 dynamic forces consistent with the fluctuation-dissipation theorem for pure CD and pure plasmons, respectively. Obviously, the first and the third terms of eq 1, which are connected with plasmons, require a more detailed consideration. Specifically, the plasma introduced in section II above is nondegenerate, nonrelativistic, practically collisionless, and possessed of an approximate symmetry in its charge-to-mass ratio: e+/m+ ≈ e-/m-, where m( are masses of the positive and negative charges. Therefore, we may view all the N charged side chains of a protein as a system of (N/2) positively charged and (N/2) negatively charged classical particles with the N/2 Hamiltonian ∑i)1 [pi2/2m+ + pi2/2m-] + ∑i,j)1N eiej/|qi - qj| N + ∑i)1 eiφext(qi,t), where i numbers the particles, pi and qi are their canonical momenta and coordinates in the phase space, and φext(qi,t) is an external potential. To describe the dynamics of such a system, it is convenient to introduce a probability distribution function f in the full phase space (plus time) that will be a function of any of the integrals of motion of the above Hamiltonian and (in the limit of high particle density and/or large Debye length) should obey the Vlasov equation ∂f/∂t + {f,H} ) 0, where {...} are Poisson brackets. Since we are looking for collective motions in the twocomponent plasma, it is straightforward to recast the Vlasov equation for the distribution functions f+(p,q,t) and f-(p,q,t) of the positively and negatively charged side chains, respectively ({p,q} stand for the momentum and position of the system’s center of mass, both of which are integrals of motion for any N-particle system)

[

]

∂Φeff ∂ ∂ p ∂ f (p, q, t) ) 0 + +e ∂t m+ ∂q ∂q ∂p +

[

]

∂Φeff ∂ ∂ p ∂ f (p, q, t) ) 0 (2) + -e ∂t m- ∂q ∂q ∂p -

where Φeff(q,t) is an effective electrostatical potential which obeys the Poisson equation:

∆Φeff )

4π (F + e[n+(q, t) - n-(q, t)]) εε0 ext n( )

∫ f((p, q, t) dp (3)

where n is the internal charge density of protein, Fext is the charge density due to an external electrostatic potential, ε is the dielectric constant, ε0 is the dielectric constant of a vacuum. Since dimensions of proteins are always much greater than their Debye radius (see section II for the estimation of the latter), we may use the quasi-electroneutrality condition n+(q,t) ≈ n-(q,t) as well. Besides, in case of the charge-to-mass symmetry, apart from the functionsf((p,q,t) themselves, their linear combinations will also obey the Vlasov equation and deliver physically reasonable solutions.50 To this end, it is possible to define the following distribution functions: F+(p,q,t) ) f+(p,q,t) + f-(p,q,t) and F-(p,q,t) ) f+(p,q,t) - f-(p,q,t), which ought to describe the acoustic and optical plasmons, respectively. Now, by substituting the above plasmon distribution functions into the third (purely plasmonic) term on the right-hand side of eq 1 and taking into account eq 2, we see that this term will have the form (∂F(/∂t + p/m ∂F(/∂q). As usually done in plasma physics, we may assume that after some external perturbation (ATP hydrolysis or other enzymatic charge redistribution reaction, for example) the distribution function can be cast as F(p,q,t) ) F0(p) + g(p,q,t) for the both types of plasmons, where F0(p) stands for the equilibrium distribution function and g for the external perturbation. Obviously, the pure

Starikov et al. plasmon contribution in eq 1 will be determined by the external perturbation only. Further, the first (“plasmon load”) term on the right-hand side of eq 1 describes contributions from both optical and acoustic plasmons giVen other diffusive motions in protein. Physically, this means that the plasmons interact with the rest of the conformational dynamical modes, in that they exert a (ponderomotive) force on the latter and, vice versa, the latter promotes a correlation between the both plasmon kinds. Since the (nondiffusive) plasmons are an integral part of the overall protein dynamics, their interaction with other (diffusive) modes is not negligible. Hence, the form of the plasmon distribution functions will change appreciably, as compared to those for the “free” plasmon modes discussed above, owing to an energy exchange among the plasmons and the rest of the protein modes, which can be represented by scattering of plasmons on diffusional modes. To treat the resulting “renormalized plasmons in a fluctuating medium” on an equal footing with the free ones, we may introduce a collision term into the right-hand side of eq 2 and thus arrive at the Vlasov-Fokker-Planck equation:

[ [

]

[

∂f((p, q, t) ∂Φeff ∂ ∂ p ∂ f((p, q, t) ) + +e ∂t m( ∂q ∂q ∂p ∂t ∂f((p, q, t) ∂t

]

coll

[

) γ β∆p f((p, q, t) + 〈p 〉 )

]

coll

∂ ( p - 〈p 〉)f((p, q, t) ∂p

∫ pf((p, q, t) dp

]

