Soft-sphere continuum solvation models for nonaqueous solvents.

Solvation effects profoundly influence the characteristics and behavior of chemical systems in liquid solutions. The interaction between solute and solvent molecules intricately impacts solubility, reactivity, stability, and various chemical processes. Continuum solvation models gained prominence in quantum chemistry by implicitly capturing these interactions and enabling efficient investigations of diverse chemical systems in solution. In comparison, continuum solvation models in condensed matter simulation are very recent. Among these, the self-consistent continuum solvation (SCCS) and the soft-sphere continuum solvation models (SSCS) have been among the first to be successfully parameterized and extended to model periodic systems in aqueous solutions and electrolytes. As most continuum approaches, these models depend on a number of parameters that are linked to experimental or theoretical properties of the solvent, or that can be tuned based on reference data. Here, we present a systematic parameterization of the SSCS model for over 100 nonaqueous solvents. We validate the model's efficacy across diverse solvent environments by leveraging experimental solvation-free energies and partition coefficients from comprehensive databases. The average root means square error over all the solvents was calculated as 0.85 kcal/mol which is below the chemical accuracy (1 kcal/mol). Similarly to what has been reported by Hille et al. (J. Chem. Phys. 2019, 150, 041710.) for the SCCS model, a single-parameter model accurately reproduces experimental solvation energies, showcasing the transferability and predictive power of these continuum approaches. Our findings underscore the potential for a unified approach to predict solvation properties, paving the way for enhanced computational studies across various chemical environments.

Agent-Based and Continuum Models for Spatial Dynamics of Infection by Oncolytic Viruses.

The use of oncolytic viruses as cancer treatment has received considerable attention in recent years, however the spatial dynamics of this viral infection is still poorly understood. We present here a stochastic agent-based model describing infected and uninfected cells for solid tumours, which interact with viruses in the absence of an immune response. Two kinds of movement, namely undirected random and pressure-driven movements, are considered: the continuum limit of the models is derived and a systematic comparison between the systems of partial differential equations and the individual-based model, in one and two dimensions, is carried out. In the case of undirected movement, a good agreement between agent-based simulations and the numerical and well-known analytical results for the continuum model is possible. For pressure-driven motion, instead, we observe a wide parameter range in which the infection of the agents remains confined to the center of the tumour, even though the continuum model shows traveling waves of infection; outcomes appear to be more sensitive to stochasticity and uninfected regions appear harder to invade, giving rise to irregular, unpredictable growth patterns. Our results show that the presence of spatial constraints in tumours' microenvironments limiting free expansion has a very significant impact on virotherapy. Outcomes for these tumours suggest a notable increase in variability. All these aspects can have important effects when designing individually tailored therapies where virotherapy is included.

Quantum Chemistry in Solution: Fiftieth Anniversary.

In this perspective, we briefly present the historical context in which, fifty years ago, dielectric continuum models were developed to incorporate solvent effects into quantum mechanical calculations. Since the first self-consistent-field equations including the solvent electrostatic potential (or reaction field) were reported in 1973, continuum models have become extremely popular in the computational chemistry community and are routinely used in a very wide range of applications.

Discrete and continuum models for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes and tumour cells.

We present an individual-based model for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes (CTLs) and tumour cells. In this model, every cell is viewed as an individual agent whose phenotypic state is modelled by a discrete variable. For tumour cells, this variable represents a parameterization of the antigen expression profiles, while for CTLs it represents a parameterization of the target antigens of T-cell receptors (TCRs). We formally derive the deterministic continuum limit of this individual-based model, which comprises a non-local partial differential equation for the phenotype distribution of tumour cells coupled with an integro-differential equation for the phenotype distribution of CTLs. The biologically relevant homogeneous steady-state solutions of the continuum model equations are found. The linear-stability analysis of these steady-state solutions is then carried out in order to identify possible conditions on the model parameters that may lead to different outcomes of immune competition and to the emergence of patterns of phenotypic coevolution between tumour cells and CTLs. We report on computational results of the individual-based model, and show that there is a good agreement between them and analytical and numerical results of the continuum model. These results shed light on the way in which different parameters affect the coevolutionary dynamics between tumour cells and CTLs. Moreover, they support the idea that TCR-tumour antigen binding affinity may be a good intervention target for immunotherapy and offer a theoretical basis for the development of anti-cancer therapy aiming at engineering TCRs so as to shape their affinity for cancer targets.

