## Introduction

This document is a tutorial for writing JSim models using the MFAX Biological Component Library (BCL). MFAX stands for a Metabolite Flow and Exchange, and contains a set of components appropriate for chemical network models in multiple compartments with convective flow and membrane transport between those compartments.

Prerequisites:

Contents:

- A Simple Example
- Using Physical Units
- Incorporating MML into BCL Models
- Setting Concentrations
- More Compontent Types
- Reaction Types
- Consumption and Production
- Membrane and Transport
- Convective Flow
- Branching Flows
- Recirculating Flows
- Injections
- Privatizing Sub-variables
- Comments or Questions?

See also:

## Simple example

The following model uses the MFAX BCL to describe the dimerization of a chemical species A in a single compartment:

// simple MFAX reaction, without units import MFAX; MFAX example1 { Time t; t.min=0; t.max=10; t.delta=0.5; Chem A, B; // this is just a comment Compartment C; C.vol = 10; MassBalReaction R(C, "2A=B"); R.kf = 1; R.kb = 0.5; }

The line "import MFAX" loads the MFAX BCL so that is can be used in this model. The block beginning "MFAX example1" gives the MFAX BCL model specifics. A BCL block must be given a name (here, example1) and is delimited by curly braces. Note that MML and BCLs are case sensitive and that white space (spaces, tabs, line breaks) is ignored. Text following double slashes (//) or between /* and */ is ignored as commentary as in Java or C++.

Within the example1 MFAX block, five components (t, A, B, C, R) are defined, each corresponding to a physical entity. Each component may have sub-variables (t.min, t.max, t.delta, C.vol, R.kf, R.kb) that must be constrained so that model is completely specified. Here are short descriptions of the example1 components.

- Time: The MFAX system requires you give a name to the variable representing time (here t). Time is implemented as an equally-spaced grid. The subvariables t.min and t.max represent the minimum and maximum values for t respectively and t.delta represents the time-point separation.
- Chem: A and B here are names for the two chemical species of interest. Chems have no associated sub-variables that must be constrained. Chem names must start with an upper-case letter.
- Compartment: example1 has a single compartment named C. Compartments in MFAX are assumed to be "well-stirred tanks" so that all chemical species are assumed to have uniform concentrations over C. The associated sub-variable C.vol represents the volume of C.
- MassBalReaction: R here represents the reversible mass-balance reaction A=2B taking place in compartment C. The associated sub-variables R.kf and R.kb represents the forward and backward equilibrium constants.

If you run example1 under JSim, the output shows variables C.A and C.B, which represent the concentrations of A and B in compartment C, to be uniformly zero. More interesting results can be obtained by changing the initial concentrations (variables C.A__init and/or C.B__init) from their default values of zero and rerunning the model.

## Using physical units

Example1 above is somewhat confusing because it is unclear what physical units are appropriate when assigning variables. For instance, is C.vol 10 liters or 10 milliliters or what? JSim supports, but does not require, attaching physical units to components and variables. In BCLs, using physical units is normally recommended. Example2 below adds physical units to the previous example:

// dimerization import MFAX; import nsrunit; unit conversion on; MFAX example1 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM, B mM; Compartment C liter; C.vol = 10; MassBalReaction R(C, "A=2B"); R.kf = 1; R.kb = 0; }

JSim does not have a default units system or policy. The line "import nsrunit" imports the distributed NSR units file for use in this model. The line "unit conversion on" turns on automated unit conversion. The unit conversion line is optional in plain MML, but is required when using units with the MFAX BCL. Additionally, four components (t, A, B and C) are declared with physical units. This is done by declaring the units after the name of each component.

Time t is declared in seconds. The sub-variables t.min, t.max and t.delta will also be in seconds.

Compartment C is declared in liters. The sub-variable C.vol will be in liters.

Chem A and B units are molar. The concentration variables C.A and C.B will be molar.

MassBalReaction R requires no unit declaration. The units for sub-variables R.kf and R.kb are calculated using the units for t, A and B. Here R.kf will have units 1/sec and R.kb units 1/molar/sec.

JSim models support custom user-defined units, however importing the distributed NSR unit file is appropriate for most users. In the JSim GUI, this file can be viewed in the Debug sub-tab of a model tab by selecting "View system units file" from the View menu. See Using Units in MML for further information.

The BCL author may make a range of choices in defining physical units, so long as they are dimensionally compatible with the components internal unit. These internal units for example2 components are:

- Time sec
- Chem molar
- Comparment liter
- MassBalReaction dimensionless

Thus Time t could have been defines as sec, min or hour, but not as meters. Chem A could be defined as mM (millimolar), uM (micromolar), nM (nanomolar), pM (picomolar), but not moles/(gram*liter). Compartment C could have be defined as ml (milliliters), m^3 (cubic meters), foot^3 (cubic feet) but not m^2. MassBalReaction R requires a dimensionless unit, and the unit assigned has no effect on model calculations.