β ) m(kBT (4)

where γ stands for the relaxation rate constant. With the above reasoning in mind, we may ask, what is the physical essence of the plasmon contributions to protein dynamics? Indeed, it is well-known that polymer blends are generally possessed of low entropies of mixing and high enthalpies of interaction, so that an effective way to render polymer blends more self-compatible is by adding charged groups to the polymeric backbone.5 Taking into account the results of a previous paper,5 we suggest that the latter modification introduces characteristic optical plasmon modes that help decrease the inter- and intrapolymer friction. Besides, extremely slow acoustic plasmon modes can create quasistatic (with respect to the most probable time scales of the conformational diffusion) deformation potentials, which break the spatial symmetry of the conventional potential of the mean force and thus promote its conversion into a productive Brownian ratchet potential. Finally, to otherwise produce unidirectional ratchet motion, special nonlinear friction terms may be designed to break the overall friction symmetry through a specific interplay between the friction and inertia,1c–e and such terms can in principle be introduced by plasmons. Reverting to the enzymatic activity of proteins, the most recent report on the combined systematical experimental and theoretical studies of substrate-free adenylate kinase dynamics51 shows that the enzyme’s larger-scale motions are not random but preferentially follow the pathways that create the configuration capable of proficient chemistry. Remarkably, this standpoint is seriously challenged in previous works,18b,c according to which the general trend in enzyme catalysis is that the best catalyst involves less motion during the reaction than the less optimal catalyst and that to obtain small electrostatic reorganization energy (the basic requirement of the most efficient catalysis18b,c) it is even necessary to invest some overall protein folding energy into the functional preorganization processes in the active site. Besides, refs 18b and c underline the huge importance of the enthalpy-entropy compensation in enzymatic

Energy Conversion in Biomacromolecular Machines catalysis, just as the present paper does (see the last paragraph of Appendix II). To sum up, as our plasmon concept is in fact closely related to the all-protein reorganization processes in response to electrostatic charge redistribution due to the chemical reaction in the enzyme’s active site, we feel that further discussions on the details of plasmon involvement into protein dynamics might be helpful in resolving the debates in question. The last, but not the least, point is that we should note the qualitative consistency of our plasmon approach with computer simulation results of enzymatic reactions.52,53 Indeed, the central finding of these simulations concerns electrostatic effects, which drive enzymatic reactions (instead of the commonly invoked sterical impetus) and consist of the fact that the conformational changes are conVerted to changes in the electrostatic interaction between enzymes and their substrate. Specifically, the main electrostatic effect was found to be due to the interaction between the substrate and the nearest-neighboring charged and polar groups, whereas the more distant charges do not play a significant role.51–53 Obviously, such highly screened, dynamical electrostatic effects can just be qualified as plasmons, according to the terminology of our present paper. This qualitative agreement of our theory with independent molecular modeling findings encourages us to propose novel effective computer simulation approaches for studying diverse phenomena of biophysical significance, where charged monomers/residues play important roles. Implications of the stochastic Newton’s second law for proteins are considered in detail in Appendix IV. In particular, it is noteworthy that eq A12 refers to the probability distribution function via the local entropy. Appendix IV outlines how the local entropy may be computed from the probability distribution function for the case when the usual conformational diffusion takes place in presence of some other dynamical process (for example, low-frequency plasmons). IV. Conclusions By and large, the sequence of events discussed in this communication, namely, charge redistribution in enzyme’s active site f nonequilibrium optical/acoustical plasmon generation in the whole enzyme molecule f excitation of the elastic vibrations and motions promoting enzymatic reactions, can pave the general pathway for the chemical-to-mechanical energy conversion in proteins. Specifically, here we advocate the following concept: Optical plasmons may pump entropy out of (or into) the protein conformational diffusion and, at the same time, help decrease protein’s intra- and/or intermolecular friction (for example, by squeezing the whole protein) or even break the symmetry of the latter, whereas the acoustic plasmons may break the spatial symmetry of the usual protein’s potential of mean force to convert it into an effective Brownian ratchet potential by applying quasistatic deformational corrections to the former. These proposals are consistent with the stochastic optimum control theory of protein dynamics. To sum up, our model highlights the power stroke produced by the ATP hydrolysis (or other kind of charge redistribution in the enzymatic active site) as a source of a nonequilibrium, activated Brownian ratchet, which disappears as soon as the desired work has been carried out and the protein approaches its equilibrium conformational state. The plasmon approach proposed here definitely has a general significance, since it can also be applied to work out efficient combinations of computer-aided simulation approaches when studying other biopolymers and their complexes, like, for

J. Phys. Chem. B, Vol. 112, No. 28, 2008 8323 example, intercalation of cationic dyes into DNA duplexes, or positively charged oligopeptide transduction through cell membranes. Appendix I. Equations of Motion To Describe Protein Dynamics First of all, a fundamental difference should be noted between the equations of motion for the conventional macroscopic mechanics22,23 and those describing mesoscopic/nanoscale machines.24 Specifically, the latter equations of motion must contain terms describing stochastic processes (thermal fluctuations) due to the environment (heat bath), which appreciably influence the states of the mesoscopic objects, as well as due to coarsegraining procedures, when modeling biomolecular machines. Therefore, equations of motion for nanoscaled machines should be cast in the mathematical form of a Fokker-Planck equation (describing the general nonequilibrium probability distribution function) combined with a Langevin equation (describing the pertinent diffusion dynamics). If we are able to single out the essential variable q (which can in general be a n-dimensional vector {q1, q2,..., qn}) in the phase space of a mesoscopic machine, the corresponding Langevin equation would read

mq¨(t) + V ′ (q(t)) ) -ηq ˙(t) + ξ(t)

