Worked Conservation and Unit Audit of a Compartmental Pathway Model

Why Audit a Complete Pathway Model?

Checking a single equation is not enough to establish that a pathway model is internally consistent. A rate can have the right units but the wrong sign; two equations can each look plausible while their transfer terms create a spurious loss when summed. A worked audit therefore follows each species through the model, checking both the dimensions of every term and the balance of material across the full pathway.

Consider a pathway with a parent species \(A\) in two well-mixed compartments and a product \(B\) in compartment 2. The parent can transfer between compartments, convert to product, and be cleared. External inputs and product elimination make this an open system. We will track the amount of the shared conserved moiety in \(A\) and \(B\), assuming the conversion is one-to-one and the product retains that moiety.

The audit assumes each compartment is well mixed, the stated amounts represent all modeled material in the moiety, and all rates use the same time unit. Inputs are external additions; clearance and product elimination are exits from the modeled moiety. These assumptions define what should balance. A dimensional check alone cannot establish them.

Write the Model as a Flux Ledger

Define \(J\) as positive transfer of \(A\) from compartment 1 to compartment 2, \(v\) as conversion of \(A_2\) to \(B_2\), and \(u_1,u_2\) as external inputs. A representative model is

Here \(PS\) is a transfer coefficient, \(CL_1\) and \(CL_2\) are clearances, \(V_{\max}\) is a maximum conversion rate, \(K_m\) is a concentration constant, and \(k_B\) is a first-order product-elimination rate constant. A positive \(J\) reduces \(A_1\) and increases \(A_2\); if \(C_2>C_1\), then \(J<0\), and the same signed terms describe transfer in the opposite direction.

The flux ledger is the first useful check: each internal process must appear with opposite signs in the equations for the material it moves or transforms. The transfer moves \(A\) between compartments, while conversion removes \(A\) and adds \(B\). Inputs and exits appear only in the equation for the species they directly affect.

Derive the Balances Before Interpreting Them

A balance for a chosen total is obtained by adding the equations for all included states. This is a direct accounting argument: a term that moves material between included states should cancel, while a term that crosses the boundary of the chosen total should remain. For the total parent amount \(A_{\mathrm{tot}}=A_1+A_2\), add the two parent equations:

The \(-J\) and \(+J\) terms cancel because transfer changes the location of parent material, not its total amount. Conversion remains as a loss from the parent pool because it moves material from \(A\) into \(B\).

Now include the product and define the total modeled moiety \(M=A_1+A_2+B_2\). Adding the product equation cancels conversion as well:

The conversion rate does not appear in this total-moiety balance because it is an internal transformation under the one-to-one moiety assumption. The total can still change through external input, parent clearance, or product elimination. If all such inputs and exits are set to zero, then \(dM/dt=0\), so the model conserves \(M\).

Locate Errors by Separating Dimensions from Balance

Dimensional consistency and conservation accounting detect different problems. A dimensional mismatch means a formula cannot be added to the other terms in the same equation. A balance mismatch means the represented flows do not agree with the intended movement of material. A term may pass one check and fail the other, so both are necessary.

Practical Checks, Alternatives, and Limits

A reliable audit can be performed in a small number of explicit passes. Writing each flux once and then placing it in the affected state equations helps prevent sign changes or duplicated flows. A separate sum for each biologically meaningful total then shows which terms cancel and which cross the system boundary.

Several diagnostic checks follow directly. When \(C_1=C_2\), the specified permeability flux is zero; a nonzero calculated transfer indicates a formula or implementation error. When \(C_1<C_2\), \(J\) is negative and the signed equations should reverse the direction of transfer without changing their structure. In a closed model with no inputs or exits, a changing total moiety signals an inconsistency in the equations, the species included in the total, or the assumed reaction stoichiometry.

An amount-based audit is often clearest when compartments have unequal volumes because each state equation tracks the amount gained or lost. A concentration-based formulation can also be used, but its rates must be converted consistently: for a fixed volume, \(dC_i/dt=(1/V_i)dA_i/dt\). The stoichiometric-matrix approach provides a systematic way to identify conserved combinations in reaction networks; the hand-summed flux ledger here makes the transfer and open-system terms explicit. These approaches are complementary rather than competing.

The audit has limits. Correct units and a correct balance do not prove that a mechanism, rate law, parameter value, or compartment structure is biologically appropriate. Conversely, an apparent failure may indicate an omitted species or an incorrectly chosen total rather than a sign error. The audit also relies on fixed compartment volumes; if a volume changes, amount and concentration rates are not interchangeable by simple division alone. Its purpose is to test internal consistency against declared states, units, and pathway assumptions.