Data-driven Efficient Solvers and Predictions of Conformational Transitions for Langevin Dynamics on Manifold in High Dimensions

We work on dynamic problems with collected data $\{\mathsf{x}_i\}$ that distributed on a manifold $\mathcal{M}\subset\mathbb{R}^p$. Through the diffusion map, we first learn the reaction coordinates $\{\mathsf{y}_i\}\subset \mathcal{N}$ where $\mathcal{N}$ is a manifold isometrically embedded into an Euclidean space $\mathbb{R}^\ell$ for $\ell \ll p$. The reaction coordinates enable us to obtain an efficient approximation for the dynamics described by a Fokker-Planck equation on the manifold $\mathcal{N}$. By using the reaction coordinates, we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of $\mathcal{N}$. Furthermore, we provide a weighted $L^2$ convergence analysis of the upwind scheme to the Fokker-Planck equation. The proposed upwind scheme leads to a Markov chain with transition probability between the nearest neighbor points. We can benefit from such property to directly conduct manifold-related computations such as finding the optimal coarse-grained network and the minimal energy path that represents chemical reactions or conformational changes. To establish the Fokker-Planck equation, we need to acquire information about the equilibrium potential of the physical system on $\mathcal{N}$. Hence, we apply a Gaussian Process regression algorithm to generate equilibrium potential for a new physical system with new parameters. Combining with the proposed upwind scheme, we can calculate the trajectory of the Fokker-Planck equation on $\mathcal{N}$ based on the generated equilibrium potential. Finally, we develop an algorithm to pullback the trajectory to the original high dimensional space as a generative data for the new physical system.Comment: 59 pages, 16 figure

Thermophilicity and catalytic efficiency in dihydrofolate reductase

This thesis presents an investigation of the hydrogen transfer reactions between dihydrofolate (H2F) and NADPH that are catalysed by the dihydrofolate reductase (DHFR) isolated from Geobacillus stearothermophilus (BsDHFR) as well as an artificial hybrid originating from the DHFRs from mesophilic Escherichia coli (EcDHFR) and hyperthermophilic Thermotoga maritima (TmDHFR). A broad spectrum of studies, focusing on the relationship between structure, thermostability and kinetics, showed that the catalytic behaviour of BsDHFR is generally similar to other monomeric DHFRs, including ones found in the mesophile Escherichia coli and the psychrophile Moritella profunda, but significantly different from the dimeric TmDHFR. The fact that all monomeric DHFRs display similar catalytic behaviour, regardless of their widely different optimal temperatures, suggests that thermostability does not directly relate to catalytic efficiency. The biophysical differences between monomeric DHFRs and TmDHFR are likely derived from the dimeric nature of the hyperthermophilic enzyme. An artificial dimeric variant of EcDHFR, Xet-3, was prepared by introducing residues at the dimer interface of TmDHFR. While thermostability of this variant is enhanced, it showed a great decrease in its steady-state and pre-steady-state rate constants. Given that the corresponding rate constants did not increase when the loops are released in the monomeric variant of TmDHFR, the lowered catalytic ability in Xet-3 is likely a consequence of geometric distortion of the active site and loss of loop flexibility that is catalytically important in EcDHFR. In contrast, the relatively poor activity of TmDHFR is not simply a consequence of reduced loop flexibility; the dimer interface of TmDHFR plays a rather complicated role in catalysis

A constitutive model for particulate-reinforced titanium matrix composites subjected to high strain rates and high temperatures

Quasi-static and dynamic tension tests were conducted to study the mechanical properties of particulate-reinforced titanium matrix composites at strain rates ranging from 0.0001/s to 1000/s and at temperatures ranging from 20 °C to 650 °C Based on the experimental results, a constitutive model, which considers the effects of strain rate and temperature on hot deformation behavior, was proposed for particulate-reinforced titanium matrix composites subjected to high strain rates and high temperatures by using Zener-Hollomon equations including Arrhenius terms. All the material constants used in the model were identified by fitting Zener-Hollomon equations against the experimental results. By comparison of theoretical predictions presented by the model with experimental results, a good agreement was achieved, which indicates that this constitutive model can give an accurate and precise estimate for high temperature flow stress for the studied titanium matrix composites and can be used for numerical simulations of hot deformation behavior of the composites

Evolution properties of the community members for dynamic networks

The collective behaviors of community members for dynamic social networks are significant for understanding evolution features of communities. In this Letter, we empirically investigate the evolution properties of the new community members for dynamic networks. Firstly, we separate data sets into different slices, and analyze the statistical properties of new members as well as communities they joined in for these data sets. Then we introduce a parameter φ to describe community evolution between different slices and investigate the dynamic community properties of the new community members. The empirical analyses for the Facebook, APS, Enron and Wiki data sets indicate that both the number of new members and joint communities increase, the ratio declines rapidly and then becomes stable over time, and most of the new members will join in the small size communities that is s≤10s≤10. Furthermore, the proportion of new members in existed communities decreases firstly and then becomes stable and relatively small for these data sets. Our work may be helpful for deeply understanding the evolution properties of community members for social networks

Uncovering the popularity mechanisms for Facebook applications

Understanding the popularity dynamics of online application(App) is significant for the online social systems. In this paper, by dividing the Facebook Apps into different groups in terms of their popularities, we empirically investigate the popularity dynamics for different kinds of Facebook Apps. Then, taking into account the influence of cumulative and recent popularities on the user choice, we present a model to regenerate the growth of popularity for different App groups. The experimental results of 917 Facebook Apps show that as the popularities of Facebook Apps increase, the recent popularity plays more important role. Specifically, the recent popularity plays more important role in regenerating the popularity dynamics for more popular Apps, and the cumulative popularity plays more important role for unpopular Apps. We also conduct temporal analysis on the growth characteristic of individual App by comparing the increment at each time with the average of historical records. The results show that the growth of more popular App tends to fluctuate more greatly. Our work may shed some lights for deeply understanding the popularity mechanism for online applications