(A1)

where m is the corresponding (effective) mass, V is the potential under which the dynamics takes place, V′ stands for the potential gradient (force acting on the mesoscopic object), η is the friction coefficient, and ξ(t) is a stochastic process due to the heat bath and/or coarse-graining. Equation A1 corresponds to the wellknown Ornstein-Uhlenbeck diffusion process. For example, Brownian ratchet dynamics can be described using this equation, if V is a periodic potential with broken spatial symmetry.24 Concerning the physical contents of the Langevin equations for molecular motors, the conventional approach is to neglect the inertial term mq¨(t), getting an oVerdamped version of eq A1.24 This facilitates mathematical analysis and, in many cases, can be a viable (thermodynamically noncontradictory) approximation. Actually, the main area of overdamped Langevin equation applicability is any kind of critical dynamics (dynamics on the verge of some phase transition). Still, in effect, one can encounter other realistic physical situations when neglecting the inertial term is inadmissible.24,25 Furthermore, the overdamped eq A1 describes a relaxation to an equilibrium state, without any idea of how the initial nonequilibrium state was prepared. Such a standpoint can be completely justified, if the initial state of the system under study has been prepared by some external factor or the very preparation of the initial state is not interesting for the modeling. But these both statements cannot apply, for example, to protein enzymatic “machines” driven by ATP hydrolysis reaction: The ATP reaction takes place at the very “heart” of the machineswithin the enzymatic active sitesand is, of course, not uninteresting from the viewpoint of deciphering detailed molecular mechanisms of enzymatic machines functioning. Hence, the very presence of the inertial term in Langevin equations turns out to be mandatory for any correct mechanistic study on enzymatic processes. Moreover, the conventional way of modeling the ATP reaction and its effect on protein motors is to invoke the conventional chemical kinetics based upon discrete chemical states and to add the corresponding periodic or stochastic timedependent corrections to V and/or to the right-hand side of eq A1.24 Meanwhile, it is long known26–29 that enzymatic reactions in general are not simple activated processes described by conventional chemical kinetics; they are rather “gated” in a

8324 J. Phys. Chem. B, Vol. 112, No. 28, 2008 complex manner by the slow conformational dynamics of the proteins involved. Indeed, the conventional activated process in chemical kinetics implies that equilibration of microstates within every discrete chemical macrostate (intrastate relaxation) is much faster than that among the different chemical species (interstate relaxation), which is obviously not the case for enzymatic proteins.26 In effect, proteins are known to be possessed of a quasicontinuum of allowed conformational states, so equilibrium microscopic internal dynamics of proteins ought to be viewed as a kind of permanent migration within one out of a huge number of conformational potential wells, interrupted by accident with spontaneous conformational transitions in such a way as to exhibit a quasicontinuum of characteristic relaxation times. To this end, there ought to be virtually no clear difference among the intra- and interstate relaxation times in proteins.29 It is clear that in such a case one has to speak of slow conformational diffusion underlying all the processes in proteins, instead of the conventional molecular-vibrational spectrum of separate normal modes; thus, it is this slow conformational diffusion that prepares initial states of enzymatic reactions, gates all of their transient states, and controls their steady states.27,28 The above-mentioned viewpoint is in full accordance with (and even significantly extends the framework of) the conventional enzymatic reaction picture.18 But, notwithstanding that the conformational diffusion is believed to govern eVery stage of enzymatic processes, the previous works26–29 consider it an overdamped stochastic process. Anyway, the above concept of slow conformational diffusion allows us to treat the very process of ATP hydrolysis, together with the Brownian motion directed by it, on equal footing, by writing up the relevant Langevin equations with the inertial term. On the basis of such equations, one ought to look for something like a Brownian ratchet created and controlled by power stroke. The role of the power stroke should then be played by the very event of the ATP hydrolysis, whereas the Brownian ratchet should result from the controlled conformational diffusion. Indeed, if we agree with the previous works26–29 that internal protein dynamics is so intimately intertwined with the enzymatic reaction kinetics, we must recognize the obvious fact that not only conformational diffusion may guide the enzymatic reaction but, vice versa, also the onset of the latter may appreciably modify the former. We propose that the physical “carrier” and “director” of the feedback between the very enzymatic reaction and the protein conformational diffusion can be the protein plasmons being the main topic of this paper. Appendix II. Stochastic Optimum Control Theory for Protein Dynamics An elegant formalization of the mechanistic principle formulated in Appendix I is possible through the optimum control theory, and since we deal with diffusion processes, this must be stochastic optimum control theory. The main direction here would be to find an optimum strategy (or, in case of a controlled stochastic process, a definite family of optimum strategies) to produce a Brownian ratchet out of the ordinary conformational diffusion, that is, to perform some directional work using the stochastic, noisy system. Interestingly, the stochastic optimum control theory has recently been successfully used to formulate a physically clear-cut efficiency criterion for Brownian motors.30 Regretably, the work30 is based upon the overdamped Langevin equation for diffusion-mediated transport processes, so that it cannot be helpful in solving our general mechanistic problem. Before considering the optimum control theory in detail, it is worth casting a look at the physical sense of the ultimately