## Incorporating MML into BCL models

MML syntax allows tweaking a number of features of BCL models. Examples include:

- adding additional variable and sub-variables to the model;
- specifying custom calculations for variables;
- specifying external inputs to a model;
- cleaning up user interface by repressing some output variables.

This tutorial focuses on BCL, so when a useful MML construct comes up it will be mentioned only briefly, and the user referred to Writing a Simple MML Model for further information.

## Setting concentrations

BCL authors may wish the initial concentrations in their models. This is accomplished via the MML "when" clause (see "Basics of MML" for more info). In example2 above, the concentration variables were C.A and C.B. Example3 below sets the initial concentrations:

// dimerization with initial conditions import MFAX; import nsrunit; unit conversion on; MFAX example3 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM, B mM; Compartment C liter; C.vol = 10; MassBalReaction R(C, "A=2B"); R.kf = 1; R.kb = 0; when (t=t.min) C.A = 1; when (t=t.min) C.B = 0; }

Another possibility is having a particular concentration set over its entire time course. In example4 below, the B concentration increases linearly because it is set by a pump in compartment C. The A concentration is determined using the B concentration and the reaction R:

// dimerization with 1 forced concentration import MFAX; import nsrunit; unit conversion on; MFAX example4 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM, B mM; Compartment C liter; C.vol = 10; MassBalReaction R(C, "A=2B"); R.kf = 1; R.kb = 0; when (t=t.min) C.A = 1; C.B = t*(2 mol/liter/sec); }

Declaring additional real variables (again, see "MML Basics") can help to modularize BCL models:

// dimerization with another forced concentration import MFAX; import nsrunit; unit conversion on; MFAX example4 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM, B mM; Compartment C liter; C.vol = 10; MassBalReaction R(C, "A=2B"); R.kf = 1; R.kb = 0; when (t=t.min) C.A = 1; real rate = 2 mole/liter/sec; C.B = t*rate; }

## More Component Types

In addition to the four component types so far described (Time, Chem, Compartment, MassBalReaction), MFAX BCL provides about another dozen for specification of other types chemical reactions and various methods of passing chemicals between compartments. Each new component type is introduced in sections below. Except as noted, a BCL model may declare any number of components of each type, allowing for creation of arbitrarily complex models. Each new component's behavior in the model will depend on how the BCL author sets or constrains the sub-variables that are associated with the component. To successfully incorporate a component, the BCL author must understand the physical meaning of the component, how to declare it in his model and the names and physical significance of its sub-variables. MFAX BCL Reference provides details each component type.

## Reaction Types

MFAX BCL provides three alternative components to specify chemical reactions MassBalReaction, FluxReaction and FastReaction.

MassBalReaction models a mass-balance reaction specified via forward and backward equilibrium constants (kf and kb). This has been covered previously.

FluxReaction models an arbitrary reaction specified by the forward net flux variable (flux) whose units are compatible with moles/second. For example, example6 below models the enzyme mediated dimerization of A using simple, irreversible, Michalis-Mentin kinetics:M

// flux-based dimerization import MFAX; import nsrunit; unit conversion on; MFAX example6 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM, B mM; Compartment C liter; C.vol = 10; FluxReaction R(C, "A=2B"); real Vmax = 5 mole/(liter*sec); real Km = 1 mole/liter; R.flux = Vmax*C.A/(Km + C.A)*C.vol; when (t=t.min) C.A = 1; when (t=t.min) C.B = 3; }

Modelers must take care so that flux is non-negative when species on the right-hand side of the equation have zero concentration, and non-positive when species on the left have zero concentration. Otherwise, negative values for the concentrations will result. This is mathematically correct behavior, although (obviously) physically unrealistic.

FastReaction models a mass-balance reaction that equilibrates instantaneously relative to MassBalReactions and FluxReactions, specified by a single equilibrium constant (k). Mathematically, it is equivalent to a MassBalReaction with very high kf and kb relative to other reactions, but much more numerically efficient and accurate. In example7 below, there are two reactions R1 and R2, with R2 equilibrating much faster than R1:

// mixed mass-balance and fast reactions import MFAX; import nsrunit; unit conversion on; MFAX example7 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem X mM, Y mM, Z mM; Compartment C liter; C.vol = 10; MassBalReaction R1(C, "X=2Y"); R1.kf = 1; R1.kb = 0; FastReaction R2(C, "Y=Z"); R2.k = 1; when (t=t.min) C.X = 1; when (t=t.min) C.Y = 0; when (t=t.min) C.Z = 0; }

