Linking Measurements to Latent Model States

From Mechanistic States to Measured Outputs

A QSP model may represent biological quantities that an experiment does not measure directly. The model can track free cytokine, cytokine bound to receptor, and intracellular cytokine, while an assay reports a single “total cytokine” concentration. It can track free and ligand-bound receptors, while an assay reports the percentage of receptors occupied. Calibration therefore needs an explicit link between each model state and the corresponding experimental readout.

That link is an observation equation. It specifies what the assay is predicted to report when the mechanistic model has a particular state. Its form depends on both the biology represented by the states and what the assay recognizes or counts. The observation equation is not automatically the same as a state variable: it may be a sum, ratio, fraction, or other function of several states.

Definition: Let $x(t;\theta)$ be the vector of model states at time $t$, given parameters $\theta$. For assay $s$, an observation equation is a function $h_s$ that maps the relevant model states (and, when needed, specified model parameters) to a predicted assay output: $$ \hat y_s(t)=h_s(x(t;\theta),\theta). $$ Here, $\hat y_s(t)$ is the model-predicted readout on the assay’s reporting scale. A measured value is data; $\hat y_s(t)$ is the prediction to which that data is compared during calibration.

The observation equation must reflect the assay’s measurement target, not merely the name of a nearby model state. For each data source, identify the material being sampled, the molecular forms the assay detects, and how its reported quantity is constructed. The earlier material on constructing an evidence map helps organize which sources and readouts need distinct mappings.

Constructing an Observation Equation

Start by listing the states that could contribute to the assay signal. Then determine which of those states are included by the assay and how they contribute. For a concentration assay, a total may be a sum of species. For an occupancy assay, the readout may be a ratio with a biologically meaningful denominator. For a signal that responds differently to different forms, a weighted sum may be more appropriate than an unweighted total.

A general representation for a measured output is:

\hat y_s(t)=h_s\bigl(x_1(t;\theta),x_2(t;\theta),\ldots,x_m(t;\theta)\bigr).

This notation does not prescribe a particular formula. The function $h_s$ follows from a stated measurement definition. If the measured quantity is total concentration across two recognized species, $h_s$ could be their sum. If it is the fraction of surface receptors that are bound, $h_s$ is a ratio. A different assay of the same biological system can require a different observation equation because it detects a different set of species or reports a different quantity.

A useful discipline is to label the observation equation with the assay or source, even when two equations happen to have the same mathematical form. This makes clear which experimental readout is being predicted and discourages silently treating distinct measurements as interchangeable. The state variables remain defined by the mechanistic model; the observation equation expresses how an assay relates to them.

Worked Example: Total Cytokine After Dissociation

Consider a hypothetical model with free extracellular cytokine $C$ and cytokine bound to a soluble binding protein, $CB$. At one sampling time, the model predicts $C=12$ pM and $CB=8$ pM. Suppose the assay preparation dissociates the complex and the assay reports the total molar amount of cytokine from both forms, with equal response per mole.

The assay’s target is cytokine molecules in either form, not free cytokine alone. Thus the observation equation is:

\hat y_{\mathrm{total}}(t)=C(t)+CB(t).

Substituting the model predictions gives:

\hat y_{\mathrm{total}}=12\ \mathrm{pM}+8\ \mathrm{pM}=20\ \mathrm{pM}.

The model predicts a total cytokine readout of 20 pM. If a separate assay instead detects only free cytokine without dissociating the complex, its observation equation is $\hat y_{\mathrm{free}}(t)=C(t)$, giving 12 pM. These are different predictions for different measurement definitions, even though they use states from the same model.

Ratios and Fractions Require an Explicit Denominator

Occupancy is a common example of a readout that is not a concentration or a single state. For a surface receptor with one ligand-binding site, let $R$ denote free surface receptor and $RL$ denote ligand-bound surface receptor. If the assay reports the fraction of those receptors that are ligand-bound, its denominator is the total surface receptor represented by these two states.

Definition: For a one-site receptor represented by free surface receptor $R$ and ligand-bound surface receptor $RL$, the occupancy fraction is the number of occupied receptors divided by the total number of represented surface receptors: $$ \hat f_{\mathrm{occ}}(t)=\frac{RL(t)}{R(t)+RL(t)}. $$ The corresponding percentage is $100\hat f_{\mathrm{occ}}(t)\%$, provided the denominator is positive.