A Hybrid Discrete-Continuum Modelling Approach to Explore the Impact of T-Cell Infiltration on Anti-tumour Immune Response.

We present a spatial hybrid discrete-continuum modelling framework for the interaction dynamics between tumour cells and cytotoxic T cells, which play a pivotal role in the immune response against tumours. In this framework, tumour cells and T cells are modelled as individual agents while chemokines that drive the chemotactic movement of T cells towards the tumour are modelled as a continuum. We formally derive the continuum counterpart of this model, which is given by a coupled system that comprises an integro-differential equation for the density of tumour cells, a partial differential equation for the density of T cells and a partial differential equation for the concentration of chemokines. We report on computational results of the hybrid model and show that there is an excellent quantitative agreement between them and numerical solutions of the corresponding continuum model. These results shed light on the mechanisms that underlie the emergence of different levels of infiltration of T cells into the tumour and elucidate how T-cell infiltration shapes anti-tumour immune response. Moreover, to present a proof of concept for the idea that, exploiting the computational efficiency of the continuum model, extensive numerical simulations could be carried out, we investigate the impact of T-cell infiltration on the response of tumour cells to different types of anti-cancer immunotherapy.

Computational methods and theory for ion channel research.

Ion channels are fundamental biological devices that act as gates in order to ensure selective ion transport across cellular membranes; their operation constitutes the molecular mechanism through which basic biological functions, such as nerve signal transmission and muscle contraction, are carried out. Here, we review recent results in the field of computational research on ion channels, covering theoretical advances, state-of-the-art simulation approaches, and frontline modeling techniques. We also report on few selected applications of continuum and atomistic methods to characterize the mechanisms of permeation, selectivity, and gating in biological and model channels.

Modelling Nuclear Morphology and Shape Transformation: A Review.

As one of the most important cellular compartments, the nucleus contains genetic materials and separates them from the cytoplasm with the nuclear envelope (NE), a thin membrane that is susceptible to deformations caused by intracellular forces. Interestingly, accumulating evidence has also indicated that the morphology change of NE is tightly related to nuclear mechanotransduction and the pathogenesis of diseases such as cancer and Hutchinson-Gilford Progeria Syndrome. Theoretically, with the help of well-designed experiments, significant progress has been made in understanding the physical mechanisms behind nuclear shape transformation in different cellular processes as well as its biological implications. Here, we review different continuum-level (i.e., energy minimization, boundary integral and finite element-based) approaches that have been developed to predict the morphology and shape change of the cell nucleus. Essential gradients, relative advantages and limitations of each model will be discussed in detail, with the hope of sparking a greater research interest in this important topic in the future.

Simulation of 3D centimeter-scale continuum tumor growth at sub-millimeter resolution via distributed computing.

Simulation of cm-scale tumor growth has generally been constrained by the computational cost to numerically solve the associated equations, with models limited to representing mm-scale or smaller tumors. While the work has proven useful to the study of small tumors and micro-metastases, a biologically-relevant simulation of cm-scale masses as would be typically detected and treated in patients has remained an elusive goal. This study presents a distributed computing (parallelized) implementation of a mixture model of tumor growth to simulate 3D cm-scale vascularized tissue at sub-mm resolution. The numerical solving scheme utilizes a two-stage parallelization framework. The solution is written for GPU computation using the CUDA framework, which handles all Multigrid-related computations. Message Passing Interface (MPI) handles distribution of information across multiple processes, freeing the program from RAM and the processing limitations found on single systems. On each system, Nvidia's CUDA library allows for fast processing of model data using GPU-bound computing on fewer systems. The results show that a combined MPI-CUDA implementation enables the continuum modeling of cm-scale tumors at reasonable computational cost. Further work to calibrate model parameters to particular tumor conditions could enable simulation of patient-specific tumors for clinical application.

Discrete and continuum models of COVID-19 virus, formal solutions, stability and comparison with real data.