Since FastReactions equilibrate instantaneously, it is possible to overspecify a system using them. Care must be taken that there is always one "free" species that is not specified by a fast reaction or forced concentration. For example, the following system is over-specified:

import MFAX; import nsrunit; unit conversion on; MFAX example8 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem X molar, Y molar; Compartment C liter; C.vol = 10; FastReaction R(C, "X=2Y"); R.k = 1; real k = 2 molar/sec; C.Xconc = k*t; C.Yconc = 2*k*t; }

## Consumption and Production

Consumption and Production are components that add or remove a chemical from a compartment. The associated subvariable "flux" must be compatible with mole/second. In the example below, A is produced by component Q1 in compartment C1 and is consumed by component Q2 in compartment C2:

// consumption and production import MFAX; import nsrunit; unit conversion on; MFAX example9 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM; Compartment C1 liter, C2 liter; C1.vol = 10; C2.vol = 5; when (t=t.min) C1.A = 0; when (t=t.min) C2.A = 1; real t0 = 1 sec; real R = 1 mole/sec; Production Q1(C1, A); Q1.flux = R*exp(-t/t0); Consumption Q2(C2, A); Q2.flux = R*exp(-t/t0); }

Consumption and Production are both "slow" processes in that FastReactions equilibrate instantaneously relative to them. If the rate variable goes negative, a Consumption component will function as a Production and vice-versa. In order to prevent negative concentrations, Consumption components turn themselves off at zero concentrations. However, due to unresolved numeric method issues, small negative concentrations may result. This can be avoided by making Consumption rates fall continuously to zero with concentration.

## Membrane and Transport

The components so far described model concentrations in one or more compartments, but do not explicitly describe passage of molecules from one compartment to another. There are two mechanisms to do this. Membrane and Transport components model transport across a membrane. Various Flow components describe convective flow between compartments.

In the following example, species A is produced in compartment C1 and is transported through a membrane to compartment C2:

// membrane transport import MFAX; import nsrunit; unit conversion on; MFAX example10 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM; Compartment C1 liter, C2 liter; C1.vol = 10; C2.vol = 5; Production Q(C1, A); Q.flux = (1 mol/sec); Membrane M(C1, C2); TransportPS T(M, A); T.PS = 1; }

The Membrane construct requires two arguments, the two compartments it connects. Membranes have no inherent functionality (and have no sub-variables), but serve as attachment points for Transports that pass species across the Membrane. A Transport construct requires two arguments: the membrane it's attached to, and the species transported. TransportPS, in which the transport rate is determined via a permiability surface-area product (sub-variable PS), is used in this example. PS has dimension liter/sec (derived from units for C1 and t). The other Transport currently available is TransportFlux, where the transport rate is determined via a flux rate. See the example below:

// flux-based transport import MFAX; import nsrunit; unit conversion on; MFAX example10f { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A molar; Compartment C1 liter, C2 liter; C1.vol = 10; C2.vol = 5; Production Q(C1, A); Q.flux = (1 mole/sec); Membrane M(C1, C2); TransportFlux T(M, A); T.flux = 1; // flux has units mole/sec }

Other Transport components will be available in future JSim versions.

## Convective Flow

Flow, FlowSource, FlowSink and related components describe convective flow between compartments. Convective flow starts at a FlowSource, may branch through any number of compartments, and must terminate at a FlowSink. The following example describes flow through two compartments, with production of species A in the first:

// flow source, flow and flow sink import MFAX; import nsrunit; unit conversion on; MFAX example11 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM; Compartment C1 liter, C2 liter; C1.vol = 10; C2.vol = 5; Production Q1(C1, A); Q1.flux = 1; FlowSource F1(C1) liter/sec; F1.flow = 1; Flow F2(C1, C2); FlowSink F3(C2); }

The FlowSource component (F1) takes a single argument, the Compartment that it flows into. The attached unit must be compatible with liter/sec. The sub-variable flow describes the rate of flow and inherits its unit from the FlowSource. "flow" may be time-variant, but negative values are not supported.

A Flow component (F2) describes flow between two compartments and thus takes two arguments, the input compartment (C1) and the output compartment (C2). F2's rate of flow in a Flow component is determined by the rate of flow out of C1 (determined by FlowSource F1) and so does not need to be (and should not be) set explicitly. Flow components do not require unit declarations.

Convective flow disappears from the system via FlowSink component, which takes as it's single argument the Compartment it drains. As with the Flow component, FlowSink's rate of flow is not set explicitly, but is determined from the inflow to the attached compartment. FlowSink components do not require unit declarations.

