Piecewise Linear Potential Research Articles

Investigation of Simultaneous and Sequential Cooperative Homotropic Inhibitor Binding to the Catalytic Chamber of SARS-CoV-2 RNA-dependent RNA Polymerase (RdRp).

In our previous report, the unique architecture of the catalytic chamber of the SARS-CoV-2 RNA-dependent RNA polymerase (RdRp), which harbours two distinctive binding sites, was fully characterized at molecular level. The significant differences in the two binding sites BS1 and BS2 in terms of binding pockets motif, as well as the preferential affinities of eight anti-viral drugs to each of the two binding sites were described. Recent Cryogenic Electron Microscopy (Cryo-EM) studies on the RdRp revealed that two suramin molecules, a SARS-CoV-2 inhibitor, bind to RdRp in two different sites with distinctive interaction landscape. Here, we provide the first account of investigating the combined inhibitor binding to both binding sites, and whether the binding of two inhibitors molecules concurrently is "Cooperative binding" or not. It should be noted that the binding of inhibitors to different sites do not necessary constitute mutually independent events, therefore, we investigated two scenarios to better understand cooperativity: simultaneous binding and sequential binding. It has been demonstrated by binding free energy calculations (MM/PBSA) and piecewise linear potential (PLP) interaction energy analysis that the co-binding of two suramin molecules is not cooperative in nature; rather, when compared to individual binding, both molecules adversely affect one another's binding affinities. This observation appeared to be primarily due to RdRp's rigidity, which prevented both ligands from fitting comfortably within the catalytic chamber. Instead, the suramin molecules showed a tendency to change their orientation within the binding pockets in order to maintain their binding to the protein, but at the expense of the ligand internal energies. Although co-binding resulted in the loss of several important key interactions, a few interactions were conserved, and these appear to be crucial in preserving the binding of ligands in the active site. The structural and mechanistic details of this study will be useful for future research on creating and developing RdRp inhibitors against SARS-CoV-2.

Mpemba effect for a Brownian particle trapped in a single well potential.

The Mpemba effect refers to the counterintuitive phenomenon of a hotter system equilibrating faster than a colder system when both are quenched to the same low temperature. For a Brownian particle trapped in a piecewise linear single well potential that is devoid of any other metastable minima, we show the existence of the Mpemba effect for a wide range of parameters through an exact solution. This result challenges the prevalent explanation of the Mpemba effect that requires the energy landscape to be rugged with multiple minima. We also demonstrate the existence of inverse and strong Mpemba effects.

Molecular Docking Study of New Sorafenib Analogues as Platelet-Derived Growth Factor Receptor Inhibitors for the Treatment of Cancer.

Cancer is a disease triggered by an uncontrolled growth of a group of cells usually from a single cell. Chemotherapy is a common and systematic therapy that involves the use of anticancer drugs also known as chemotherapeutical agents to treat cancer. Tyrosine kinases are a subset of protein kinases that are a family of over 90 enzymes that selectively phosphorylate tyrosine residues in various substrates. Receptors with internal tyrosine kinase activity mediate the actions of several growth factors, differentiation factors, and hormones, resulting in the reproduction and differentiation of the affected cells. In the fight against cancer, the platelet-derived growth factor receptor has emerged as a novel target via inhibition of this receptor resulting in the inhibition of tyrosine kinase cascade. Docking investigations were conducted using the Genetic Optimization for Ligand Docking (GOLD) Suite (v. 5.7.1) from the Cambridge Crystallographic Data Center. A high-definition X-ray crystallography of the platelet-derived growth factor protein [Protein Data Bank (PDB) ID 6JOL] was downloaded from the website PDB with a resolution of 2 A. Compounds II, III, VII, and VIII have greater binding energies than the GOLD standard medication sorafenib, which gives Piecewise Linear Potential (PLP) fitness value (85.3). Other ligands exhibit good inhibitory action and docking scores comparable to that of the reference ligand sorafenib.

