N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

This tutorial clarifies the axiomatic definition of (v(α); i(β)) circuit elements via a lookup table dubbed an A-pad, of admissible (v; i) signals measured via Gedanken probing circuits. The (v(α); i(β)) elements are ordered via a complexity metric. Under this metric, the memristor emerges naturally as the fourth element, characterized by a state-dependent Ohm's law. A logical generalization to memristive devices reveals a common fingerprint consisting of a dense continuum of pinched hysteresis loops whose area decreases with the frequency ω and tends to a straight line as ω ~ ∞, for all bipolar periodic signals and for all initial conditions. This common fingerprint suggests that the term memristor be used hence-forth as a moniker for memristive devices.

The Fourth Element.

From a pedagogical point of view, the memristor is defined in this tutorial as any 2-terminal device obeying a state-dependent Ohm’s law. This tutorial also shows that from an experimental point of view, the memristor can be defined as any 2-terminal device that exhibits the fingerprints of “pinched” hysteresis loops in the v–i plane. It also shows that memristors endowed with a continuum of equilibrium states can be used as non-volatile analog memories. This tutorial shows that memristors span a much broader vista of complex phenomena and potential applications in many fields, including neurobiology. In particular, this tutorial presents toy memristors that can mimic the classic habituation and LTP learning phenomena. It also shows that sodium and potassium ion-channel memristors are the key to generating the action potential in the Hodgkin-Huxley equations, and that they are the key to resolving several unresolved anomalies associated with the Hodgkin-Huxley equations. This tutorial ends with an amazing new result derived from the new principle of local activity, which uncovers a minuscule life-enabling Goldilocks zone, dubbed the edge of chaos, where complex phenomena, including creativity and intelligence, may emerge. From an information processing perspective, this tutorial shows that synapses are locally-passive memristors, and that neurons are made of locally-active memristors.

Memristor, Hodgkin-Huxley, and Edge of Chaos.

This exposition shows that the potassium ion-channels and the sodium ion-channels that are distributed over the entire length of the axons of our neurons are in fact locally-active memristors. In particular, they exhibit all of the fingerprints of memristors, including the characteristic pinched hysteresis Lissajous figures in the voltage-current plane, whose loop areas shrink as the frequency of the periodic excitation signal increases. Moreover, the pinched hysteresis loops for the potassium ion-channel memristor, and the sodium ion-channel memristor, from the Hodgkin-Huxley axon circuit model are unique for each periodic excitation signal. An in-depth circuit-theoretic analysis and characterizations of these two classic biological memristors are presented via their small-signal memristive equivalent circuits, their frequency response, and their Nyquist plots. Just as the Hodgkin-Huxley circuit model has stood the test of time, its constituent potassium ion-channel and sodium ion-channel memristors are destined to be classic examples of locally-active memristors in future textbooks on circuit theory and bio-physics.

Brains Are Made of Memristors.

Measures of microbial growth, used as indicators of cellular stress, are sometimes quantified at a single time-point. In reality, these measurements are compound representations of length of lag, exponential growth-rate, and other factors. Here, we investigate whether length of lag phase can act as a proxy for stress, using  a number of model systems (Aspergillus penicillioides; Bacillus subtilis; Escherichia coli; Eurotium amstelodami, E. echinulatum, E. halophilicum, and E. repens; Mrakia frigida; Saccharomyces cerevisiae; Xerochrysium xerophilum; Xeromyces bisporus) exposed to mechanistically distinct types of cellular stress including low water activity, other solute-induced stresses, and dehydration-rehydration cycles. Lag phase was neither proportional to germination rate for X. bisporus (FRR3443) in glycerol-supplemented media (r2 = 0.012), nor to exponential growth-rates for other microbes. In some cases, growth-rates varied greatly with stressor concentration even when lag remained constant. By contrast, there were strong correlations for B. subtilis in media supplemented with polyethylene-glycol 6000 or 600 (r2 = 0.925 and 0.961), and for other microbial species. We also  analysed data from independent studies of food-spoilage fungi under glycerol stress (Aspergillus aculeatinus and A. sclerotiicarbonarius); mesophilic/psychrotolerant bacteria under diverse, solute-induced stresses (Brochothrix thermosphacta, Enterococcus faecalis, Pseudomonas fluorescens, Salmonella typhimurium, Staphylococcus aureus); and fungal enzymes under acid-stress (Terfezia claveryi lipoxygenase and Agaricus bisporus tyrosinase). These datasets also exhibited diversity, with some strong- and moderate correlations between length of lag and exponential growth-rates; and sometimes none. In conclusion, lag phase is not  a reliable measure of stress because length of lag and growth-rate inhibition are sometimes highly correlated, and sometimes not at all.