The denominator is part of the measurement definition. If internalized receptor is represented by another state, it belongs in the denominator only if the assay’s protocol and readout include that receptor pool. Likewise, if a model includes receptor bound to more than one ligand, each state must be included or excluded according to what the assay calls “occupied.” The equation cannot be selected from the word “occupancy” alone.

For a single binding site, the ratio can also be written in terms of the occupied fraction $f_{\mathrm{occ}}$ and the total surface receptor $R_{\mathrm{tot}}=R+RL$. Since $RL=f_{\mathrm{occ}}R_{\mathrm{tot}}$ and $R=(1-f_{\mathrm{occ}})R_{\mathrm{tot}}$, dividing occupied receptor by total receptor returns $f_{\mathrm{occ}}$. This confirms that the ratio measures a fraction of the specified pool rather than an absolute amount of bound receptor.

Worked Example: Predicting Receptor Occupancy

In a hypothetical QSP model, free surface receptor is $R=30$ fmol and ligand-bound surface receptor is $RL=70$ fmol at a specified time. Assume the assay reports the fraction of surface receptor occupied and excludes internalized receptors.

The relevant pool is $R+RL$, so the predicted occupancy is:

\hat f_{\mathrm{occ}}=\frac{RL}{R+RL} =\frac{70\ \mathrm{fmol}}{30\ \mathrm{fmol}+70\ \mathrm{fmol}} =\frac{70}{100}=0.700.

The predicted occupancy is 0.700, or 70.0%. It is not 70 fmol: that value is the amount of bound receptor, whereas the assay reports a fraction. If the model also tracked 20 fmol of internalized receptor, adding it to the denominator would change the prediction to $70/(30+70+20)=0.583$, but that would answer a different measurement question and would be inappropriate under the stated assay definition.

Weighted Signals and Assay-Specific Readouts

Not every assay reports an exact molecular total or a fraction. An assay may produce a signal in which different forms contribute unequally. If those contributions are known or defined by the measurement model, an observation equation can express them as weights. The weights must correspond to the particular readout and its scale; they should not be added solely because a more elaborate equation seems more realistic.

Worked Example: A Phosphorylated-Protein Fraction

Suppose a hypothetical intracellular model tracks unphosphorylated protein $U$ and phosphorylated protein $P$. A cell-based assay is defined to report the fraction of the measured protein pool that is phosphorylated. At one time, the model predicts $U=42$ fmol and $P=18$ fmol. Assume these are the only two forms included in the assay’s denominator.

The observation equation is:

\hat f_{\mathrm{phos}}(t)=\frac{P(t)}{U(t)+P(t)}.

Substituting the predicted amounts gives:

\hat f_{\mathrm{phos}}=\frac{18\ \mathrm{fmol}}{42\ \mathrm{fmol}+18\ \mathrm{fmol}} =\frac{18}{60}=0.300.

The predicted phosphorylated fraction is 0.300, or 30.0%. The state $P$ alone is 18 fmol, not 30.0%; the assay’s reported fraction requires the total measured pool in the denominator. If the assay instead reported the absolute phosphorylated amount, the appropriate equation would be $\hat y_P(t)=P(t)$, and the prediction would be 18 fmol.

Justifying and Checking the Mapping

Each observation equation makes assumptions about how the assay relates to model states. The equation for a total assumes the relevant forms are included and contribute equally on the stated molar scale. The occupancy equation assumes that its numerator and denominator refer to the intended receptor pool, and that the states count compatible receptor quantities. The phosphorylated fraction assumes that the two states exhaust the measured pool. These assumptions should be checked against the assay definition and the model’s state definitions before the equation is used.

A practical construction process is:

Define the readout.
Write down what the assay reports: for example, total cytokine concentration, occupied-receptor fraction, or phosphorylated-protein fraction.

Identify the contributing states.
Specify which model states are present in the sampled material and which forms the assay includes in its numerator, denominator, or total.