Docking Study, ADMET Profiling of the Designed 3-Chlorobenzo[b]thiophene-2-Carbonyl Chloride Derivatives: Promising Anti-Breast Cancer Activity

Globally, breast cancer accounted for approximately 700٫000 deaths in 2020. Estrogen receptor alpha (ERα) is the primary pathway for breast cancer treatment. Tamoxifen (TAM) is the most extensively used drug for estrogen receptor (+) breast cancer. However, it is implicated in endometrial carcinoma, pulmonary thrombus, stroke, and breast cancer resistance. This research aims to overcome these issues in selectivity and side effects. The study was conducted in a new drug design using Structure Based Drug Design (SBDD) approach, molecular docking with ERα and then predicting their ADMET (Absorption, Distribution, Metabolism, Excretion and Toxicity) profile. GOLD (Genetic Optimization for Ligand Docking, v.5.7.1), SwissADME and BIOVIA Discovery Studio Visualizer 2020 applications used for the predicting and analyzing of the results. It has shown that all of our compounds (MFA1-8) where a higher PLP (the Piecewise Linear Potential) fitness scores with a range (66.21-77.20) than (60.96) for TAM with different molecular interactions such as H- bondings and non-covalent hydrophobic interactions with the pose. They were all exhibited better conformation to prevent helix-12 (H-12) from repositioning over the opening of the binding pocket. This work generated fresh insight into the significant hydrophobic bonding with LEU428. MFA1-8 showed better pharmacokinetics, drug-likeness, and toxicity profiles than TAM. These findings suggest that these novel 3-chlorobenzo[b]thiophene-2-carbonyl chloride derivatives could serve as the lead compounds to fight breast cancer by inhibiting the ERα pathway.

A high quality, industrial data set for binding affinity prediction: performance comparison in different early drug discovery scenarios.

We release a new, high quality data set of 1162 PDE10A inhibitors with experimentally determined binding affinities together with 77 PDE10A X-ray co-crystal structures from a Roche legacy project. This data set is used to compare the performance of different 2D- and 3D-machine learning (ML) as well as empirical scoring functions for predicting binding affinities with high throughput. We simulate use cases that are relevant in the lead optimization phase of early drug discovery. ML methods perform well at interpolation, but poorly in extrapolation scenarios-which are most relevant to a real-world application. Moreover, we find that investing into the docking workflow for binding pose generation using multi-template docking is rewarded with an improved scoring performance. A combination of 2D-ML and 3D scoring using a modified piecewise linear potential shows best overall performance, combining information on the protein environment with learning from existing SAR data.

Reducing mean first passage times with intermittent confining potentials: a realization of resetting processes

During a random search, resetting the searcher’s position from time to time to the starting point often reduces the mean completion time of the process. Although many different resetting models have been studied over the past ten years, only a few can be physically implemented. Here we study theoretically a protocol that can be realised experimentally and which exhibits unusual optimization properties. A Brownian particle is subject to an arbitrary confining potential v(x) that is switched on and off intermittently at fixed rates. Motion is constrained between an absorbing wall located at the origin and a reflective wall. When the walls are sufficiently far apart, the interplay between free diffusion during the ‘off’ phases and attraction toward the potential minimum during the ‘on’ phases give rise to rich behaviours, not observed in ideal resetting models. For potentials of the form v(x) = k|x − x 0| n /n, with n > 0, the switch-on and switch-off rates that minimise the mean first passage time (MFPT) to the origin undergo a continuous phase transition as the potential stiffness k is varied. When k is above a critical value k c, potential intermittency enhances the target encounter: the minimal MFPT is lower than the Kramer time and is attained for a non-vanishing pair of switching rates. We focus on the harmonic case n = 2, extending previous results for the piecewise linear potential (n = 1) in unbounded domains. We also study the non-equilibrium stationary states emerging in this process.

Path-integral treatment of quantum bouncers

The one-sided bouncer (1SB) and the symmetric bouncer (SB) involve a one-dimensional particle in a piecewise linear potential. For such problems, the time-dependent quantum mechanical propagator cannot be found in closed form. The semiclassical Feynman path integral (PI) is a very appealing approach, as it approximates the propagator by a closed-form expression (a sum over a finite number of classical paths). In this paper we solve the classical path enumeration problem. We obtain closed-form expressions for the initial velocity, bounce times, focal times, action, van Vleck determinant, and Morse index for each classical path. We calculate the propagator within the semiclassical approximation. The numerical results agree with eigenfunction expansion results away from caustics. We derive mappings between the 1SB and SB, which explains why each bounce of the 1SB increases the Morse index by 2 and results in a phase change of π. We interpret the semiclassical Feynman PI to obtain visualizations of matter wave propagation based on interference between classical paths, in analogy with the traditional visualization of light wave propagation as interference between classical ray paths.