Any number of Compartment, Flow and FlowSink components may be specified, however currently JSim only supports a single FlowSource. The network must be such that any Compartment with an attached Flow, FlowSink or FlowSource must have exactly one input flow and one output flow.

## Branching Flows

Diverging and converging flow streams are supported via the FlowJunc (flow junction) component. FlowJunc represents a volumeless junction in which one or more input flows may be mixed and one or more output flows may be split. A FlowJunc may be specified in place of a Compartment in FlowSource, Flow and FlowSink declarations. FlowJunc components do not require unit declarations. A FlowJunc must have at least one inflow and at least one outflow. In the following example one FlowJunc (J1) splits the incoming flow to pass through two compartments (C1 and C2), and another FlowJunc (J2) merges the two compartment outflows before pass to a FlowSink (F6):

// flow junctions import MFAX; import nsrunit; unit conversion on; MFAX example12 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM; FlowJunc J1, J2; Compartment C1 liter, C2 liter; C1.vol = 10; C2.vol = 5; Production Q1(C1, A); Q1.flux = 1; FlowSource F1(J1) liter/sec; F1.flow = 1; Flow F2(J1, C1); Flow F3(J1, C2); Flow F4(C1, J2); Flow F5(C2, J2); FlowSink F6(J2); J1.F2wgt = 2; J1.F3wgt = 1; }

Notice the two FlowJunc weighting variables J1.F2wgt and J1.F3wgt. These specify what proportion of J1's output will pass to F2 and F3 respectively. The weights have a default value of 1, which results in an even split between all paths. In example12 however, the weights are set so that twice as much (2/3 of the total) flow will pass through F2.

## Recirculating Flow

The MFAX BCL supports recirculating flows, however a FlowSource is always required to specify the flow rate. To allow this, a FlowSource may be declared with two arguments (input and output) instead of the one (output only) which has been used in previous examples. Using FlowSource with two arguments makes it like a Flow component, but with the subvariable flow settable. The following example shows recirculating flow between two compartments (C1 and C2). Notice that a FlowSink is not required:

// recirculating flow import MFAX; import nsrunit; unit conversion on; MFAX example13 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM; Compartment C1 liter, C2 liter; C1.vol = 10; C2.vol = 5; Production Q1(C1, A); Q1.flux = 1; FlowSource F1(C2, C1) liter/sec; F1.flow = 1; Flow F2(C1, C2); }

## Injections

The Inject component models an injection of a chemical species into a Flow or FlowSource. The Inject component requires two arguments: the Flow or FlowSource to inject into, and the Chem to inject. The Inject component is specified without units. The rate of injection is determined by the "fluw" subvariable whose units are compatible with moles/sec. The exact units are dependent upon the units of the Chem and Time components. The example below shows flow into and out of a single compartment (C1), with an injection of species A attached to the FlowSource:

// injection import MFAX; import nsrunit; unit conversion on; MFAX example14 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM; Compartment C1 liter; C1.vol = 10; FlowSource F1(C1) liter/sec; F1.flow = 1; Inject I1(F1,A); I1.flux = t*(2 mol/sec^2); FlowSink F3(C1); }

The injection rate variable (here I1.Arate) has no default value, and so must be constrained for the model to be completely specified. It is often useful to allow the user to specify the injection rate function at run-time from an external data source. The BCL author accomodates this via the general MML mechanism for specifying a variable as external (see Writing a Simple MML Model for further information):

// externally controlled injection import MFAX; import nsrunit; unit conversion on; MFAX example15 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM; Compartment C1 liter; C1.vol = 10; FlowSource F1(C1) liter/sec; F1.flow = 1; Inject inj(F1,A); extern inj.flux; FlowSink F3(C1); }

## Privatizing sub-variables

A concentration sub-variable is created for each Chem inside each Compartment. Depending on model dynamics, some of these concentrations may always be zero, and thus not useful to look at. JSim is not yet smart enough to repress these variables in the run-time environment, however the BCL author may do this using the general MML mechanism for specifying a variable as private (see Writing a Simple MML Model ). In the example below, species B never appears in Compartment C1, so the concentration there is privatized:

// production, reaction with flow import MFAX; import nsrunit; unit conversion on; MFAX example16 { Time t sec; t.min=0; t.max=10; t.delta=1; Chem A mM, B mM; Compartment C1 liter, C2 liter; C1.vol = 10; C2.vol = 5; Production Q1(C1, A); Q1.flux = 1; MassBalReaction Q2(C2, "A=B"); Q2.kf = 1; Q2.kb = 1; FlowSource F1(C1) liter/sec; F1.flow = 1; Flow F2(C1, C2); FlowSink F3(C2); private C1.B; }

This mechanism can be used to privatize any variable, not just concentration within compartments.

Comments or Questions?

**Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants:** NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.