Write the function.
Use a sum for a defined total, a ratio for a fraction, or a weighted combination when the measurement definition supports it.

Evaluate and sanity-check.
Substitute model states and check the resulting scale, sign, range, and meaning against the stated readout.

The sanity check often reveals a mapping error. A predicted fraction should be between 0 and 1 when its numerator is a nonnegative subset of its denominator and the denominator is positive. A predicted total should include each intended form once, not omit a bound pool or count it twice. A predicted absolute amount should not be reported as a percentage. If a ratio has a zero denominator, the ratio is undefined; the model or the measurement definition must address that case rather than assigning a value by arithmetic convention.

The mapping also affects what can be learned from calibration. If two states enter an assay only through their sum, that assay alone may constrain the sum without distinguishing the individual states. A second readout that responds differently to those states can add a distinct constraint. This is one reason to preserve the distinction between biological states and assay predictions: a good fit to an aggregate measurement does not, by itself, establish that every contributing state is separately well determined.

Common Pitfalls and Alternatives

Choice	When it is appropriate	Typical failure
Use a state directly as the prediction	The assay reports that exact state on the same defined scale.	Using free cytokine for an assay that measures free plus bound cytokine.
Sum states	The assay measures the combined amount of the specified forms.	Summing forms the assay excludes, or counting the same molecules in multiple states.
Use a ratio	The readout is a fraction with a defined numerator and denominator.	Using an incomplete or incorrect denominator, or interpreting the ratio as an absolute amount.
Use a weighted signal	The assay definition supports unequal contributions from model states.	Adding arbitrary weights without a measurement-based justification.

A common mistake is to assume that the state whose name most closely resembles the assay label must be the observation equation. For example, a measured “total cytokine” value is sometimes mapped directly to the free cytokine state because that is the model’s most familiar cytokine variable. The correction is to identify every cytokine form included by the assay and write the total as the corresponding function of those states. Conversely, an assay that is explicitly selective for free cytokine should not be mapped to a total simply because a total is more comprehensive biologically.

The main alternative to an explicit observation equation is to treat the measured output as a model state, or to compare data with an informal proxy chosen for convenience. That shortcut is valid only when the state and the measured quantity genuinely coincide. Explicit mapping requires more care, but it makes the biological and measurement assumptions visible and lets different assays connect to the same mechanistic model without pretending they measure the same thing.

Observation equations do not remove uncertainty about whether the assay detects the proposed forms or whether the model includes all relevant contributors. Nor can a mapping make an unmeasured state directly observed: a predicted output may constrain a combination of states while leaving their individual values ambiguous. The mapping is therefore a necessary part of calibration, not proof that the chosen biological interpretation is uniquely established.

Key takeaway: A QSP observation equation is a stated, assay-specific function of model states. Define the measured quantity first, then map the relevant states to it using the correct sum, ratio, or justified signal function.

Check Your Understanding

Answer each question by stating the measurement definition and the corresponding prediction. Solutions follow the questions.

A hypothetical assay dissociates a complex and reports total analyte. The model predicts 5 nM free analyte and 3 nM complex. What is the observation equation and prediction?
A surface-receptor assay reports the fraction occupied. The model predicts 24 fmol free receptor and 36 fmol bound receptor. What percentage does it predict?
An assay reports the absolute amount of phosphorylated protein, not the phosphorylated fraction. Which state should be used if that amount is represented directly in the model?
Why is it generally incorrect to include internalized receptor in an occupancy denominator when the assay is defined to measure surface receptor occupancy?
An assay reports the sum of two states, $A+B$. What can that readout establish about $A$ and $B$ without additional information or a distinct measurement?

Solutions. (1) The mapping is $\hat y=C+CB$, so $\hat y=5\ \mathrm{nM}+3\ \mathrm{nM}=8\ \mathrm{nM}$. (2) The prediction is $36/(24+36)=36/60=0.600$, or 60.0%. (3) Use the phosphorylated-amount state directly, because the defined readout is an absolute amount rather than a fraction. (4) Internalized receptor is outside the assay’s defined surface-receptor pool, so including it changes the denominator and predicts a different quantity. (5) The readout constrains the sum $A+B$; by itself it does not generally determine the separate values of $A$ and $B$.