Connecting Biological Scales Without Duplicating States

Why Connecting Scales Requires Care

A QSP model may represent a molecular interaction, the response of a cell, a tissue-level biomarker, and a whole-body clinical measurement. Connecting these levels is not simply a matter of placing more variables in one system. Each connection must specify what quantity crosses the interface, how it is converted, and which model component owns it. Otherwise, a model can count the same biological material or process twice, or pass a quantity whose meaning changes unnoticed between scales.

The earlier material on integrating evidence across biological scales motivates using different types of evidence together. Here the focus is model architecture: defining explicit mappings between representations. The earlier material on compartment boundaries provides the idea of an exchange interface; a scale interface also needs to connect quantities such as per-molecule occupancy, per-cell activity, and tissue-level amount or concentration.

A state is a quantity the model tracks as changing over time, such as the amount of a biomarker in a compartment. A mapping may instead produce a derived quantity, such as the concentration calculated from an amount and volume, or a per-cell molecular signal passed to a cell-response function. A parameter, such as the number of cells in a fixed tissue sample, specifies a model property rather than a changing biological pool. These distinctions help determine whether a quantity should be tracked, calculated, or supplied from outside the model.

Specify Ownership, Units, and Direction

For each biological quantity, assign one primary owner: the component in which that quantity is represented as a state. Other components may receive a mapped value, but should not independently track the same pool unless the model explicitly defines how the representations relate. For example, an amount and the concentration calculated from that amount are two representations of one pool, not two independent pools.

A useful interface record states: (1) the source and receiving scales; (2) the quantity being passed; (3) the source and receiving units; (4) the conversion or aggregation rule; and (5) whether the receiving quantity is an input, a derived quantity, or a new state. If material physically transfers between modeled compartments, the interface should also make clear which component loses it and which gains it. A signal that influences a response, by contrast, need not represent material transfer.

Write the mapping explicitly, even when it is simple. In general, if the source quantities are collected in \(x\), an interface can be written as \(z=g(x)\), where \(g\) is the defined mapping and \(z\) is the quantity received. The mapping must preserve the meaning needed by the receiving component. It need not preserve the numerical value: a number of molecules per cell and a total number of molecules in a tissue are different quantities linked by a cell count.

From Cells to Tissue: Map Rates Before Concentrations

When cells produce or consume a substance, the scale interface often maps a per-cell rate into a total tissue rate. The units provide a direct check: multiplying a rate per cell by a cell count should produce an amount per time. If only a fraction reaches the receiving pool, that fraction must be included and its interpretation stated. A tissue concentration then requires a further conversion using the volume of the pool in which the amount is represented.

This separation prevents a frequent ambiguity: a production rate is not a concentration, and a concentration is not an amount. The total rate can be passed as a source term to a tissue amount balance. If concentration is the tissue state instead, the corresponding source term must be divided by the represented volume. These are alternative representations of the same source contribution, not two sources to add together.

Aggregation Across Tissue Regions

A whole-body or pooled measurement may summarize several tissue regions. For concentrations, a simple arithmetic average is generally inappropriate when the regions have different volumes. The conserved quantity is amount: for a region with concentration \(C_i\) and volume \(V_i\), its amount is \(V_iC_i\). If the regions are disjoint and the measured pool combines them, total amount is the sum of regional amounts. Dividing by total volume gives the pooled concentration.

The expression follows from amount balance. Region \(i\) contains amount \(A_i=V_iC_i\). For disjoint regions, \(A_{\text{total}}=\sum_i A_i=\sum_i V_iC_i\), while \(V_{\text{total}}=\sum_i V_i\). By the definition \(C_{\text{pool}}=A_{\text{total}}/V_{\text{total}}\), the weighted-average formula follows. The weights are volumes because each concentration is amount per unit volume.

Diagnostics, Alternatives, and Limitations

A scale interface is ready for use when its units cancel or convert as intended, its quantities refer to compatible biological pools, and its mapping agrees with the model’s state definitions. Useful checks include tracing each state to its owner, checking whether mapped rates have the receiving state’s required units, and verifying that aggregation uses non-overlapping populations. For an amount balance, compare material entering and leaving the represented pool; for an aggregate measurement, recompute it from component amounts rather than averaging incompatible summaries.

Three common interface choices have different uses. Passing a derived quantity, such as occupancy, is appropriate when the receiving component needs that signal but should not independently track the underlying molecular pool. Passing a flux or rate is appropriate when material enters or leaves a separately represented compartment. Passing an empirical mapping may be useful when evidence supports a relationship between scales but does not support a detailed mechanistic conversion. The empirical option should not be presented as a mechanistic mechanism, and it must still specify input, output, units, and applicable context.

A mapping is not automatically valid outside the conditions used to define it. Uniform cell density, constant cell count, complete transfer, and compatible measurement pools are assumptions, not universal truths. If these assumptions fail, the mapped quantity may be biased or may not represent the intended biology. Also, matching units is necessary but not sufficient: two variables can share units while referring to different pools, times, or populations.

A final audit is to ask whether every cross-scale quantity has a defined source, receiving role, transformation, and unit, and whether any quantity is represented as an independent state in more than one component. This does not determine whether the biological boundary is adequate for every perturbation; it makes the chosen boundary explicit enough to inspect and test.