# SBML Level 3 Geometry

## Contents |

## Suggestions for a SBML standard for spatial modeling

### Some thoughs from Jim Schaff

I'd like to bring up some not so biological issues associated with spatial modeling of chemical and transport processes (oriented toward continuous modeling). In no particular order.

1) chemical kinetics can no longer be described as substance/time if it can be spatial heterogeneous (must use concentration/time).

2) when doing space, dimensionality matters ... reactions on membranes are dimensionally equivalent to fluxes (mass/area/time). Always!

3) boundary conditions are absolutely required. a) concentrations are a given function of solution and x,y,z,t b) fluxes are a given function of solution and x,y,z,t c) distribution functions for stochastic variables must be specified also at boundaries.

4) There is a continuous analogy to discrete particle trafficking - directed transport along microtubules can be treated as motion along a velocity fields that is a function of motors and polymer substrate orientation and density. (yet another thing that I've worked out 85% and have nothing to point to except a poster abstract).

5) rather than use "outside compartment" (ok, I know that I asked for this one many years ago), we need adjacency lists. This enables us to encode non-tree (cyclic) topologies and those that have no parent/child relationships. This comes up a lot in multicellular contexts.

6) spatial operators (if we are going to allow "rate rules" for spatial models) du/dt = div(D*grad(u)).

7) we need a spatial domain (xmin,ymin,zmin,xmax,ymax,zmax) and associated coordinate system (with reserved symbols for spatial coordinates) or some other notion to size/origin of world.

8) what about anisotropic diffusion or conductivity (used in cardiac modeling all the time).

Of course we need to scope this project ... especially because of the limited resources.

Also, I'd like to propose one important ground rule:

1) meshes are not part of the model, rather part of the solution. However a separate mechanism for mesh interchange would be very useful.

Thoughts on possible geometric representations:

1) geometric primatives with constructive solid geometry (union/intersection/difference of spheres, cones, planes, ...)

2) implicit surfaces/volumes using inequalities in x,y,z

3) implicit surfaces/volumes using level sets.

4) a set of piecewise analytic functions (e.g. polygonal surfaces ... WHICH IS TREATED AS A GEOMETRIC DEFINITION AND NOT A MESH) might provide an alternative mechanism for interchange that is easy to understand and unambiguous.

5) images are not geometries (without further interpretation). Under some circumstances, the uncertainty of surface location from images may be tolerable ... but rarely.

6) One could provide services to generate (4) from (1)(2)(3)(5).

### Some thoughs from Le Novère

There are actually two topics hidden behind "spatial" SBML extensions, that should belong to two different extensions: the support of spatial models and the encoding of geometry.

Spatial:

We need to extend SBML so it can support PDE, but also finite elements, single-particle models etc.

Examples: The differential location of a given species in subcellular compartments affects its function, e.g. the CaMKII differential phosphorylation in post-synaptic density and in cytosol or the hysteresis generated by the differential location of kinases and phosphatases. Another example is the MAPK cascade from the membrane to the nucleus.

That requires the encoding of initial conditions (specific coordinates - link to the array proposal? - or distributions) and movement laws.

Geometry:

We need to extend SBML to describe the geometry of physical objects, whether compartments or species.

Examples: the morphology of neuronal compartments is central to the treatment of signals, whether electrical or calcium diffusion. The topology of supra-macromolecular structures in the post-synaptic junction affects the signal transduction.

That requires the encoding of physical entity topology - link with the complex proposals? - and the encoding of deformation laws.

### Needs from the Le Novère Compneur group

Meredys, our mesoscopic simulator, represents the movements of individual molecules. Any biological entity of interest, such as a protein molecule, is represented by a software object which propagates in continuous simulation space according to Brownian Dynamics algorithms. Several diffusion spaces can be defined (static, free diffusion, within membrane, above membrane, blow membrane). Two types of diffusion are considered, rotational and translational. The diffusion is controlled by two sets of equations, the unrestricted brownian motion and the membrane diffusion described by Saffman and Delbruck. Dominic Tolle is rewriting the diffusion-collision algorithm to use Smoldyn algorithm.