Very recently, various mathematical models, for the dynamics of COVID-19 with main contribution of suspected-exposed-infected-recovered people have been proposed. Some models that account for the deceased, quarantined or social distancing functions were also presented. However, in any local space the real data reveals that the effects of lock-down and traveling are significant in decreasing and increasing the impact of this virus respectively. Here, discrete and continuum models for the dynamics of this virus are suggested. The continuum dynamical model is studied in detail. The present model deals with exposed, infected, recovered and deceased individuals (EIRD), which accounts for the health isolation and travelers (HIT) effects. Up to now no exact solutions of the parametric-dependent, nonlinear dynamical system NLDS were found. In this work, our objective is to find the exact solutions of a NLDS. To this issue, a novel approach is presented where a NLDS is recast to a linear dynamical system LDS. This is done by implementing the unified method (UM), with auxiliary equations, which are taken coupled linear ODE's (LDS). Numerical results of the exact solutions are evaluated, which can be applied to data in a local space (or anywhere) when the initial data for the IRD are known. Here, as an example, initial conditions for the components in the model equation of COVID-19, are taken from the real data in Egypt. The results of susceptible, infected, recovered and deceased people are computed. The comparison between the computed results and the real data shows an agreement up to a relative error 1 0 - 3 . On the other hand it is remarked that locking-down plays a dominant role in decreasing the number of infected people. The equilibrium states are determined and it is found that they are stable. This reveals a relevant result that the COVID-19 can be endemic in the case of a disturbance in the number of the exposed people. A disturbance in the form of an increase in the exposed number, leads to an increase in the number of infected people. This result is, globally, valid. Furthermore, initial states control is analyzed, where region of initial conditions for infected and exposed is determined. We developed a software tool to interact with the model and facilitate applying various data of different local spaces.

Multiscale Models for Light-Driven Processes.

Multiscale models combining quantum mechanical and classical descriptions are a very popular strategy to simulate properties and processes of complex systems. Many alternative formulations have been developed, and they are now available in all of the most widely used quantum chemistry packages. Their application to the study of light-driven processes, however, is more recent, and some methodological and numerical problems have yet to be solved. This is especially the case for the polarizable formulation of these models, the recent advances in which we review here. Specifically, we identify and describe the most important specificities that the polarizable formulation introduces into both the simulation of excited-state dynamics and the modeling of excitation energy and electron transfer processes.

Bridging the gap between individual-based and continuum models of growing cell populations.

Continuum models for the spatial dynamics of growing cell populations have been widely used to investigate the mechanisms underpinning tissue development and tumour invasion. These models consist of nonlinear partial differential equations that describe the evolution of cellular densities in response to pressure gradients generated by population growth. Little prior work has explored the relation between such continuum models and related single-cell-based models. We present here a simple stochastic individual-based model for the spatial dynamics of multicellular systems whereby cells undergo pressure-driven movement and pressure-dependent proliferation. We show that nonlinear partial differential equations commonly used to model the spatial dynamics of growing cell populations can be formally derived from the branching random walk that underlies our discrete model. Moreover, we carry out a systematic comparison between the individual-based model and its continuum counterparts, both in the case of one single cell population and in the case of multiple cell populations with different biophysical properties. The outcomes of our comparative study demonstrate that the results of computational simulations of the individual-based model faithfully mirror the qualitative and quantitative properties of the solutions to the corresponding nonlinear partial differential equations. Ultimately, these results illustrate how the simple rules governing the dynamics of single cells in our individual-based model can lead to the emergence of complex spatial patterns of population growth observed in continuum models.

Mathematical modelling of the restenosis process after stent implantation.

The stenting procedure has evolved to become a highly successful technique for the clinical treatment of advanced atherosclerotic lesions in arteries. However, the development of in-stent restenosis remains a key problem. In this work, a novel two-dimensional continuum mathematical model is proposed to describe the complex restenosis process following the insertion of a stent into a coronary artery. The biological species considered to play a key role in restenosis development are growth factors, matrix metalloproteinases, extracellular matrix, smooth muscle cells and endothelial cells. Diffusion-reaction equations are used for modelling the mass balance between species in the arterial wall. Experimental data from the literature have been used in order to estimate model parameters. Moreover, a sensitivity analysis has been performed to study the impact of varying the parameters of the model on the evolution of the biological species. The results demonstrate that this computational model qualitatively captures the key characteristics of the lesion growth and the healing process within an artery subjected to non-physiological mechanical forces. Our results suggest that the arterial wall response is driven by the damage area, smooth muscle cell proliferation and the collagen turnover among other factors.

Using Experimental Data and Information Criteria to Guide Model Selection for Reaction-Diffusion Problems in Mathematical Biology.