Identification of potential inhibitory compounds of an essential nucleotide biosynthesis enzyme, 6‐Phosphogluconate Dehydrogenase in Plasmodium Vivax

The erythrocytic infection and onset of malaria is a result of the protozoan parasitism from the Plasmodium genus. In 2018, there were an estimated 228 million cases of malaria worldwide largely attributed to P. falciparum and P. vivax infections. Current chemotherapy efforts against the Plasmodium parasites target mitochondrial function (Malarone), the 30s ribosomal subunit (Vibramycin) and prevention of the parasite's abilities to degrade the host's hemoglobin (Chloroquine), however, rising mutations have decreased their therapeutic effects. The essentiality of 6-phosphogluconate dehydrogenase (6PGDH) is demonstrated in its oxidation of 6-phosphogluconate (6PG) to ribulose-5-phosphate and the reduction of nicotinamide adenine dinucleotide phosphate (NADP+) to provide precursors in nucleic acid synthesis. Here, the homologous 3D structure of 6PGDH from P. falciparum was used for molecular docking studies since there is not a published structure of P. vivax 6PGDH in the Protein Data Bank (PDB). The structures of 6PGDH from P. falciparum and P. vivax display 80.30% sequence identity. The apo form of Pf6PGDH (PDB ID: 6FQX) was aligned with the structures containing the 6PG substrate (PDB ID: 6FQZ) and NADP+ coenzyme (PDB ID: 6FQY) ligands in order to define the active site. The 6PGDH protein was virtually screened against 307,735 ligands from several diverse compound libraries using the molecular docking program Genetic Optimization Ligand Docking (GOLD) and ranked for their predicted binding fitness based on empirical scoring functions of the employed genetic algorithm. The highest binding compound, ZINC55224876, received a GOLD Fitness score of 91.1 and demonstrated strong hydrophobic interactions as quantified by the piecewise linear potential (PLP) sub-score. The ligands gathered to act as controls were obtained from studies conducted on Trypanosoma brucei 6PGDH that demonstrated a reduction in protein function to act as positive controls. The screening of the positive and negative control ligands against the Pf6PGDH structure culminated in GOLD Fitness scores of 74.76 for the highest to the lowest score of 41.75. The top 10% of the highest overall scoring library, the Zinc ChemBridge library (124,345 ligands) resulted in a range of GOLD Fitness scores from 91.1- 59.73, whereas the lowest overall scoring library, Fragment (3,963 ligands) received scores ranging from 59.84-20.88. The resulting top scoring ligands may further be used for future enzyme assays in the process of drug discovery.

STiDi-BP: Spike time displacement based error backpropagation in multilayer spiking neural networks

Error backpropagation is the most common approach for direct training of spiking neural networks. However, the non-differentiability of spiking neurons makes the backpropagation of error a challenge. In this paper, we introduce a new temporal learning algorithm, STiDi-BP, in which we ignore backward recursive gradient computation, and to avoid the non-differentiability of SNNs, we use a linear approximation to compute the derivative of latency with respect to the potential. We apply gradient descent to each layer independently based on an estimation of the temporal error in that layer. To do so, we calculate the desired firing time of each neuron and compare it to its actual firing time. In STiDi-BP, we employ the time-to-first-spike temporal coding, one spike per neuron, and use spiking neuron models with piecewise linear postsynaptic potential which provide large computational benefits. To evaluate the performance of the proposed learning rule, we run three experiments on the XOR problem, the face/motorbike categories of the Caltech 101 dataset, and the MNIST dataset. Experimental results show that the STiDi-BP outperforms traditional BP in terms of accuracy and/or computational cost.

In silico investigation on alkaloids of Rauwolfia serpentina as potential inhibitors of 3-hydroxy-3-methyl-glutaryl-CoA reductase.