https://pureadmin.qub.ac.uk/ws/files/202812251/s41598_020_62552_4.pdf

Microbial lag phase can be indicative of, or independent from, cellular stress.

Here we have systematically studied the cooperative binding of substrate molecules on the active sites of a single oligomeric enzyme in a chemiostatic condition. The average number of bound substrate and the net velocity of the enzyme catalyzed reaction are studied by the formulation of stochastic master equation for the cooperative binding classified here as spatial and temporal. We have estimated the entropy production for the cooperative binding schemes based on single trajectory analysis using a kinetic Monte Carlo technique. It is found that the total as well as the medium entropy production show the same generic diagnostic signature for detecting the cooperativity, usually characterized in terms of the net velocity of the reaction. This feature is also found to be valid for the total entropy production rate at the nonequilibrium steady state. We have introduced an index of cooperativity, C, defined in terms of the ratio of the surprisals or equivalently, the stochastic system entropy associated with the fully bound state of the cooperative and non-cooperative cases. The criteria of cooperativity in terms of C is compared with that of the Hill coefficient and gives a microscopic insight on the cooperative binding of substrate on a single oligomeric enzyme which is usually characterized by macroscopic reaction rate.

Entropic estimate of cooperative binding of substrate on a single oligomeric enzyme: An index of cooperativity

Here we have studied the nonequilibrium thermodynamic response of a voltage-gated Shaker potassium ion channel using a stochastic master equation. For a constant external voltage, the system reaches equilibrium indicated by the vanishing total entropy production rate, whereas for oscillating voltage the current and entropy production rates show dynamic hysteretic behavior. Here we have shown quantitatively that although the hysteresis loop area vanishes in low and high frequency domains of the external voltage, they are thermodynamically distinguishable. In the very low frequency domain, the system remains close to equilibrium, whereas at high frequencies it goes to a nonequilibrium steady state (NESS) associated with a finite value of dissipation function. At NESS, the efficiency of the ion conduction can also be related with the nonlinear dependence of the dissipation function on the power of the external field. Another intriguing aspect is that, at the high frequency limit, the total entropy production rate oscillates at NESS with half of the time period of the external voltage.

Entropy hysteresis and nonequilibrium thermodynamic efficiency of ion conduction in a voltage-gated potassium ion channel

Motivated by the single molecule enzymatic experiments, we have provided a master equation description of enzyme catalysis in a chemiostatic condition for an immobilized oligomeric molecule with many equivalent active sites. The random attachment and detachment of substrate molecules on the various active sites of the oligomeric enzyme is studied in terms of the classical parameters of the Michaelis-Menten type process. In the limit of single molecule process, the master equation approach gives the result of waiting time distribution. On the other hand, for a large number of equivalent active sites or a few numbers of active sites with large Michaelis constant, the master equation gives a Poisson distribution in the nonequilibrium steady state. For the oligomeric enzyme, the net rate of the reaction in the nonequilibrium steady state is multiplied by the number of active sites which is further enhanced by more than two orders of magnitude with the application of external force of 10-100 pN through the techniques of atomic force microscopy. Substrate flux and reaction rate constants have interesting consequences on the dynamics and at nonequilibrium steady state which can be the controlling factors for macroscopic biochemical processes.

Biswajit Das

Papers

Entropic estimate of cooperative binding of substrate on a single oligomeric enzyme: An index of cooperativity

Entropy hysteresis and nonequilibrium thermodynamic efficiency of ion conduction in a voltage-gated potassium ion channel

Master equation approach to single oligomeric enzyme catalysis: Mechanically controlled further catalysis

Stochastic theory of interfacial enzyme kinetics: A kinetic Monte Carlo study

Nonequilibrium response of a voltage gated sodium ion channel and biophysical characterization of dynamic hysteresis.