Reaction-diffusion models describing the movement, reproduction and death of individuals within a population are key mathematical modelling tools with widespread applications in mathematical biology. A diverse range of such continuum models have been applied in various biological contexts by choosing different flux and source terms in the reaction-diffusion framework. For example, to describe the collective spreading of cell populations, the flux term may be chosen to reflect various movement mechanisms, such as random motion (diffusion), adhesion, haptotaxis, chemokinesis and chemotaxis. The choice of flux terms in specific applications, such as wound healing, is usually made heuristically, and rarely it is tested quantitatively against detailed cell density data. More generally, in mathematical biology, the questions of model validation and model selection have not received the same attention as the questions of model development and model analysis. Many studies do not consider model validation or model selection, and those that do often base the selection of the model on residual error criteria after model calibration is performed using nonlinear regression techniques. In this work, we present a model selection case study, in the context of cell invasion, with a very detailed experimental data set. Using Bayesian analysis and information criteria, we demonstrate that model selection and model validation should account for both residual errors and model complexity. These considerations are often overlooked in the mathematical biology literature. The results we present here provide a straightforward methodology that can be used to guide model selection across a range of applications. Furthermore, the case study we present provides a clear example where neglecting the role of model complexity can give rise to misleading outcomes.

Lattice and continuum modelling of a bioactive porous tissue scaffold.

A contemporary procedure to grow artificial tissue is to seed cells onto a porous biomaterial scaffold and culture it within a perfusion bioreactor to facilitate the transport of nutrients to growing cells. Typical models of cell growth for tissue engineering applications make use of spatially homogeneous or spatially continuous equations to model cell growth, flow of culture medium, nutrient transport and their interactions. The network structure of the physical porous scaffold is often incorporated through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a square grid lattice to demonstrate the importance of explicitly modelling the network structure of the porous scaffold and compare results from this model with those from a modified continuum model from the literature. We capture two-way coupling between cell growth and fluid flow by allowing cells to block pores, and by allowing the shear stress of the fluid to affect cell growth and death. We explore a range of parameters for both models and demonstrate quantitative and qualitative differences between predictions from each of these approaches, including spatial pattern formation and local oscillations in cell density present only in the lattice model. These differences suggest that for some parameter regimes, corresponding to specific cell types and scaffold geometries, the lattice model gives qualitatively different model predictions than typical continuum models. Our results inform model selection for bioactive porous tissue scaffolds, aiding in the development of successful tissue engineering experiments and eventually clinically successful technologies.

Microscale water distribution and its effects on organic carbon decomposition in unsaturated soils.

Microscale water distribution in the subsurface is key to many geochemical and biogeochemical reactions. This study investigated microscale water distribution and movement in unsaturated soils using micro-continuum hydrodynamic models, and examined the effect of microscale water distribution on organic carbon (C) decomposition using a micro-continuum biogeochemical reaction model. The micro-continuum hydrodynamic model that relates capillary pressure to porosity captured the measured water imbibition curve at the core scale, and exhibited reasonable water distribution and movement at the microscale. The simulations of organic C decomposition illustrate that microscale water distribution strongly affected the distribution of C decomposition rates by regulating the availability of dissolved organic C and oxygen. Particularly, changes in water distribution altered the location and intensity of reactive hotspots and thereby CO2 flux from soils. The microscale interactions between water content and organic C decomposition rate provide underlying mechanisms for explaining macroscale phenomenon observed in laboratory and fields. Overall, this study presents a useful tool for explicating hydro-biogeochemical behaviors in the subsurface by integrating micro-continuum hydrodynamic and biogeochemical reaction modeling.

Continuum and discrete approach in modeling biofilm development and structure: a review.

The scientific community has recognized that almost 99% of the microbial life on earth is represented by biofilms. Considering the impacts of their sessile lifestyle on both natural and human activities, extensive experimental activity has been carried out to understand how biofilms grow and interact with the environment. Many mathematical models have also been developed to simulate and elucidate the main processes characterizing the biofilm growth. Two main mathematical approaches for biomass representation can be distinguished: continuum and discrete. This review is aimed at exploring the main characteristics of each approach. Continuum models can simulate the biofilm processes in a quantitative and deterministic way. However, they require a multidimensional formulation to take into account the biofilm spatial heterogeneity, which makes the models quite complicated, requiring significant computational effort. Discrete models are more recent and can represent the typical multidimensional structural heterogeneity of biofilm reflecting the experimental expectations, but they generate computational results including elements of randomness and introduce stochastic effects into the solutions.

On the role of micro-inertia in enriched continuum mechanics.