Present work aimed to investigate the in silico activity of the alkaloids of roots of Rauwolfia serpentina as inhibitors of 3-hydroxy-3-methyl-glutaryl-CoA reductase (HMGCR). For this purpose, the three-dimensional (3D) structure of the protein HMGCR (PDB ID: 1HW9) was downloaded from Protein Data Bank (PDB) database, as a target enzyme. The structures of twelve alkaloids from the roots of R. serpentina were selected as ligands and docked with the selected HMGCR enzyme using Molegro Virtual Docker (MVD) software. The software ‘MVD’ computes the binding (atom) energies of selected protein (enzyme) and each ligand at minimum energetic conformation state by using the PLP (Piecewise Linear Potential) scoring mechanism. Docking results of twelve tested alkaloids showed that five alkaloids including compound 1 (ajmalicine), 2 (reserpine), 3 (indobinine), 4 (yohimbine), and 5 (indobine) have displayed the highest MolDock scores and best fit within the prominent active site residues (positioned between 684 and 692 of cis-loop) of HMGCR. According to the lowest MolDock energies obtained through non-covalent interactions of alkaloids with HMGCR, these are characterized to be the potential inhibitors of HMGCR. Therefore, the alkaloids from R. serpentina can effectively suppress the cholesterol biosynthesis pathway through inhibition of HMGCR and can serve as potential lead compounds for the development of new drugs for the treatment of hyperlipidaemia.

Mathematical modeling of rotating disk states

We consider the problem of a rapidly rotating disk in the elastic-plastic state. The piecewise linear plasticity condition in general form is chosen. It is believed that all plastic curves have the common point of intersection which corresponds to uniaxial tension. For external parameters, we obtain the conditions that determine the probability of inception of plastic zones. It is shown that plastic zones could incept in the center of the disk and/or on the boundary of it. The problem in the plastic zone is statically determinate. The case when the plastic zone occupies some central part (core) of the disk, where one regime of plastic condition is fulfilled, is considered. In order to estimate the stress state inside the elastic zone of the disk, equivalent stress which is equal to the chosen plasticity function is defined. In order to define the relationship between plastic deformations and stresses, the piecewise linear plastic potential being equal to the plasticity function is chosen. The plastic incompressible body is considered. The associated flow rule can be integrated so that the problem of getting displacements turns into quasistatic one. The problem of determining displacements in the plastic region leads to a first-order differential equation with respect to the radial component of the displacement vector. Therefore the continuity condition for displacements at the elastic-plastic boundary and the assumption that the displacements in the center of the disk are equal to zero leads to an overdetermined problem. So, only the continuity condition for displacements at the elasto-plastic boundary is accepted. It is assumed that plastic deformations at the elastic-plastic boundary are equal to zero. It is shown that displacements at the center of the disk are equal to zero automatically for all piecewise linear conditions of plasticity apart from the Tresca yield criterion. For the Schmidt–Ishlinskii yield criterion, all deformations at the center of the disk attain finite values. Meanwhile, for other piecewise linear conditions, plastic deformations at the center of the disk attain infinitely large values. This explains the discontinuity of displacements at the center of the disk for the Tresca yield criterion. The calculation results are presented as graphs of stresses, displacements, and deformations.

Stochastic Resonance and First Arrival Time for Excitable Systems

We study the noise induced thermally activated barrier crossing of Brownian particles that hop in a piecewise linear potential. Using the exact analytic solutions and via numerical simulations not only we explore the dependence for the first passage time of a single particle but also we calculate the first arrival time for one particle out of N particles. The first arrival time decreases as the number of particles increases as expected. We then explore the thermally activated barrier crossing rate of the system in the presence of time varying signal. The dependence of signal to noise ratio SNR as well as the power amplification ( $$\eta $$ ) on model parameters is explored. $$\eta $$ and SNR depict a pronounced peak at particular noise strength. In the presence of N particles, $$\eta $$ is considerably amplified as N steps up showing the weak periodic signal plays a vital role in controlling the noise induced dynamics of the system. Moreover, for the sake of generality, the viscous friction $$\gamma $$ is considered to decrease exponentially when the temperature T of the medium increases ( $$\gamma =Be^{-A T}$$ ) as proposed originally by Reynolds (Philos Trans R Soc Lond 177:157, 1886).

In Russian

The review of the piecewise linear potential reliefs approximation of actual and model nanodevices (Brownian motors) rectifying chaotic Brownian motion in systems with broken reflexion symmetry under the action of an external fluctuation perturbation in the absence of macroscopic driving forces is given. Relations are derived to carry out this approximation.

Kramers problem for a dimer: Effect of noise correlations.

