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[MATH] known as a D [MATH] -brane. For a target space with metric [MATH] and anti-symmetric 2-form [MATH] the sigma model action is |
[EQUATION] with [MATH] the [MATH] gauge field coupling to the end-points of the open string and [MATH] denoting the tangent directions to the worldvolume of the D [MATH] -brane. In conformal gauge, [MATH] , and lightcone coordinates, [MATH] , we have |
[EQUATION] Varying the action with respect to the fields [MATH] (to obtain the equations of motion) one encounters a boundary term leading to Dirichlet and (generalised) Neumann directions provided the metric splits orthogonally: |
[EQUATION] with [MATH] the directions normal to the D [MATH] -brane and where we introduced the Abelian field strength [MATH] . The classical energy momentum tensor of the sigma model is given by |
[EQUATION] and in light cone coordinates we have, [EQUATION] It is straightforward to see that on the boundary, imposing either Dirichlet conditions ( 121 ) or generalised Neumann conditions ( 122 ), the energy-momentum tensor satisfies |
[EQUATION] which we will call the (classical) conformal boundary condition. Hence, there is no momentum flow through the boundary (although [MATH] and [MATH] charge can be interchanged). If we now summarise the Dirichlet and Neumann conditions ( 121 122 ) using a map [MATH] that combines them as, |
[EQUATION] then the Dirichlet conditions correspond to [MATH] eigenvalues of [MATH] while the generalised Neumann conditions are described by all other eigenvalues. The classical conformal boundary condition eq. ( 125 ) then requires that [MATH] preserves the target space metric [MATH] |
[EQUATION] The dynamics of the D [MATH] -brane with tension [MATH] is governed by the DBI action (throughout this paper we ignore the scalar fields parameterizing the fluctuations transversal to the brane), |
[EQUATION] where [MATH] is the induced metric on the worldvolume and [MATH] is the gauge-invariant worldvolume flux given by [EQUATION] |
with [MATH] the induced anti-symmetric 2-form. Appendix B WZW models and conventions In this appendix we collect a number of conventions together with a short review concerning (boundary) WZW models. The WZW model is a non-linear sigma model of maps [MATH] from a 1+1 dimensional Riemann surface [MATH] (with or without ... |
Before writing down the action let us make our conventions clear. We pick for the Lie algebra a basis of hermitian generators [MATH] , with [MATH] , that satisfy [MATH] . The ad-invariant metric on the Lie algebra is given by [MATH] with [MATH] the index of the representation [MATH] . The left and right-invariant Maure... |
The WZW action Witten:1983ar is [EQUATION] where [MATH] is the closed torsion 3-form (locally satisfying [MATH] ) given by [EQUATION] |
with [MATH] the extension of [MATH] into [MATH] such that [MATH] . There are two topological obstructions for the consistency of the definition of the WZW action and its quantum theory . First, the existence of [MATH] is guaranteed only when the second homology group [MATH] is empty. Second, the path integral based on ... |
[EQUATION] to a worldsheet model ( 120 ) we have units in which [MATH] , crucial for [MATH] when [MATH] The WZW model is invariant under a global [MATH] action leading to an infinite-dimensional symmetry group described by the chirally conserved holomorphic Kac-Moody currents |
[EQUATION] in the conventions [MATH] and [MATH] with the Euclidean worldsheet coordinates [MATH] and lightcone coordinates [MATH] . At the quantum level the current algebra takes the form, |
[EQUATION] and analogous for the [MATH] OPE (hence the sign difference in the definition ( 133 )). The exact conformal invariance is established through the energy-momentum tensor obtained via the Sugawara construction based on the current algebra DiFrancesco:1997nk |
[EQUATION] where [MATH] is the dual Coxeter number of [MATH] (here we assumed for simplicity [MATH] to be semi-simple). The central charge of the theory can then be found to be, |
[EQUATION] and all analogous for [MATH] When considering a boundary in the WZW model one will seek boundary conditions that preserve its exact conformal invariance. Since the CFT is easily described in terms of the chiral currents ( 133 ) it is convenient to express these boundary conditions as a class of gluing condit... |
[EQUATION] with [MATH] . To preserve the exact conformal invariance the gluing condition should satisfy the conformal boundary condition [MATH] . This translates into the condition that the gluing map [MATH] should be an isometry of the ad-invariant Lie algebra metric. One could further require [MATH] to be an algebra ... |
Stanciu:1999id Alekseev:1998mc Felder:1999ka or of ‘type D’ Stanciu:1999id , a terminology that we will adapt. Another possibility analysed in Stanciu:1999id |
Stanciu:1999nx , where it was dubbed ‘type N’, is to consider, [EQUATION] with [MATH] a metric-preserving automorphism. They preserve the conformal invariance but do not preserve the current algebra which makes them somewhat more difficult to analyse. However, as suggested in section they seem to be related by generali... |
From the sigma model point of view the WZW action ( 130 ) is necessarily modified when the Riemann surface [MATH] has a boundary [MATH] |