In this paper, the role of gradient micro-inertia terms [Formula: see text] and free micro-inertia terms [Formula: see text] is investigated to unveil their respective effects on the dynamic behaviour of band-gap metamaterials. We show that the term [Formula: see text] alone is only able to disclose relatively simplified dispersive behaviour. On the other hand, the term [Formula: see text] alone describes the full complex behaviour of band-gap metamaterials. A suitable mixing of the two micro-inertia terms allows us to describe a new feature of the relaxed-micromorphic model, i.e. the description of a second band-gap occurring for higher frequencies. We also show that a split of the gradient micro-inertia [Formula: see text], in the sense of Cartan-Lie decomposition of matrices, allows us to flatten separately the longitudinal and transverse optic branches, thus giving us the possibility of a second band-gap. Finally, we investigate the effect of the gradient inertia [Formula: see text] on more classical enriched models such as the Mindlin-Eringen and the internal variable ones. We find that the addition of such a gradient micro-inertia allows for the onset of one band-gap in the Mindlin-Eringen model and three band-gaps in the internal variable model. In this last case, however, non-local effects cannot be accounted for, which is a too drastic simplification for most metamaterials. We conclude that, even when adding gradient micro-inertia terms, the relaxed micromorphic model remains the best performing one, among the considered enriched models, for the description of non-local band-gap metamaterials.

First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model.

In this paper, we propose the first estimate of some elastic parameters of the relaxed micromorphic model on the basis of real experiments of transmission of longitudinal plane waves across an interface separating a classical Cauchy material (steel plate) and a phononic crystal (steel plate with fluid-filled holes). A procedure is set up in order to identify the parameters of the relaxed micromorphic model by superimposing the experimentally based profile of the reflection coefficient (plotted as function of the wave-frequency) with the analogous profile obtained via numerical simulations. We determine five out of six constitutive parameters which are featured by the relaxed micromorphic model in the isotropic case, plus the determination of the micro-inertia parameter. The sixth elastic parameter, namely the Cosserat couple modulus µc , still remains undetermined, since experiments on transverse incident waves are not yet available. A fundamental result of this paper is the estimate of the non-locality intrinsically associated with the underlying microstructure of the metamaterial. We show that the characteristic length Lc measuring the non-locality of the phononic crystal is of the order of [Formula: see text] of the diameter of its fluid-filled holes.

A Hybrid Model for the Computationally-Efficient Simulation of the Cerebellar Granular Layer.

The aim of the present paper is to efficiently describe the membrane potential dynamics of neural populations formed by species having a high density difference in specific brain areas. We propose a hybrid model whose main ingredients are a conductance-based model (ODE system) and its continuous counterpart (PDE system) obtained through a limit process in which the number of neurons confined in a bounded region of the brain tissue is sent to infinity. Specifically, in the discrete model, each cell is described by a set of time-dependent variables, whereas in the continuum model, cells are grouped into populations that are described by a set of continuous variables. Communications between populations, which translate into interactions among the discrete and the continuous models, are the essence of the hybrid model we present here. The cerebellum and cerebellum-like structures show in their granular layer a large difference in the relative density of neuronal species making them a natural testing ground for our hybrid model. By reconstructing the ensemble activity of the cerebellar granular layer network and by comparing our results to a more realistic computational network, we demonstrate that our description of the network activity, even though it is not biophysically detailed, is still capable of reproducing salient features of neural network dynamics. Our modeling approach yields a significant computational cost reduction by increasing the simulation speed at least 270 times. The hybrid model reproduces interesting dynamics such as local microcircuit synchronization, traveling waves, center-surround, and time-windowing.

Accuracy assessment of the linear Poisson-Boltzmann equation and reparametrization of the OBC generalized Born model for nucleic acids and nucleic acid-protein complexes.

The generalized Born model in the Onufriev, Bashford, and Case (Onufriev et al., Proteins: Struct Funct Genet 2004, 55, 383) implementation has emerged as one of the best compromises between accuracy and speed of computation. For simulations of nucleic acids, however, a number of issues should be addressed: (1) the generalized Born model is based on a linear model and the linearization of the reference Poisson-Boltmann equation may be questioned for highly charged systems as nucleic acids; (2) although much attention has been given to potentials, solvation forces could be much less sensitive to linearization than the potentials; and (3) the accuracy of the Onufriev-Bashford-Case (OBC) model for nucleic acids depends on fine tuning of parameters. Here, we show that the linearization of the Poisson Boltzmann equation has mild effects on computed forces, and that with optimal choice of the OBC model parameters, solvation forces, essential for molecular dynamics simulations, agree well with those computed using the reference Poisson-Boltzmann model.