Starikov et al. generalized (canonically invariant) Langevin equation.31 The latter is derived from the general Liouville equation by “tagging” some special system (for example, a protein molecule as a whole) and declaring the surrounding of this tagged molecule to be a “heat bath”. This heat bath is assumed to consist of a huge number of oscillators with random initial phases (The source of noise in the system!) and characteristic frequencies much higher than those of the tagged molecule. Thus, a separation of fast and slow motions becomes possible, and the coupling of the tagged molecule to its heat bath can be thought of as being sufficiently weak so that one can safely construct special projection operators that separate the fast and slow degrees of freedom (the Mori-Zwanzig approach). To carry out the latter separation, a molecule-bath coupling Hamiltonian has to be written down in an explicit form. Usually, it is assumed that there is only a pairwise linear interaction among the coordinates of the tagged molecule qi and those of the bath N oscillators yi in the form Hcoup ) ∑i qi[∑a)1 Aiayia]. But the previous work31 calls for treating the total canonical phase space (molecule + bath) on an equal footing and assumes that not only coordinates but also momenta of the molecule pi and bath ˙yai take part in the coupling (the coupling can, of course, be N not necessarily linear): Hcoup ) ∑i∑a)1 [AaiGi1(q,p)yai + i 2 i BaGi (q,p)˙ya]. After applying the conventional Mori-Zwanzig procedure, the work31 arrives at the following canonically invariant Langevin equation for any dynamical variable A(q,p):

˙ A ) κ{A, H} +

∑ {A, Gj}(ξj(t) + {Gj, H})

(A2)

j

where H is the Hamiltonian of the tagged molecule, {A,B} ) ∑i(∂A/∂qi ∂B/∂pi - ∂A/∂pi ∂B/∂qi) are Poisson brackets, κ stands for the inverse-time constant, Gj (j ) 1, 2,..., R) denotes the specific combinations of the functions G1i (q,p) and G2i (q,p), and the noise ζj(t) should be white and Gaussian. Equation A1 can be obtained from eq A2 by letting κ ) 1 and setting A(q,p) ) {qj,pj} and Gj ) -qj for all the j. If κ ) 0 and Gj ) pj, then we have the overdamped version of eq A1. It should be noted that, for proteins, the dimensions of eqs A1 and A2 can be dramatically reduced by employing the socalled dynamical principal components32 or hydrodynamical collective coordinates,33–35 which can also be justified for proteins.36 Interestingly, the Langevin equation for concentrated polyelectrolyte solutions in hydrodynamical collective coordinates allows plasmon modes, which ought to make the system nondiffusive.5,35 Besides, if we define a time-dependent probability distribution function P(qj,pj,t), and let A(q,p) ) P(qj,pj,t), we arrive at the generalized Fokker-Planck equation (more exactly, with κ ) 1 and Gj ) -qj we get the Kramers equation for inertial diffusions, whereas with κ ) 0 and Gj ) pj we get the conventional Fokker-Planck equation for overdamped diffusions).31 Therefore, one may say that the Langevin and Fokker-Planck equations are both contained in eq A2. From the conventional standpoint, the mathematical procedures to derive both Langevin and Fokker-Planck equations consist of projecting out the fast bath variables plus aVeraging over them to get the former and just eliminating them to get the latter.37 Physically, the Langevin and the Fokker-Planck equations represent two viewpoints concerning one and the same diffusion process; the former describes a continuous diffision, whereas the latter describes discrete random walks/jumps.38 Besides, there are also two very important standpoints concerning time eVolution of diffusion processes according to eq A2. The conventional idea is to assume that the past is known