Klimcik:1996hp Gawedzki:1999bq Figueroa-OFarrill:2000lcd Stanciu:2000fz . The image of [MATH] is a (D-brane) submanifold [MATH] of [MATH] on which a two-form [MATH] lives such that the restriction of [MATH] on [MATH] coincides with [MATH] . Locally the two-form coincides with the gauge-invariant worldvolume flux [MATH]... |
Figueroa-OFarrill:2000lcd . The action of the boundary WZW model is, [EQUATION] where [MATH] with boundary [MATH] and [MATH] . Note that only the boundary equations of motion will depend on the two-form [MATH] . Demanding that the boundary conditions obtained from the gluing conditions that preserve conformal invarianc... |
Stanciu:2000fz Gawedzki:1999bq Again, there are two topological obstructions for the consistency of the definition of the boundary WZW action and its quantum theory (for a detailed exposition see Klimcik:1996hp |
Gawedzki:1999bq Figueroa-OFarrill:2000lcd ) . The existence of [MATH] and [MATH] is guaranteed only when the second relative homology [MATH] vanishes. The path integral is insensitive to the choice of [MATH] and [MATH] provided that the third relative cohomology class [MATH] is integral. As seen in section this conditi... |
Bachas:2000ik . For [MATH] the position of the D [MATH] -strings will not be constrained by this particular topological obstruction; however, the D [MATH] -strings carry a natural quantisation descending from the Gauss constraint of two-dimensional gauge theory on the brane Bachas:2000fr Witten:1995im |
# Source: arxiv 1806.10889 # Title: Orchestral: a lightweight framework for parallel simulations of cell-cell communication # Sections: all # Downloaded: 2026-03-02T08:44:09.922960+00:00 |
Orchestral: a lightweight framework for parallel simulations of cell-cell communication Abstract We develop a modeling and simulation framework capable of massively parallel simulation of multicellular systems with spatially resolved stochastic kinetics in individual cells. By the use of operator-splitting we decouple ... |
Index Terms: Computational systems biology, high performance and parallel computing, distributed computing, cloud computing Introduction |
One important insight gained in computational systems biology is that intrinsic molecular noise in cellular regulatory networks can have important consequences for the functioning of the system |
. This has led to the development of a wide range of methods and simulation software capable of simulating the stochastic dynamics of intracellular kinetics, both in a well-mixed and in a spatial setting |
In the case that one can assume a well-mixed system, i.e. when the rate of chemical reactions can be assumed to be independent of the spatial location of the molecule, Gillespie’s Stochastic Simulation Algorithm is a popular algorithm to generate trajectories of the involved chemical species, who are assumed to follow ... |
Cellular regulation kinetics are inherently spatial though, with reactions frequently taking place in a diffusion-limited regime where the spatial homogeneity assumption breaks down. One popular alternative that simulates stochastic kinetics with spatial resolution is the Reaction Diffusion Master Equation (RDME) |
On an even finer scale, molecules are assumed to be continuously diffusing hard spheres and interact according to Smoluchowski diffusion limited theory. The Green’s Function Reaction Dynamics (GFRD) framework |
is a popular simulation software in that setting. There is therefore a well established hierarchy for single-cell modeling techniques, with ordinary differential equation (ODE) based methods on the coarser end of the spectrum and detailed particle dynamics methods on the other end. |
However, in complex multicellular systems, cells do not act in isolation of each other, on the contrary, they interact with each other in many different ways. Through direct contact, where cells physically push or pull each other, changing position and shape over time, and through chemical signaling, where signals are ... |
Several studies have focused on coupling single-cell and multicellular interactions. Cell mechanics have been modeled with agent-based models |
, center based models vertex based models or cellular Potts models . These models have been coupled with ODEs , boolean networks |
, and more recently the RDME framework . Non-spatial deterministic models remain the preferred framework due to their relatively low complexity and low computational cost. Similar to the insights gained from using stochastic models in the single-cell setting though, enabling the use of more detailed stochastic models o... |
Contrary to methods used for single-cell models, for multicellular systems there is no unifying underlying mathematical theory that would provide an analytical solution to be compared with. Also, there is no well-established model hierarchy, and models are often only compared qualitatively to each other |
So far, cell mechanics models have often been coupled with coarse, simple internal cell dynamics, such as ad hoc rules, ordinary and delayed differential equations, or simple stochastic dynamics |
. It remains, however, unclear how much information is lost in using these coarse approximations. In other words, using more detailed models and software for simulating intracellular reaction networks, such as GFRD |
, Smoldyn or URDME , could uncover new insights. Handling such detailed simulation remains highly challenging due to their prohibitive computational cost. |