The Kramers problem for a dimer in a bistable piecewise linear potential is studied in the presence of correlated noise processes. The effect of such a correlation is to redistribute the thermal power between the dynamical degrees of freedom, and this leads to significant deviations in the dynamics of the system from the case of independent noise processes. The distribution of first passage times from one minima to the basin of attraction of the other minima is found to have exponentially decaying tails with the parameter dependent on the amount of correlation and the coupling between the particles. The strong coupling limit of the problem is analyzed using adiabatic elimination, where it is found that the initial probability density relaxes towards a stationary value on the same time scale as the mean escape time when the noise intensity of the system is low. For higher noise fluctuations, the relaxation towards the stationary state is slower in comparison to escape times. In the extreme limit of perfect anticorrelation, the random dynamical system behaves as a deterministic system in a steady state in which the center of mass starting from the unstable maxima moves down the hill and gets trapped in the potential minima. The implications for polymer dynamics in a potential are discussed.

Enhancing energy harvesting by coupling monostable oscillators

The performance of a ring of linearly coupled, monostable nonlinear oscillators is optimized towards its goal of acting as energy harvester---through piezoelectric transduction---of mesoscopic fluctuations, which are modeled as Ornstein--Uhlenbeck noises. For a single oscillator, the maximum output voltage and overall efficiency are attained for a soft piecewise-linear potential (providing a weak attractive constant force) but they are still fairly large for a harmonic potential. When several harmonic springs are linearly and bidirectionally coupled to form a ring, it is found that counter-phase coupling can largely improve the performance while in-phase coupling worsens it. Moreover, it turns out that few (two or three) coupled units perform better than more.

Exact results on diffusion in a piecewise linear potential with a time-dependent sink

The Smoluchowski equation with a time-dependent sink term is solved exactly. In this method, knowing the probability distribution P(0, s) at the origin, allows deriving the probability distribution P(x, s) at all positions. Exact solutions of the Smoluchowski equation are also provided in different cases where the sink term has linear, constant, inverse, and exponential variation in time.

Transport of interacting self-propelled Brownian particles in a common ratchet potential

Transport of self-propelled particles coupled via springs is studied in the presence of a piecewise linear potential. By the means of numerical simulations, we find that the interaction between particles plays a key role in the transport. On increasing the spring constant, the average velocity decreases, and finally tends to zero. There exists an optimal value of the self-propelled velocity at which the average velocity takes its maximal value. When the number of particles tends to infinity, the ratcheting transport disappears. The results we have presented contribute further to understanding the transport of interacting active particles.

Fluctuation-induced transport of two coupled particles: effect of the interparticle interaction.

We consider a system of two coupled particles fluctuating between two states, with different interparticle interaction potentials and particle friction coefficients. An external action drives the interstate transitions that induces reciprocating motion along the internal coordinate x (the interparticle distance). The system moves unidirectionally due to rectification of the internal motion by asymmetric friction fluctuations and thus operates as a dimeric motor that converts input energy into net movement. We focus on how the law of interaction between the particles affects the dimer transport and, in particular, the role of thermal noise in the motion inducing mechanism. It is argued that if the interaction potential behaves at large distances as x(α), depending on the value of the exponent α, the thermal noise plays a constructive (α > 2), neutral (α = 2), or destructive (α < 2) role. In the case of α = 1, corresponding piecewise linear potential profiles, an exact solution is obtained and discussed in detail.

Quantum anharmonic oscillator: The airy function approach

New and simple numerical method is being reported to solve anharmonic oscillator problems. The method is setup to approach the real potential V(x) of the anharmonic oscillator system as a piecewise linear potential u(x) and to solve the Schrödinger equation of the system using the Airy function. Then, solutions continuity conditions lead to the energy quantification condition, and consequently, the energy eigenvalues. For testing purpose, the method was applied on the sextic and octic oscillators systems. The proposed method is found to be realistic, computationally simple, and having high degrees of accuracy. In addition, it can be applied to any form of potential. The results obtained by the proposed method were seen closely agreeing with results reached by other complicated methods.

Speeding up the first-passage for subdiffusion by introducing a finite potential barrier

We show that for a subdiffusive continuous time random walk with scale-free waiting time distribution the first-passage dynamics on a finite interval can be optimized by introduction of a piecewise linear potential barrier. Analytical results for the survival probability and first-passage density based on the fractional Fokker–Planck equation are shown to agree well with Monte Carlo simulations results. As an application we discuss an improved design for efficient translocation of gradient copolymers compared to homopolymer translocation in a quasi-equilibrium approximation.