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and the future develops from the past according to some certain probabilistic laws (the so-called “Itoh convention”). But it is much more natural to treat the past and the future on an equal footing and thus to maintain the complete symmetry between them during diffusion processes. The latter can be achieved in the so-called “Stratonovich sense”, that is, all the system’s phase space functions in the right-hand side of eq A2 should be evaluated as an average of their discretized values in the past and in the future. Obviously, eq A2 is fully compliant with the Stratonovich convention. Meanwhile, eq A2 in the Itoh representation is solely the Langevin equation in the future (forward time evolution), whereas in the past (backward time evolution) one more term, T{Gj,{Gj,A}} (T is temperature in the conventional thermodynamical sense), will be added to eq A2. This means that one and the same diffusion process will be described by two Langevin equations in the Itoh representation (see eqs 5 and 6 in ref 31) or by one Langevin equation (eq A2) in the Stratonovich sense. From the viewpoint of physics, the difference between the Stratonovich and Itoh approaches is purely technical: The former allows one to single out an explicit representation of the stochastic force contribution in the form of noise, whereas the latter paves the way to treating the stochastic component implicitly, that is, via thermodynamical concepts. It should be noted to this end that a detailed, constructive, and rigorous mathematical theory of the timesymmetrical forward and backward diffusion kinematics is presented in the book by Nelson39 and offers the third standpoint on the diffusion problems. Interestingly, it is the Nelson representation for stochastic differential equations analogous to eqs A1 and A2 that has proved very fruitful for formulating a rigorous variational theory for inertial diffusions and, as a result, a rigorous stochastic optimum control theory for them.40–43 Here we would like to recall a number of the results that are important for any physically correct theory of protein functioning. First of all, we introduce R “dynamical principal components” of protein dynamics in the conventional sense,32 that is, in form of some pertinent linear combinations of the true canonical variables of protein atoms. Then, the new set of coordinates and impulses thus obtained will correspond to the dynamical variables A(q,p) ) {xj,yj} (j ) 1, 2,..., R) in eq A2. Without going into detail on the functions Gj, it is convenient to recast eq A2 in the following generalized Ornstein-Uhlenbeck form, by singling out the explicit stochastic force contribution in the conventional way

dxi(t) ) V(xi, yi, t) dt

xi(0) ) x0i

dyi(t) ) b+(xi, yi, t) dt + σ(t) dWi

yi(0) ) y0i (A3)

where V and b+ are a generalized drift and a generalized force, respectively, W is a multidimensional stochastic Wiener process, whereas a(t) ) σ(t) σ*(t) stands for a generalized diffusion coefficient (a positive definite continuous matrix function).42 Since in proteins xi are simply displacements of some more or less stiff structural domains,26–29 one may safely state that V(xi,yi,t) ≡ yi ≡ Vi. Further, the structure of the right-hand side of eq A231 allows us to distinguish between two main terms in the term b+, namely, a conservative force due to the gradient of some multidimensional (intra/inter)molecular potential φ(xi), as well as a dissipative friction force f(xi,Vi). Bearing all this in mind, eq A3 can be rewritten as follows:41

dx ) v dt dv ) [bf(x,v) - M-1 ∇ φ(x)] dt + σ dW

(A4)

where M is a diagonal matrix containing the effective masses of protein dynamical domains.

There are two important fundamental questions connected with eq A4. The first and foremost question is whether eq A4 is consistent with the laws of equilibrium thermodynamics, a prerequisite also for the formulation of a correct nonequilibrium thermodynamical theory. The “thermodynamical consistency” means that the Fokker-Planck equation corresponding to the Langevin equation under study must have its stationary solution in the form of the Maxwell-Boltzmann distribution function PMB,

PMB ) Z exp(-H/kBT)

Z≡

(∫∫∫ P Ω

MB

(x) +

-1

)

dΩ

H≡φ

1〈 v, Mv 〉 (A5) 2

(where kB is the Boltzmann constant, T is the temperature, H is the Hamiltonian, Z is the partition function, Ω is the total phase space volume, and 〈...〉 denotes here a quadratic form), just in accordance with the Gibbs hypothesis central to the statistical thermodynamics. The second problem is a controllability of the system described by eq A4 in the presence of the noise contribution represented by the σ dW term. The controllability is a Very significant issue as concerns protein machines; they cannot perform their prescribed work if they are not controllable in some definite sense. A rigorous mathematical analysis of both these posers was carried out previously.40,41 By and large, to be thermodynamically consistent, eq A4 must satisfy the following three rigorous conditions:41 (a) f(x,v) ) -Bv, with B being a matrix of friction coefficients; (b) a(t) ) σ(t) σ*(t) ) kBTM-1(B + B*), which is a matrix form of the Einstein relation expressing the fluctuation-dissipation theorem (a dissipation in a thermodynamical system must efficiently neutralize deviations from equilibrium due to thermal fluctuations); and (c) the matrix B must be symmetrical to ensure that Newton’s second law holds (B is usually taken to be diagonal, but this requires one take into account some hydrodynamics36 of the system). Additionally, the simultaneous fulfillment of all these conditions simplifies the mathematical appearance of eq A4 to a great extent. Further, the controllability of the Langevin system described by eq A4 is based upon the mechanical stability of the corresponding dissipative dynamical system in the absence of the fluctuating random force.41 Reference 41 investigates in detail the stability of the linear system, assuming that ∇φ(x) ≡ Kx, where K is the pertinent force-constant matrix. Generally, if the potential φ(x) is possessed of minima, this analysis guarantees that the eq A4 should be controllable for all the trajectories in the vicinity of these minima. Even more generally: To produce a controllable Langevin system, the potential φ(x) (in combination with the damping term -Bv) ought to be possessed of stable attractors (they can represent not only points in the phase space but also limit cycles, that is, some stable periodic trajectories, as well as inVariant tori, that is, some stable quasiperiodic trajectories or, in other words, multidimensional limit cycles) but should not have strange attractors (where the “deterministic chaos” comes into play). However, the latter statements would still require detailed mathematical analysis, which definitely goes beyond the scope of the present paper. Now it is possible to formulate an optimum control theory for the Ornstein-Uhlenbeck system described by the equations