Given this diversity in both single-cell and multicellular simulation frameworks, there is a need for tool to easily combine different software into complex simulations that includes single-cell resolution and cell-cell signaling. Due to the computational cost, such a framework needs to be able to leverage modern distr... |
The aim of this paper is therefore to provide such a unifying framework making it possible to construct composite multicellular models from state-of-the-art single-cell simulators and from multicellular modeling frameworks. To accommodate as many current and future modeling approaches as possible, the framework needs t... |
or BSim ) our framework is independent of any language or technology, and focuses on combining different tools into massively parallel simulations rather than to provide new tools for individual components of a simulation. Rather than a new simulation software, or a generic workflow library, our framework is a tool to ... |
, from the computational biology field on the one hand, and from parallel computing on the other hand The rest of the paper is organized as follows: first, we introduce our framework and its components. We then move on to the scaling study and detail our simple biological test-model and how we have implemented our fram... |
II A parallel framework for multiscale simulation of cells Our main objective is to design a general framework that makes it possible to a) combine different existing tools for single cell reaction kinetics simulation and multicellular mechanics models into multi-fidelity multicellular simulations, and b) enable to sim... |
Three main design assumptions are critical to meet these requirements. First, we assume that a multicellular simulation can be broken down into principal parts, or simulation modules, e.g: intracellular dynamics, cell signaling, and mechanics. Second, we assume that each such simulation module can be simulated one afte... |
Third, we assume that the simulation software used to implement the different simulation modules can be called from the command line, with a single input file and a single output file as arguments (SISO). This assumption makes our framework agnostic to specific technology, software or library to handle the simulation m... |
Figure illustrates the architecture of our proposed framework. Blue nodes represent our two simulation modules, one for internal spatial details ( [MATH] ), and the other for external cell interactions ( [MATH] ). Red nodes are the programs responsible for providing cell interactions with summarized internal spatial da... |
We point out that all the above requirements are very weak assumptions that are easily satisfied. On the one hand, the splitting should work provided the coupling implemented by the user (i.e. the way data are summarized and shared between simulation modules) is correct and makes senses, physically speaking. On the oth... |
In the following subsections we describe the main components of the framework. Our implementation is available at II-A Simulation modules |
In this paper, we restrict the scope to study simulation modules for intracellular dynamics and cell-cell signaling. II-A Internal cell dynamics module |
This module focuses on simulating chemical kinetics inside cells, e.g. the interaction between proteins and genes. In this paper it will be denoted by the Greek letter [MATH] There is a wide range of models that can be used here, from ordinary differential equation models to more detailed particle based algorithms. One... |
II-A Cell signaling module Here pairwise signaling is considered, that is, all molecular interactions involving exactly two cells. This can for example be binding of a ligand in one cell to a receptor at an adjacent cell. All pairs being independent, this step is also naturally embarrassingly parallel. We will use the ... |
II-B Translation scripts The translation scripts connect the simulation modules. Before a simulation module is run, an input file is data computed previously by the other simulation modules (Figure ). The translation scripts ensure compatibility between the simulation modules. If such a module is replaced by a new one ... |
Specifically, when translating from an internal dynamics model to a signaling model [MATH] ), the script takes a pair of cells as input. The script can then be used to filter out molecules that are close to the common boundary of the two cells and put them in the signaling model input file. When translating from a sign... |
II-C Orchestration Finally, the main module of our framework is the orchestrator, which is responsible for executing simulation modules and translation scripts, and for efficiently distributing the computations across the computing grid. Again, this module is completely independent from the simulation modules. Replacin... |
We chose to use Dask a python library for parallel and distributed computing. However, we point out that this is only one of many available alternatives, and that switching to any other library could be done very easily. In Dask, tasks can be described and hierarchized as a directed acyclic graph (DAG) using a simple P... |
The orchestrator takes two input files as parameters. The first file describes the cell network, i.e. the position of each cell, the id of their neighbors, and model specific simulation parameters. An example of such a file, in |
json format, is given in Figure . The data is encoded as a dictionary, where unique cell identification numbers are associated to their parameters. It is possible to fine tune the parameters so as to have various models for each cell. Only |
