Why Initial Conditions Must Match Conserved Totals
An ODE model can have correctly specified reaction rates and still start from a biologically impossible state. For a closed reaction network, the initial species amounts must agree with the conserved totals implied by its reactions. If a model is meant to contain 10 nmol of a conserved ligand moiety, for example, its initial free and bound ligand amounts must add to 10 nmol—not 9 nmol or 12 nmol.
This tutorial treats initial-condition construction as a constrained accounting problem. The goal is not merely to assign a value to every state variable, but to find a set of nonnegative values that satisfies the known totals, the specified measurements or baseline values, and the model's amount-versus-concentration conventions.
The entries of a conserved-moiety row state how much of that moiety is carried by each species. For example, if free receptor \(R\) and receptor–ligand complex \(RL\) each contain one receptor, their receptor-moiety coefficients are 1 and 1. If free ligand \(L\) and \(RL\) each contain one ligand, the ligand-moiety coefficients are also 1 and 1. The conservation tutorial material establishes how such moieties are derived from a stoichiometric matrix; here the key task is to use them to construct the state at time zero.
From Conservation Laws to Feasible Initial States
For a closed network, write the amount balance as \(\dot{x}=Sv(x)\), where \(S\) is the stoichiometric matrix and \(v(x)\) is the vector of reaction rates. A conserved-moiety matrix satisfies \(LS=0\). Therefore,
Thus \(Lx(t)\) is constant in time, and its value is set by the initial state: \(Lx(t)=Lx_0=b\). This explains why conservation constraints are not optional checks added after choosing initial values. They determine which initial values are compatible with the closed-system model and its stated totals.
When only some species are specified, partition the state into specified amounts \(x_K\) and amounts to be determined \(x_U\). If the corresponding columns of \(L\) are \(L_K\) and \(L_U\), then the unknown amounts must satisfy
A solution must satisfy both the equations and nonnegativity. If several unknown amounts remain, conservation equations alone may not determine them uniquely. Additional measured amounts, baseline assumptions, or other model conditions are then needed; choosing arbitrary values is not a substitute for stating those conditions.
List each known conserved total and the species that contribute to it. Keep the amount units explicit.
Insert specified measurements or baseline amounts into the conservation equations before solving for the remaining species.
Check that the equations are mutually consistent and that every resulting species amount is nonnegative.
Substitute the complete state back into every conserved-total equation, with units. This catches arithmetic and bookkeeping errors.
Worked Constructions
Worked Example: Initializing a Receptor–Ligand System
A closed binding compartment contains free receptor \(R\), free ligand \(L\), and complex \(RL\). The total receptor amount is 10 nmol, and the total ligand amount is 7 nmol. A baseline measurement specifies 4 nmol of complex. Construct the other initial amounts.
Each receptor-bearing species contributes one receptor, and each ligand-bearing species contributes one ligand. The constraints are \(R_0+RL_0=10\) nmol and \(L_0+RL_0=7\) nmol. Substituting \(RL_0=4\) nmol gives
The constructed state is \(R_0=6\) nmol, \(L_0=3\) nmol, and \(RL_0=4\) nmol. It is nonnegative, and direct checks give \(6+4=10\) nmol of receptor and \(3+4=7\) nmol of ligand. The initial state therefore satisfies both stated totals.
Worked Example: Initializing a Simple Closed Reaction Network
Consider the reaction \(A+B\rightleftharpoons C\). Its conserved totals are \(A+C\) and \(B+C\). Suppose these totals are 12 μmol and 9 μmol, respectively, and an assay specifies \(C_0=3\) μmol. Find \(A_0\) and \(B_0\).
The equations for the unknown amounts are
Substitute the measured complex amount into each equation:
All three amounts are nonnegative. Substitution verifies the conserved totals: \(9+3=12\) μmol and \(6+3=9\) μmol. Notice that the two totals are distinct accounting constraints: \(C\) contributes to both because it contains one unit of each moiety.
Worked Example: Allocating a Conserved Amount Across Unequal Volumes
A ligand can move between two well-mixed compartments and bind to a receptor in compartment 2. The total ligand amount across both compartments is 30 nmol. Compartment 1 has volume 0.5 L, compartment 2 has volume 2 L, and the bound complex in compartment 2 is specified as 4 nmol. The free ligand concentration in compartment 1 is 20 nmol/L. Find the free ligand concentration in compartment 2 that satisfies the total.
First convert the specified concentration in compartment 1 to an amount:
The total ligand constraint includes the free ligand amounts in both compartments and the bound ligand amount. Therefore, the free ligand amount in compartment 2 must be
Convert this amount to concentration using the volume of compartment 2:
The check is \(10+16+4=30\) nmol. The concentration values, 20 nmol/L and 8 nmol/L, are not added to obtain the total: they refer to different volumes, and the bound complex is already an amount. Conservation applies to the amount of ligand.