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Starikov et al.

dx ) v dt dv ) [-Bv - M

-1

-1

∇ φ(x)] dt + (2kBTM B)

1⁄2

dW (A6)

Specifically,42 making use of the Nelson’s forward and backward derivatives,39 eq A6 may be recast in the supermatrix “forward + backward” form as follows:

() (

D+

v x ) -1 -M ∇xφ(x) - Bv v

()

D-

(

)

x ) v

)

v (A7) M ∇xφ(x) - Bv - 2kBTM-1B∇v log P(x, v, t) -1

where the P(x,v,t) denotes nonequilibrium probability distribution function (a general solution of the correspondent Kramers/ Fokker-Planck equation). The mean-forward D+ and meanbackward D- derivatives have the following definitions.39,42 If z(t) is a continuous stochastic process, with A+ and A- being the sets of all its forward and backward realizations, respectively, then for h > 0

D+z(t) ) 98 h f 0+

E(z(t + h) - z(t)|A+) h

D-z(t) ) 98 h f 0+

E(z(t) - z(t - h)|A-) (A8) h

where the operation E(...|...) denotes a mathematical expectation (averaging over some special stochastic process realization) and hf0+ means that h is approaching zero from the right. It is also possible to define a symmetrized derivative

1 Dz(t) ) (D+ + D-)z(t) 2

(A9)

in this same sense. Further, we define the local (microscopic) entropy S(x,v,t) ) -kB log P(x,v,t) (Boltzmann formula), with the conventional thermodynamical (macroscopic) entropy being simply the mathematical expectation (average) of its local counterpart, namely Smacro(t) ) -kB∫∫P(x,v,t) log P(x,v,t) dx dv ≡ -kBE(S(x,v,t)) (the Shannon formula for information entropy). To this end, the stochastic optimum control theory dictates to find the optimum control force u ∈U by minimizing the functional42

J(x, v, u, t) )

{∫ L(x, v, u, s) ds + S(x, v, T)|x(t) ) x, v

1 E kB

T

t

(t) ) v}

L(x, v, u, s) ) 2TMB-1|-M-1∇xφ(x) - Bv + 2kBTM-1B∇vS(x, v, s) - u|2 + B - 2kBTBM-1∆vS(x, v, s) dx ) v dt dv ) u dt + (2kBTM-1B)1⁄2 dW (A10) where L is the Lagrangian functional and ∆ stands for the Laplace operator. Notice that, mathematically, the local entropy S is the Value function of the optimum control functional J. As a result, we get the optimum control force in the following closed form:

Mu* ) -∇xφ(x) - MBv + TB∇vS(x, v, t)

(A11)

Comparing eq A11 with eqs A7 and A9, we immediately notice that Mu* ≡ MDv, where Dv ) ((D+D- + D- D+)/2)x

stands for the symmetrized acceleration in the Nelson sense, so we now arrive at the so-called pathwise Newton’s second law:

MDv ) -∇xφ(x) - MBv + TB∇vS(x, v, t)

(A12)

Strikingly,42 eq A10 is identical to the well-known OnsagerMachlup functional, which lies at the heart of linear nonequilibrium thermodynamics (when the deviations from thermodynamical equilibria are relatively small) and is also well-known to be just the Lagrangian delivering the family of the most probable pathways of a diffusion process.44a Equation A12 is then describing the stochastic dynamics within this pathway family in full accordance with the linear nonequilibrium thermodynamics. Besides, it is noteworthy that, in fact, the concept of “the most probable pathways of a diffusion process” behind eq A12 is explicitly used in biomacromolecular modeling, where one works with statistical ensembles of all the possible moleculardynamical trajectories connecting potential energy bassins (or “hidden structures”) in multidimensional coordinate spaces by establishing isomorphisms between these ensembles and the thermodynamical properties (for a comprehensive review, see, for example, ref 44b). On the other hand, the above-mentioned isomorphisms are embodied in the Jarzynski-Crooks theorem,44c,d which expresses equilibrium free energy differences between the hidden structures A and B in terms of finite-time measurements of work performed on the system as it is switched from A to B. Most recently, the isomorphisms in question have been shown to be a direct manifestation of the general enthalpy-entropy compensation principle in the form of microphase transitions.44e In accordance with this, molecular machines obeying the pathwise Newton’s second law given by eq A12 ought to exhibit maximum possible efficiency quotients equal to those of the corresponding Carnot cycles.44e Appendix III. Physical Sense of Stochastic Newton’s Second Law for Proteins Physically, the right-hand side of eq A12 shows that the optimum controlled diffusion is governed by a conservative force due to the inter/intramolecular potential (the first term), a dissipative friction force (the second term), and the generalized Onsager thermodynamical force (the third term). In the thermodynamical equilibrium, the second and third terms will cancel each other, thanks to the fluctuation-dissipation theorem, so that this balance, ∇vSeq(x,v) ) 1/TB-1MBV, defines the equilibrium local time-independent entropy Seq and the conservative pathwise Newton’s second law MDv ) -∇xφ(x). If B is diagonal, as usually assumed, then T∇vSeq(x,v) ) MV is just the total mechanical impulse of the system in compliance with the equilibrium (Maxwell-Boltzmann) probability distribution, whereas hydrodynamic interaction effects (nonzero off-diagonal B elements) would only rescale the masses of the dynamical subsystems. To investigate what are the explicit effects of some external force acting on a diffusive system, we have basically to revert to eq A4, by setting f(x,v) ) -Bv+ g(x,v), where the g(x,v) is a generalized external load. A mesoscopic thermodynamical theory of eq A4 in terms of entropy production was earlier presented25 (and extended45) in an effort to theoretically describe the phenomenon of velocity-dependent feedback control used to reduce the thermal noise of a cantilever in atomic force microscopy.46 It is interesting to note that the “entropy pumping rate” (defined in ref 25 as an amount of entropy pumped out or into the system by an external agent exerting the control force)