position and neighbors are used by the orchestrator, all other parameters are fed to the internal dynamics module when the simulation is launched. |
The second file is a configuration file that provides the syntax of the command line programs used to execute the simulation modules and translation script, as well as global simulation parameters, such as the number of time steps to run. An example of such a file is given in Figure . Each command line is written so th... |
III Results In order to demonstrate the usefulness of our framework and to test its scalability, we implement a simplistic but biologically realistic model of cell-cell signalling based on the single-cell simulation software eGFRD |
. The objective here is to provide a proof-of-concept model useful to study parallel scalability and to demonstrate how to use the framework. We conduct strong scaling experiments on a modern multicore machine and weak scaling experiments over a virtual cluster deployed in an OpenStack community cloud. |
III-A Setup III-A Model The model implemented is a simplified model of the Notch signaling pathway . The cells’ membranes contain [MATH] receptors and [MATH] ligands (molecules binding to the receptor). When cells are in contact, [MATH] and [MATH] proteins can bind and this will release [MATH] ’s intracellular domain (... |
will then diffuse into this cell and potentially influence gene expression. In our model, [MATH] acts as a repressor of [MATH] , i.e. it suppresses the creation of new [MATH] protein. A single DNA molecule is placed in the center of each cell, modeling the [MATH] promoter. When not occupied by [MATH] , it transcribes [... |
To model cell-cell signalling, we assume that when [MATH] molecules are within a distance [MATH] of the cell membrane (boundary of the cube), they can be mirrored to the cell on the other side of the membrane and turn into NICD at rate [MATH] . NICD then diffuses inside the receiving cell and potentially associates to ... |
transcription. NICD unbinds from DNA in bound state at rate [MATH] . Except DNA, all species degrade at the same rate [MATH] . The parameter values used in our simulation are summed up in Table . These values are We tuned the model to illustrate our framework with interesting features. In the case one would want to per... |
We model cells as identical cubes, arranged in a 2D layer. Hence, each cell has at most four neighbors. The chemical reaction turning [MATH] into [MATH] is handled by the signaling model, while all other reactions are handled as internal cell dynamics. Coupling is done by transferring all [MATH] molecules within [MATH]... |
single face of the cube to the associated signaling model. Coupling in the other direction is done by simply reporting the molecules from the signaling model back into the appropriate cell. |
Figure shows the output of a simulation of 1024 cells after 10 timesteps, each red dot is a [MATH] molecule, each blue dot is a [MATH] molecule, and DNA is shown in black, either in free state (circles) or bound state (triangles). Suppressed cells show a loss in [MATH] production and appear as white, while cells with a... |
and remain red. Under some specific parameter regimes, this model is known to exhibit a so-called checkerboard pattern, where free cells are arranged in a regular pattern. The more noisy the system becomes, the less regular the pattern |
III-A Implementation We implement the single-cell model using eGFRD . We use its custom simulation feature to both describe our model and set up reading and writing input and output files. The signaling model is implemented with a custom made python script. All translation scripts are implemented as python scripts. |
III-B Scaling A key feature of the framework, as implemented using Dask, is the ability to seamlessly leverage a wide range of parallel and distributed computing resources. To demonstrate this flexibility we here tested the performance and scalability when executed on a single high-end multicore machine and b) using va... |
For the multicore machine, a single node of the Rackham system provided by the Uppsala Multidisciplinary Center for Advanced Computational Science (UPPMAX) was used. The used node had two 10-core Xeon E5-2630 V4 processors running at 2.2 GHz (turbo 3.1 GHz). The SSC is a community cloud based on OpenStack, and in the s... |
We adapted the MOLNs orchestration toolkit from the StochSS suite to deploy virtual Dask clusters of variable sizes ( ). MOLNs deploy an SSHFS shared filesystem which we use to share all the intermediate input-output data files of the simulation. |
We begin with a strong scaling study on a single node, where we study the computation time as a function of increased number of cores with a fixed model size. We simulate an [MATH] grid of cells for ten seconds simulation time. We set up Dask to use the threaded single machine scheduler and ran the simulation on 1, 4, ... |
Using a single core, the simulation took between 40 and 140 minutes. Using the full 20 cores of the shared memory node, the simulation took between 2 and 8 minutes, depending on [MATH] , thus demonstrating the usability of our framework for modeling purposes. Our framework hence shows very good parallel efficiency for ... |
Next we assessed the weak scaling properties of our implementation. We used [MATH] VM instances each with 8 VCPUs and 16GB RAM and kept the number of cells simulated per core in the cluster constant as we scaled the size of the distributed cluster. We start from a grid of 100 cells for one VM (8 VCPUs), up to 3200 cell... |
