Termination of Biological Function at Low Temperatures: Glass or Structural Transition?

Abstract

Energy of life is produced by electron transfer in energy chains of respiration or photosynthesis. A small input of free energy available to biology puts significant restrictions on how much free energy can be lost in each electron-transfer reaction. We advocate the view that breaking ergodicity, leading to violation of the fluctuation-dissipation theorem (FDT), is how proteins achieve high reaction rates without sacrificing the reaction free energy. Here we show that a significant level of nonergodicity, represented by a large extent of the configurational temperature over the kinetic temperature, is maintained in the entire physiological range for the cytochrome c electron transfer protein. The protein returns to the state consistent with the FDT below the crossover temperature close to the temperature of the protein glass transition. This crossover leads to a sharp increase in the activation barrier of electron transfer and is displayed by a kink in the Arrhenius plot for the reaction rate constant.

Fractional electron transfer kinetics and a quantum breaking of ergodicity

“…Each particle crosses the boundary huge many times during this equilibration process. An especially insightful and beautiful result is obtained when both asymptotics, (27) and (28), coincide. Then, GME (25) can approximately be written as a fractional master equation [61,111]…”

“…smaller than τ i,ad entering formally the same approximate (for the initial times) expression for F (ens) i (t) with the only difference: τ i,ad instead of τ i,sgl . Furthermore, to be more general and to go beyond a very restrictive case of coinciding (27) and (28), we should use a different from τ i,ad = 4τ r r 2 1,2 expression for τ i,ad . Namely, one should use the one stemming from the short-time/large−s asymptotics in Eq.…”

“…. By a Tauberian theorem [120] this yields (27). In the numerical studies of this paper, we consider a symmetric ET with α = 0.5, r 1 = r 2 = 2.5 and η 0 = 0.1.…”

“…Generally, short and long time asymptotics in (A3) and (A4) are very different even for α = 0.5, since c 1 and c 2 can differ strongly, in general. For α = 0.5, one must approximately satisfy c 1 ≈ c 2 , or η0 2 (e r − 1) ≈ 2r/ √ π, for the approximation (27) to work uniformly. This can be done only in a symmetric case.…”

Ergodicity Breaking in Thermal Biological Electron Transfer? Cytochrome C Revisited II

“…10,23,24,[27][28][29][30][31] Due to the incomplete relaxation of the protein on the timescale of the ET event, the Stokes reorganization free energy λ st is no longer the same as the variational λ var related to thermal fluctuations of the vertical energy gap but significantly smaller, λ st λ var , resulting in a lowering of the ET activation free energy by a factor λ var /λ st for parabolic free energy profiles. 10 The non-ergodic effect described above was demonstrated for oxidation of the Fe-heme cofactor in cytochrome c, where unusually high values for λ var , as large as 3.0 eV, were reported, while estimates for λ st were significantly smaller, 1.3 eV [28][29][30] Motivated by these curious results, we investigated the same system at various levels of theory including (i) classical MD simulations with additive and electronically polarizable force fields, (ii) hybrid QM(PMM)/MM calculations based on the perturbed matrix method (PMM) and ZINDO/S electronic-structure calculations, and (iii) QM(DFT)/MM calculations with the QM region treated by full electrostatic DFT embedding. 32 All approaches provided consistent results and similar values for the two reorganization free energies, λ st = 0.9 -1.0 eV, λ var = 0.9 -1.3 eV with electronically non-polarizable force fields.…”