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is governed by the ∇vbg(x,v) term and that the entropy production formula (eq 3 in ref 25) is an immediate consequence of the stochastic optimum control theory for a generalized Langevin equation (see eq 2.11 in ref 42). Therefore, any external load will modify only the thermodynamic force in the pathwise Newton’s second law (eq A12 here), so the total local entropy out of equilibrium will be just the conventional diffusional-fluctuational entropy plus the externally pumped entropy. In other words, the total nonequilibrium distribution function P(x,vt) will be a product of the conventional diffusional distribution function and the distribution function correction owing to the external load. By reverting to proteins, such a modeling standpoint means that the intraprotein diffusional effects and the perturbations caused by the external load force are statistically independent of each other. However, this cannot in principle be a sufficient background for modeling unique self-controlling systems, like, for example, the consequences of a power stroke due to ATP hydrolysis (or even other charge redistribution reactions ubiquitous in biochemistry) in protein motors, since dynamical effects of the conventional conformational diffusion and dynamical conformational reconstructions stemming from enzymatic charge redistribution reactions are intimately intertwined!26–29 Specifically, in a self-controlling system we would need not only a dynamical factor to modify the Onsager thermodynamic force term in eq A12 but also a proper modification of the conservative force term to (virtually) create a Brownian ratchet potential. For efficiency sake, there can also be a mechanism to (virtually) decrease the friction, for example, by somehow confining the macromolecule in the 3D space.47 The above reflections urge us to cast a closer look at the third term on the right-hand side of eq A12, which contains the reference to the probability distribution function via the local entropy. Our problem now is to find a probability distribution function for the case when the usual conformational diffusion takes place in the presence of some other dynamical process (for example, low-frequency plasmons) and use it to calculate the local entropy. The fundamental standpoint in physics which enables one to treat correlations resulting from some interactions is the Bayesian approach based upon joint probabilities of correlated occurences.48 From probability theory it is known that Bayes’ theorem is valid in particular for probability density functions of continuous random variables X and Y, namely:49

f(X ) x|Y )y) )

f(Y )y|X )x) f(X )x) f(Y )y)

(A13)

where f(X)x|Y)y) is an a posteriori probability density of the occurrence X ) x given Y ) y, whereas f(X)x) is an a priori probability of the occurrence X ) x. Translating eq A13 into the language of protein conformational diffusion, we get

f(CD|plasmon) )

f (plasmon|CD) f (CD) f (plasmon)

(A14)

where CD denotes “conformational diffusion”. The distribution functions f(CD|plasmon) and f(plasmon|CD) can physically be understood as the respective a priori distribution functions corrected by interaction between the conformational diffusion with plasmons. Therefore, it is tempting to interpret the f(CD) distribution function as that describing the conventional fluctuation-dissipation-like conformational diffusion in the absence of plasmons. Then, the f(plasmon|CD) distribution function ought to underlie an additional “plasmon load” term

gplasmon(x,v) (ponderomotive force) in eq A4, like it was done in ref 25 to modify only the Onsager thermodynamic force term in eq A12. Along with this, there should be a “purely plasmon” influence on the conservative potential force term and/or the dissipative friction force term of eq A12, owing to the distribution function f(plasmon). To gain more insight into how the above suggestion will work, let us notice that the Onsager’s thermodynamical force of eq A12 in the presence of plasmons ought to read TB∇vS(x,v,t) ) -kBTB∇v log f(CD|plasmon), and taking into account eq A14, we have