Again, our experiments demonstrate good parallel efficiency, with more than 50% efficiency across all tested timesteps on 16 VMs. Using 32 VMs, the efficiency drops as we approach the hardware limits of the cluster underlying the private cloud. It should be noted that the weak scaling properties depends on [MATH] , wit... |
The weak scaling efficiency of our multicellular simulations with Orchestral shown in Figure should be seen in this context. For [MATH] , individual tasks associated with updating the internal state of cells by running eGFRD vary due to stochasticity and particle counts in individual cells, but upon manual inspection i... |
IV Discussion and conclusion In this paper, we have presented a new framework for constructing and simulating high-fidelity models of multicellular systems from existing frameworks for single-cell simulation. The driving motivation for Orchestral is a need to be able to conveniently combine the many existing frameworks... |
into a multicellular model where cells signal each other via a simplistic model of the Notch-Delta pathway over adjacent boundaries of individual cells. |
The parallel scalability was demonstrated both in a strong scaling experiment on a shared memory system, and in weak scaling experiments over a multi-tenant science cloud IaaS infrastructure. The experiments serve to highlight the dependency of parallel efficiency on the splitting time step [MATH] , since this timestep... |
Orchestral is primarily aimed at modelers with basic scripting experience. Due to its modular design, it is easy to reuse existing parts and patterns from a previous successful set up. Our framework puts no constraint on the technology used by the solvers, the translation scripts, or the orchestrator. All these modules... |
Finally, the orchestrator engine is also easily replaceable for a user with experience in distributed parallel computing. It is our intention for Orchestral to evolve with the help of the community to support a wide range of orchestrators tuned to specific platforms, for example optimized for shared memory desktop comp... |
which could potentially help us to push performance for low-latency requirements on distributed infrastructures. In fact, Orchestral could be used to create a comparative benchmark of all these libraries. |
Here we have presented a proof of concept of our framework based on eGFRD as single-cell simulator. Future work on the modeling side involves exploring and comparing new combinations of simulators, in particular adding a module for cell-mechanics. |
Finally, within our setup it is already possible to use different models to update different areas of the cell population, e.g. to use highly detailed simulation modules (such as eGFRD) at the boundary of a tissue (or generally where cells are actively engaged in signaling) and coarser modules (such as the RDME or even... |
Acknowledgments The authors would like to thank S. Mathias and F. Coulier for constructive criticism of the manuscript. This work has been funded by the Swedish research council (VR) under award no. 2015-03964 and by the eSSENCE strategic collaboration of eScience. Cloud computing resources were provided by the Swedish... |
# Source: arxiv 1806.10898 # Title: Solitary waves in the Ablowitz-Ladik equation with power-law nonlinearity # Sections: all # Downloaded: 2026-03-02T08:55:59.288190+00:00 |
Solitary waves in the Ablowitz–Ladik equation with power-law nonlinearity Abstract We introduce a generalized version of the Ablowitz-Ladik model with a power-law nonlinearity, as a discretization of the continuum nonlinear Schrödinger equation with the same type of the nonlinearity. The model opens a way to study the ... |
Introduction The study of models of the discrete nonlinear-Schrödinger (DNLS) type has been a focal point within the broader theme of nonlinear dynamical lattices over the past two decades book . First, DNLS systems are obviously relevant to the discretization of the ubiquitous continuum nonlinear Schrödinger equations... |
A model that plays a critical role in the understanding of DNLS dynamical lattices is their integrable sibling, namely, the Ablowitz-Ladik (AL) equation ablolad (see also ablowitz ). In addition to offering the unique integrability structure ablowitz , the AL model is a reference point for the examination of numerous f... |
In the present work, we consider a variant of the AL model with a power-law nonlinearity instead of the particular case of the cubic one. This is motivated by many previous studies —see e.g. sulem and references therein for the continuum case, as well as Refs. |
Turitsyn malomed weinstein ourDNLS Kladko for the discrete case— of the generalized NLS model with nonlinear term [MATH] instead of the usual cubic one corresponding to the particular value of [MATH] . In particular, in the one-dimensional (1D) case the Lee-Huang-Yang correction to the mean-field nonlinearity in two-co... |
Grisha1 Grisha2 . The model with a general nonlinear term is particularly appealing because it offers, at the continuum level and for one spatial dimension, the potential to study the transition from integrability (at [MATH] ) and the existence of stable solitary waves (at [MATH] ) to the emergence of catastrophic self... |
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