Statistical Thermodynamics of General Minimal Diffusion Processes: Constuction, Invariant Density, Reversibility and Entropy Production

The solution to nonlinear Fokker-Planck equation is constructed in terms of the minimal Markov semigroup generated by the equation. The semigroup is obtained by a purely functional analytical method via Hille-Yosida theorem. The existence of the positive invariant measure with density is established and a weak form of Foguel alternative proven. We show the equivalence among self-adjoint of the elliptic operator, time-reversibility, and zero entropy production rate of the stationary diffusion process. A thermodynamic theory for diffusion processes emerges.Comment: 23 page

Stochastic Dynamics of Electrical Membrane with Voltage-Dependent Ion Channel Fluctuations

Brownian ratchet like stochastic theory for the electrochemical membrane system of Hodgkin-Huxley (HH) is developed. The system is characterized by a continuous variable $Q_m(t)$, representing mobile membrane charge density, and a discrete variable $K_t$ representing ion channel conformational dynamics. A Nernst-Planck-Nyquist-Johnson type equilibrium is obtained when multiple conducting ions have a common reversal potential. Detailed balance yields a previously unknown relation between the channel switching rates and membrane capacitance, bypassing Eyring-type explicit treatment of gating charge kinetics. From a molecular structural standpoint, membrane charge $Q_m$ is a more natural dynamic variable than potential $V_m$; our formalism treats $Q_m$-dependent conformational transition rates $\lambda_{ij}$ as intrinsic parameters. Therefore in principle, $\lambda_{ij}$ vs. $V_m$ is experimental protocol dependent,e.g., different from voltage or charge clamping measurements. For constant membrane capacitance per unit area $C_m$ and neglecting membrane potential induced by gating charges, $V_m=Q_m/C_m$, and HH's formalism is recovered. The presence of two types of ions, with different channels and reversal potentials, gives rise to a nonequilibrium steady state with positive entropy production $e_p$. For rapidly fluctuating channels, an expression for $e_p$ is obtained.Comment: 8 pages, two figure

Sensitivity Amplification in the Phosphorylation-Dephosphorylation Cycle: Nonequilibrium steady states, chemical master equation and temporal cooperativity

A new type of cooperativity termed temporal cooperativity [Biophys. Chem. 105 585-593 (2003), Annu. Rev. Phys. Chem. 58 113-142 (2007)], emerges in the signal transduction module of phosphorylation-dephosphorylation cycle (PdPC). It utilizes multiple kinetic cycles in time, in contrast to allosteric cooperativity that utilizes multiple subunits in a protein. In the present paper, we thoroughly investigate both the deterministic (microscopic) and stochastic (mesoscopic) models, and focus on the identification of the source of temporal cooperativity via comparing with allosteric cooperativity. A thermodynamic analysis confirms again the claim that the chemical equilibrium state exists if and only if the phosphorylation potential $\triangle G=0$, in which case the amplification of sensitivity is completely abolished. Then we provide comprehensive theoretical and numerical analysis with the first-order and zero-order assumptions in phosphorylation-dephosphorylation cycle respectively. Furthermore, it is interestingly found that the underlying mathematics of temporal cooperativity and allosteric cooperativity are equivalent, and both of them can be expressed by "dissociation constants", which also characterizes the essential differences between the simple and ultrasensitive PdPC switches. Nevertheless, the degree of allosteric cooperativity is restricted by the total number of sites in a single enzyme molecule which can not be freely regulated, while temporal cooperativity is only restricted by the total number of molecules of the target protein which can be regulated in a wide range and gives rise to the ultrasensitivity phenomenon.Comment: 42 pages, 13 figure

Holomorphic harmonic analysis on complex reductive groups

We define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties like the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of $K$-admissible measures. We prove that $K$-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of $K$-admissible measures.Comment: 15 pages, revision of a preprint by the first author in 200

AAANE: Attention-based Adversarial Autoencoder for Multi-scale Network Embedding

Network embedding represents nodes in a continuous vector space and preserves structure information from the Network. Existing methods usually adopt a "one-size-fits-all" approach when concerning multi-scale structure information, such as first- and second-order proximity of nodes, ignoring the fact that different scales play different roles in the embedding learning. In this paper, we propose an Attention-based Adversarial Autoencoder Network Embedding(AAANE) framework, which promotes the collaboration of different scales and lets them vote for robust representations. The proposed AAANE consists of two components: 1) Attention-based autoencoder effectively capture the highly non-linear network structure, which can de-emphasize irrelevant scales during training. 2) An adversarial regularization guides the autoencoder learn robust representations by matching the posterior distribution of the latent embeddings to given prior distribution. This is the first attempt to introduce attention mechanisms to multi-scale network embedding. Experimental results on real-world networks show that our learned attention parameters are different for every network and the proposed approach outperforms existing state-of-the-art approaches for network embedding.Comment: 8 pages, 5 figure