TB∇vS(x, v, t) ) -kBTB(∇v log f(plasmon|CD) + ∇v log f(CD) - ∇v log f(plasmon)) (A15) The three terms on the right-hand side of eq A15 ought to be the plasmon load gplasmon(x,v) mentioned above and the conventional Onsager thermodynamic force consistent with the fluctuation-dissipation theorem for pure CD and pure plasmons, respectively. Appendix IV. Stochastic Optimum Control Theory of Biopolymer Conformational Diffusion and New Molecular Modeling Approaches Indeed, the theory worked out in the present paper helps to develop a new effective molecular modeling concept. Here we shall consider its general algorithmic framework only, whereas the relevant specific results will be presented and discussed in detail elsewhere. To point out the general character of our theory, we shall consider here two phenomena having nothing to do with protein enzymatic activity. First, we consider the process of kinetical recognition of AT-rich DNA duplexes by cationic ruthenium complexes, experimentally studied in ref 54. The recognition consists of a slow intercalation of cationic ruthenium complexes into DNA duplexes, with the latter process being much faster for AT-rich DNAs than for others. Still, even the fastest intercalation events take minutes to hours to be complete, so that it is absolutely impossible to model them within the frame of the conventional molecular dynamical or Monte Carlo techniques. A plasmon hypothesis similar to that considered in the present paper was advocated earlier in a DNA case (see, for example, ref 3). It is then clear that the cationic ruthenium complexes under study in ref 54 will strongly perturb DNA plasmons upon the external binding, because their electrostatic charge is +4. Indeed, it is the external binding of such dyes to DNA that would first trigger some nonequilibrium plasmons in the latter. These nonequlibrium motions will then simply repump their energy into the conventional lowest-frequency bending modes of DNA duplexes. Mechanically, this ought to result in an increased bending amplitude of a duplex, and the maximum possible amplitude increase should be significantly dependent on the DNA base-pair sequence. With the AT-rich DNAs, such a maximum possible bending amplitude ought to be enough to trigger the slow dye intercalation, unlike with other base-pair sequences. With this in mind, we can suggest the following molecular mechanical/molecular dynamical algorithm to model the slow intercalation process: (1) docking of the dye cationic molecule under study to the surface of the DNA duplex in question, so that external binding of the dye is ensured; (2) placing the complex created during the previous step into the water box and carefully minimizing its energy, because from the trivial

8328 J. Phys. Chem. B, Vol. 112, No. 28, 2008 electrostatics standpoint the DNA should be cooperatively bent toward the bonded dye cationic molecule, as all of the DNA phosphate anions will tend to move nearer to the dye; (3) for the structure of the “dry” complex resulting from the previous calculation, calculating the lowest-frequency vibrational modes using, for example, the elastic-network model;55a (4) of much functional interest will be both of the cooperative bending modes of the DNA-dye complex, which will obviously be its lowestfrequency modes anyway; (5) analyzing different amplitudes of the both bending modes, to see which amplitude is the most appropriate one to be the sterically most favorable way of “pushing” the dye cation into the inside of the DNA duplex; (6) most probably, analyzing a combination of the both bending deformations in the sense of the previous point; (7) as soon as the proper deformation has been chosen, placing the resulting complex into the water box again and carefully minimizing the energy; (8) based upon the result of the previous computation, performing a free-energy cycle of the conventional MD simulations by gradually pushing the dye cation into the inside of the properly bent DNA duplex. In such a way, enthalpic and entropic contributions to the slow intercalation process can be estimated and compared with the experimental data. Every step of the above-mentioned molecular modeling algorithm can be accomplished using the pertinent conventional software and requires realistic periods of time for this purpose. Still, as concerns points 2 and 7 above, it should be noted that it is rather difficult to perform energy minimization in multidimensional systems of solvated charges (due to their enormous dimensionality). Moreover, one has also to take into account the fact that any charge redistribution like that in the ATP hydrolysis (which triggers collective oscillations of the plasmon type, as we here suggest) should consequently result in a negative entropic contribution to the overall free energy (positive T∆S). This means that the frequencies of the phonon modes other than the plasmons ought to increase in order to compensate the latter effect. Bearing this in mind, the simulations embodied in the steps 1-8 above would probably require the use of a restraint release approach of the type used in ref 1b and in entropy calculations of the type used in ref 55b It is, of course, extremely important to establish a connection between the present plasmon formulations and the conventional physicalchemical language and approaches, but this seems to be a very difficult topic, and we would like to postpone its thorough discussion to our further publications. Interestingly, some very similar approach can also be used to simulate the transduction of short positively charged oligopeptides via cell membranes, because the presence of positively charged amino acid side chains in the transducible oligopeptides has been found to be very important for the optimum efficiency of the process.56 In this case, we may consider the whole communion of the membrane lipid and protein charged groups and its nonequilibrium plasmons, which ought to be generated as soon as an oligopeptide with the proper positive charge is bound on the external surface of the membrane. This would stimulate acoustic waves on the membrane surface, which would help catching up to and, finally, “swallowing” the externally bonded peptide. References and Notes (1) (a) Ross, J. J. Phys. Chem. B 2006, 110, 6987–6990. (b) Sambongi, Y.; Iko, Y.; Tanabe, M.; Omote, H.; Iwamoto-Kihara, A.; Ueda, I.; Yanagida, T.; Wada, Y.; Futai, M. Science 1999, 286, 1722–1724. (c) Zolotaryuk, A. V.; Christiansen, P. L.; Norde´n, B.; Savin, A. V.; Zolotaryuk, Y. Phys. ReV. E 2000, 61, 3256–3259. (d) Norde´n, B.; Zolotaryuk, Y.; Christiansen, P. L.; Zolotaryuk, A. V. Phys. ReV. E 2001, 65, 011110. (e)

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