Fourth-Order Differential Equation for a Two-Stage Absorbing Markov Chain with a Stochastic Forward Transition Probability
The problem of averaging the kinetics of a two-stage absorbing Markov chain over random fluctuations in its forward transition probability approximated by the symmetric dichotomous stochastic process is solved exactly. It is shown that the temporal behavior of the population of chain’s transient state obeys a fourth-order differential equation with the tetra-exponential form of a solution given the finite frequency and mean amplitude of fluctuations. In the limit of frequent fluctuations, this tetra-exponential solution reduces to a simple bi-exponential form typical of the deterministic two-stage decay process lacking fluctuations in its transition probability. Rather, in the limit of rare fluctuations, the tetra-exponential solution, while simplifying to the tri- and bi-exponential solutions, becomes specific both for the low amplitude and the resonance amplitude fluctuations, respectively. Furthermore, there is a stochastic resonance point, where the forward transition probability is in resonance with the mean fluctuation amplitude, whereas the backward transition probability, decay transition probability, and fluctuation frequency are negligibly small. In result, the stochastic immobilization of the two-stage absorbing Markov chain in its initial state occurs at this point.
R.A. Alberty, W.G. Miller. Integrated rate equations for isotopic exchange in simple reversible reactions. J. Chem. Phys. 26, 1231 (1957).
R.K. Bohn, K.H. Casleton, Y.V.C. Rao, G.W. Flynn. Vibrational energy transfer in laser-excited carbonic difluoride. Infrared fluorescence from the intermediate mode 4. J. Phys. Chem. 86, 736 (1982).
A.N. Macpherson, J.B. Arellano, N.J. Fraser, R.J. Cogdell, T. Gillbro. Efficient energy transfer from the carotenoid S(2) state in a photosynthetic light-harvesting complex. Biophys. J. 80, 923 (2001).
N. Kanamaru, J. Tanaka. Nanosecond laser photolysis of n-methylindole in acetonitrile. Bull. Chem. Soc. Jpn. 59, 569 (1976).
N. Wermuth. International Encyclopedia of Statistical Science, edited by M. Lovric (Springer, 2011), part 7, p. 618.
J.R. Alcala, E. Gratton, F.G. Prendergast. Fluorescence lifetime distributions in proteins. Biophys. J. 51, 597 (1987).
A.I. Burshtein. Non-Markovian theories of transfer reactions in luminescence and chemiluminescence and photoand electrochemistry. Adv. Chem. Phys., edited by S.A. Rice (Wiley, 2004), 129, p. 105.
E.J. Crampin, S. Schnell, P.E. McSharry. Mathematical and computational techniques to deduce complex biochemical reaction mechanisms. Prog. Biophys. Mol. Biol. 86, 77 (2004).
K. Banerjee, K. Bhattacharyya. Enzyme efficiency: An open reaction system perspective. J. Chem. Phys. 143, 235102 (2015).
M.J. Lawson, L. Petzold, A. Hellander. Accuracy of the Michaelis–Menten approximation when analysing effects of molecular noise. J. R. Soc. Interface 12, 20150054 (2015).
P. Suppes. Stimulus – response theory of finite automata. J. Math. Psychol. 6, 327 (1969).
W.M.L. Holcombe. Algebraic Automata Theory (Cambridge Univ. Press, 1982) [ISBN: 0521604923].
D.L. Andrews, D.S. Bradshaw, M.M. Coles. Perturbation theory and the two-level approximation: A corollary and critique. Chem. Phys. Lett. 503, 153 (2011).
C. Csajka, D. Verota. Pharmacokinetic-pharmacodynamic modelling: history and perspectives. J. Pharmacokin. Pharmacodyn. 33, 227 (2006).
V.I. Teslenko, O.L. Kapitanchuk, Y. Zhao. Controlling cooperativity of a metastable open system coupled weakly to a noisy environment. Chin. Phys. B 24, 028702 (2015).
G.H. Weiss, J. Masoliver. Statistics of dwell times in a reaction with randomly fluctuating rates. Physica A 296, 75 (2001).
E.G. Petrov, V.I. Teslenko. Kinetic equations for a quantum dynamical system interacting with a thermal reservoir and a random field. Theor. Math. Phys. 84, 986 (1990).
V.I. Teslenko, E.G. Petrov, A. Verkhratsky, O.A. Krishtal. Phys. Rev. Lett. 104, 178105 (2010).
E.G. Petrov. Coarse-grained kinetic equations for quantum systems. Eur. Phys. J. Spec. Top. 216, 205 (2013).
V.I. Teslenko, O.L. Kapitanchuk. Theory of kinetics of multistep ligand–receptor assembly in dissipating and fluctuating environments. Int. J. Mod. Phys. B 27, 1350169 (2013).
O.L. Kapitanchuk, O.M. Marchenko, V.I. Teslenko. Hysteresis of transient populations in absorbing-state systems. Chem. Phys. 472, 249 (2016).