The specificity of the simulator comes from the fact that simulated entities contain crude geometric information, such as the relative position of reactive surfaces to the objects centre of mass. Molecular entities can react to form transient complexes. This molecular resolution allows us to address questions found in the field of molecular interactions and signal transduction of biological systems, where local concentration effects and the geometry of interactions are often important.

### Suggestions from the MesoRD-group

We would like that the standard describes (1) The 3D geometry of SBML compartments, including boundary conditions. (2) The diffusion rate constants of different species in different compartments. (3) The transition rates of molecules over compartment boundaries.

Currently we add this information in annotation blocks in the species and compartment descriptions. The geometry of individual compartments is defined using Constructive Solid Geometry descriptions. MesoRD can do set operations on rotated, translated and scaled boxes, sphers, cyliners, cones, and other compartments. For the next version we will also have a 3D-mesh primitive. See more information on http://mesord.sourceforge.net/ and in particular Ch. 3 and 4 of the on-line manual http://mesord.sourceforge.net/man/mesord/docbook/index.html

Example - Min oscillation:

SBML file: http://www.people.fas.harvard.edu/~elf/MinWT.xml

General comments: We think it is a good idea to describe geometry independent of the particular algorithm that will be used to analyze the SBML model. In particular, we would not like if a specific spatial discretization is given in the SBML model.

Although it is not currently needed in MesoRD we would be happy to Smoluchowski-supportsee

### Some views from E-Cell group

One of the major focuses of the next major version of E-Cell System (Version 4) will be multi-spatial representation.

There are several different classes of representations of space, most of which are useful in different contexts. In other words there is no single type of representation that can do everything. For the forthcoming space support in SBML to be truly useful, it will also be necessary to account for 'multi-spatial representation', in which interactions between entities in sub-modules with different types and/or coordinates of spatial representations can be defined.

The following paper summarizes some aspects relevant topics. (See figure 1 for some types of spatial representations, for example)

http://www.e-cell.org/software/documentation/takahashi2005.pdf

Another point raised in the paper was the strengths and limitations of the pure-Brownian approximation, particularly in the context of roles of 'macro-molecular crowding' in biochemical dynamics and kinetics, which is supposed to be the most obvious but critical features of all cellular systems after all.

(Koichi Takahashi, The Molecular Sciences Institute)

## Original working group members

Steve Andrews ssandrews @ lbl.gov

Johan Elf elf @ fas.harvard.edu

Thierry Emonet emonet @ uchicago.edu

Mathilde foglieri @ embl.de

Akira Funahashi funa @ symbio.jst.go.jp

Michael Hucka mhucka @ caltech.edu

Nicolas Le Novere lenov @ ebi.ac.uk

Karen Lipkow kl280 @ cam.ac.uk

Mike Mc Collum jmccoll2 @ utk.edu

Poul Nielsen p.nielsen @ auckland.ac.nz

Michael North north @ anl.gov

Greg Peterson gdp @ utk.edu

Andrzej Przekwas ajp @ cfdrc.com

Emmanuele Raineri raineri @ embl.de

Wayne Rindone wrindone @ arep.med.harvard.edu

Takeshi Sakurada sakurada @ sfc.keio.ac.jp

Vijay Saraswat saraswat @ cse.psu.edu

Jim Schaff schaff @ neuron.uchc.edu

Luis Serrano serrano @ embl.de

Tom Shimizu tshimizu @ mcb.harvard.edu

Joel Stiles stiles @ psc.edu

Koichi Takahashi ktakahashi @ molsci.org

Dominic Tolle dominic @ ebi.ac.uk

Yann Dublanche dublanch @ embl.de

purvis @ mail.med.upenn.edu

(This list of members might be out of date.)

## Preliminary summary of some tools that support spatial modeling

The following link presents a summary of some tools that represent geometry/support spatial modeling. Summary of tools that support spatial modeling/geometries.

## Preliminary efforts on SBML Geometry/Spatial representations

The following link proposes a preliminary representation for Geometry in SBML (Level 3 Spatial/Geometry) which is based on the Virtual Cell geometry representation.