When the Constraints Are Inconsistent
Worked Example: Detecting an Impossible Set of Totals
For \(A+B\rightleftharpoons C\), suppose the stated totals are \(A+C=10\) μmol and \(B+C=8\) μmol. A separate measurement specifies \(A+B=20\) μmol. Determine whether a nonnegative initial state can satisfy all three statements.
Add the first two conservation equations. The left-hand side becomes \(A+B+2C\), so the stated totals require
But \(A+B=20\) μmol, which would require \(20+2C=18\), or \(C=-1\) μmol. A negative species amount is not feasible. Thus no nonnegative initial state satisfies the three specifications. The appropriate response is to investigate the totals, units, species definitions, or measurement assumptions—not to silently truncate \(C\) to zero, since that would leave the equations unsatisfied.
Units, Closed-System Assumptions, and Diagnostics
These constructions rely on the conservation relation applying to the model being initialized. A moiety is conserved only if the modeled reactions and transfers do not create or remove it from the system boundary. In an open model with input, elimination, or exchange with an unmodeled external pool, a quantity that would be conserved in the closed network may change over time. Do not impose a closed-system total unless it is appropriate to the chosen model boundary and its specified inputs and outputs.
For multi-compartment systems, transfer between included compartments redistributes an amount but does not remove it from the overall system. When volumes differ, convert each concentration \(C_i\) to an amount \(V_iC_i\) before applying an amount-based total. Bound species must also be counted according to the moieties they contain. Keep track of whether a reported total is an amount, a concentration, or a sum over a defined set of compartments.
Useful warning signs include a negative solved amount, totals that fail when the proposed state is substituted back into the equations, or a concentration used directly where the constraint requires an amount. A conflict can also arise when specified values overdetermine the constraints. Conversely, fewer independent constraints than unknown amounts may leave multiple feasible states; that is underdetermination, not inconsistency. State which additional information selects a particular state.
| Situation | What it means | Appropriate response |
|---|---|---|
| All constraints hold and all amounts are nonnegative | A feasible initial state has been constructed | Record the assumptions and verify each total with units |
| Equations conflict or imply a negative amount | The supplied totals or specifications are incompatible | Check units, measurements, moiety coefficients, and model boundary |
| Several nonnegative states satisfy the constraints | The information does not uniquely determine the initial state | Supply additional data or an explicit baseline assumption |
The main alternative to solving directly for unknown species is to specify all initial species amounts independently and then calculate implied totals. That approach is useful when the species amounts are measured, but it does not ensure that the values match a separately asserted total. Constrained construction starts with the trusted totals and solves for compatible values; checking a fully specified state instead asks whether its implied totals match the stated ones. Neither approach can resolve contradictory information without a decision about which input is unreliable.
Common Pitfall
A frequent error is to assign plausible values to every species and assume that the model will “sort out” any conservation mismatch during simulation. In a closed network, the dynamics preserve the total implied by the actual initial state; they do not make that state agree with a different total written in a specification. Another error is to add concentrations from compartments of unequal volume. The correction is to formulate each conservation equation in amounts, convert concentrations using the corresponding volumes, and verify the resulting state against every stated total before using it.
Check Your Understanding
Answer each question, then compare with the solutions.
- For \(P+Q\rightleftharpoons D\), the total \(P+D\) is 15 nmol and \(D_0=6\) nmol. What is \(P_0\)?
- A closed two-compartment model has \(V_1=1\) L and \(V_2=3\) L. Free concentrations are 5 nmol/L and 4 nmol/L, and there is 2 nmol of bound ligand. What is the total ligand amount?
- For \(A+B\rightleftharpoons C\), the totals are \(A+C=8\) μmol and \(B+C=7\) μmol, with \(C_0=9\) μmol. Is the resulting state feasible?
- Why is a conserved total from a closed network not automatically a valid constraint when the modeled system has an external input?
- If three unknown species amounts are subject to only one independent conserved-total equation, does that equation generally determine a unique initial state?
Solutions. (1) \(P_0=15-6=9\) nmol. (2) The amounts are \((1\ \text{L})(5\ \text{nmol/L})=5\) nmol and \((3\ \text{L})(4\ \text{nmol/L})=12\) nmol; adding the 2 nmol bound amount gives \(5+12+2=19\) nmol. (3) The equations give \(A_0=8-9=-1\) μmol and \(B_0=7-9=-2\) μmol, so the specifications are not feasible for nonnegative amounts. (4) An external input can add the moiety to the modeled system, so its total need not remain constant; use a constraint only if it matches the system boundary and input assumptions. (5) No. One equation generally leaves multiple possible combinations of the three unknown amounts, so additional information is needed to select a state.