A. Barchielli, M. Gregoratti. Quantum continuous measurements: The stochastic Schrodinger equations and the spectrum of the output. Quant. Measur. Quant. Metrol. 1, 34 (2013).
C.W. Gardiner, P. Zoller. Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer, 2004) [ISBN 978-3-540-22301-6].
T. Holstein. Studies of polaron motion: Part II. The "small" polaron. Ann. Phys. 8, 343 (1959).
E.G. Petrov, V.I. Teslenko. Kinetics of quasi-isoenergetic transition processes in biological macromolecules. Chem. Phys. 375, 243 (2010).
V.I. Teslenko, O.L. Kapitanchuk. Dynamics of transient processes in irreversible kinetic models. Ukr. J. Phys. 57, 573 (2012).
V.I. Teslenko, E.G. Petrov. Regularization of environmentinduced transitions in nanoscopic systems. Ukr. J. Phys. 61, 627 (2016).
E. Petrov, V. Teslenko. Kinetics framework for nanoscale description of environment-induced transition processes in biomolecular structures. In Nanobiophysics: Fundamentals and Applications, edited by V.A. Karachevtsev (Pan Stanford Publishing, 2015), Chapter 9, p. 267.
N.G. van Kampen. Stochastic Processes in Physics and Chemistry (North Holland, 1981) [ISBN: 0-444-89345-0].
J.C. Anderson. Quantum Monte Carlo: Origins, Development, Applications (Oxford Univ. Press, 2007) [ISBN: 978-0-19-531010-8].
S.E. Jackson. How do small single-domain proteins fold? Fold. Des. 3, R81 (1998).
M. Bixon. Polymer dynamics in solution. Ann. Rev. Phys. Chem. 27, 65 (1976).
V.E. Shapiro, V.M. Loginov. "Formulae of differentiation" and their use for solving stochastic equations. Physica A 91, 563 (1978).
V.M. Loginov. Simple mathematical tool for statistical description of dynamical systems under random actions. Pt. I. Acta Physica Polonica B 27, 693 (1996).
H. Frauenfelder, P.G. Wolynes. Rate theories and puzzles of hemoprotein kinetics. Science 229, 337 (1985).
A. Nitzan, J. Ross. A comment on fluctuations around nonequilibrium steady states. J. Stat. Phys. 10, 379 (1974).
I.A. Goychuk, E. Petrov, V. May. Bridge-assisted electron transfer driven by dichotomically fluctuating tunneling coupling. J. Chem. Phys. 103, 4937 (1995).
A.M. Berezhkowsky, G.H. Weiss. Detailed description of a two-state non-Markov system. Physica A 303, 1 (2002).
A.M. Berezhkowsky, A. Szabo, G.H. Weiss. Theory of single-molecule fluorescence spectroscopy of two-state systems. J. Chem. Phys. 110, 9145 (1999).
A. Hurwitz. Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt. Math. Ann. 46, 273 (1895).
D. Jacewicz, A. Lapi’nska, A. D¸abrowska, L. Chmurzy’nski. A stopped-flow study on the kinetics and mechanism of CO2 uptake by the cis-[Cr(1,10-phenanthroline)2(OH2)2] 3+ complex ion. Transition Met. Chem. 31, 111 (2006).
D. Jacewicz, M. Szkatu la, A. Chylewska, A. D¸abrowska, M. Wo’zniak, L. Chmurzy’nski. Coordinate cis-[Cr(C2O4)(pm)(OH2)2] + cation as molecular biosensor of pyruvate's protective activity against hydrogen peroxide mediated cytotoxity. Sensors 8, 4487 (2008).
I.C. Kleppe, H.P.C. Robinson. Determining the activation time course of synaptic AMPA receptors from openings of colocalized NMDA receptors. Biophys. J. 77, 1418 (1999).
E.A. Moelwyn-Hughes. Physical Chemistry (Cambridge Univ. Press, 1940), Appendix 9, p. 633.
S. Sergeyev. Activated polarization pulling and decorrelation of signal and pump states of polarization in fiber Raman amplifier. Opt. Express 19, 24268 (2011).
M.F. Bukhori, S. Roy, A. Asenov. Simulation of statistical aspects of charge trapping and related degradation in bulk MOSFETs in the presence of random discrete dopants. IEEE Trans. Electron Devices 57, 795 (2010).
T. Wellens, V. Shatochin, A. Buchleitner. Stochastic resonance. Rep. Prog. Phys. 67, 45 (2004).
D.G. Luchinsky, P.V.E. McClintock. Irreversibility of classical fluctuations studied in analogue electrical circuits. Nature 389, 463 (1997).
M.I. Dykman, G.P. Golubev, I.Kh. Kaufman, D.G. Luchinsky, P.V.E. McClintock, E.A. Zhukov. Noise-enhanced optical heterodyning in an all-optical bistable system. Appl. Phys. Lett. 67, 308 (1995).
M.I. Dykman, P.M. Hunt. Large fluctuations and optimal paths in chemical kinetics. J. Chem. Phys. 100, 5737 (1994).