Modelling a biological system: rate equations, parameters and fit
A mechanistic model turns a pathway diagram into rate equations. Its parameters have to be estimated from data, and its behaviour has to be checked against held-back data. Before drawing any conclusion from a fit, ask what the data can identify.
What a learner can do afterwards
- Write rate equations for a small pathway and state the units of every parameter
- Distinguish a parameter the data can identify from one that only trades off against another
- Use a sensitivity analysis to say which parameter the model's conclusion actually depends on
1 · Read
Let A and B be concentrations and t be time. dA/dt means the rate of change of A. At fixed volume, assume constant production s into A, one-for-one conversion A to B at rate kA, and removal of B at rate hB. Then dA/dt = s - kA and dB/dt = kA - hB. Each conversion removes from A what it adds to B. Expression estimates from single-cell analysis are not direct measurements of these concentrations or rates.
With A and B in µM and t in hours, each rate is in µM/h. Thus s is in µM/h and k and h are in h^-1: concentration times inverse time is concentration per time. For example, s = 9 µM/h, k = 0.5 h^-1 and A = 6 µM give kA = 3 µM/h and dA/dt = 6 µM/h. Matching units is necessary, but it does not prove the model is correct.
For population P, the logistic model is dP/dt = rP(1 - P/K), with r,K > 0. Here r has inverse-time units and K has population units. When P is far below K, 1 - P/K is close to 1, so early growth is nearly rP. Sparse, noisy early data often constrain K less than r; neither estimate is guaranteed precise. A steady plateau helps estimate K, while the changing part of the record helps estimate r.
Identifiability asks whether observations can distinguish parameter values within a stated model. For dX/dt = s - kX with s,k > 0, a steady concentration obeys 0 = s - kX*, so X* = s/k. Steady-level data alone cannot separate s and k: scaling both by c > 0 leaves the ratio unchanged. At X = 0 the slope is s, and return speed depends on k, so matching steady levels need not mean matching transients. Limited or noisy data can make estimation harder.
With s in µM/h and k in h^-1, both pairs predict X* = 2 µM. Their steady levels agree even though their rates differ.
Choose an error score and reserve suitable test data before fitting. Fit parameters on training data. Freeze the model and parameters before scoring the test data. Tuning with those scores makes another untouched test necessary. Reserve later times or separate runs to assess future predictions. A perfect training fit proves neither the mechanism nor predictive accuracy on unseen data. Held-back agreement supports prediction under tested conditions, not unique parameter values.
Sensitivity measures how a chosen prediction responds to parameter changes. State the baseline, conditions and perturbations; hold other settings fixed. Compare equal fractional changes when scales differ. A prediction starts at 10: separately raising a by 5% gives 13, while raising b by 5% gives 10.2. The prediction changes are (13 - 10)/10 = 30% and (10.2 - 10)/10 = 2%. This local comparison proves neither exclusive dependence on a nor identification of either parameter.
Each parameter increases by 5% in a separate test. The bars show the resulting percentage changes in the prediction.
For constant k > 0, X(t) = X(0)exp(-kt) solves dX/dt = -kX. X(0) is the starting concentration; exp is the exponential function. Differentiating exp(-kt) multiplies it by -k, so dX/dt = -kX(t). At t = 0, exp(0) = 1 gives X(0). For X > 0, the rate is negative and X decreases. Compare code with this formula under the same assumptions. A mismatch calls for checks of signs, units, initial conditions and implementation; it does not identify one fault.
Fit s in dX/dt = s - kX using steady training measurements and known k. A candidate predicts X* = s/k. Subtract each observation from that prediction, square each difference and add. Choose the candidate with the smallest sum. For k = 0.5 h^-1 and observations 5 and 7 µM, s = 2, 3 or 4 µM/h predicts 4, 6 or 8 µM. The error sums are 10, 2 and 10 µM², so s = 3 µM/h fits best among these candidates. This score does not prove a unique mechanism.
Each bar shows the sum of squared concentration errors in µM². The smallest score selects s = 3 µM/h among these candidates.
Balance rates and units. Check what observations identify, test untouched data, and report sensitivity for a stated prediction and parameter change.
Lesson, questions and flashcards written for this dataset, including the candidate-fitting example. OpenStax Calculus Volume 2 was consulted for differential-equation facts and teaching structure; no textbook passages are reproduced.
Four original readings written for this dataset. Flashcards were written or revised for this dataset. OpenStax Calculus Volume 2 and Biology were consulted for equation facts and population-model assumptions; no textbook passages are reproduced.
More to read
Build a model from its transfers
Choose the quantities before choosing parameter values. Consider two well-mixed pools, A and B, at fixed volume. Material enters A at a constant rate s. A converts one-for-one into B at rate kA, and B leaves at rate hB. These assumptions give dA/dt = s - kA and dB/dt = kA - hB.
The transfer kA appears with opposite signs because the conversion removes A and supplies B. Adding the two equations cancels that internal transfer: d(A + B)/dt = s - hB. This is a useful bookkeeping check for the stated one-for-one model.
If concentrations are measured in nM and time in minutes, s has units nM/min. Both k and h have units min^-1. Different rate laws can require different parameter units, so derive them from the complete term.
Before fitting, list the assumptions and the observations. A changing volume would alter these concentration balances. A reverse conversion would add terms. Single-cell expression estimates may suggest which processes to investigate, but they do not directly measure the pool concentrations or transfer rates used here.
Ask what the observations can distinguish
A fitted value can look precise even when other values explain the measurements nearly as well. To investigate, vary the parameters and compare the resulting predictions with the observations, taking measurement error into account.
For logistic growth, dP/dt = rP(1 - P/K), the observation range matters. When all measured populations are far below K, the crowding factor is nearly one. Many sufficiently large capacities may then predict almost the same early growth. More precise observations or a wider observation range can help; sparse early measurements do not guarantee a precise estimate of r either.
Some ambiguities persist even with exact measurements of the chosen kind. For dX/dt = s - kX, an exact steady concentration gives the ratio s/k. Doubling both parameters preserves that concentration. Repeating the same steady-level measurement more precisely cannot separate those parameters by itself.
A measurement during a transient can supply different information. For example, the rate at X = 0 equals s. Experimental design should consider which observations distinguish the competing parameter values, not just how many measurements can be collected.
Separate fitting from the final prediction test
Decide which prediction the model will be used to make. Predicting a later time period calls for a test that preserves the time order. Predicting another experimental run calls for observations from a run that did not guide model development. Nearby measurements from the same run may share errors and conditions.
Choose the error score and reserve suitable observations before fitting. Training data guide the parameter estimates. Final test data assess predictions after the equations and parameters have been fixed. If the test score guides further adjustments, those observations have joined the development process. Another untouched test is needed for the revised model.
Poor predictions have several possible causes. Parameters may have followed training noise, environmental conditions may have changed, or the equations may omit a process that matters. Compare these possibilities with evidence rather than assigning every prediction failure to overfitting.
Report the tested conditions and the observed errors. Successful predictions support use under those conditions. They do not show that every parameter has a unique value or that alternative mechanisms have been ruled out.
Check a known limit before interpreting a fit
Suppose turning off production reduces a model to dX/dt = -kX. With constant k > 0 and starting concentration X0, the solution is X(t) = X0exp(-kt). Differentiating gives -kX0exp(-kt), which is exactly -kX(t). At t = 0 the formula returns X0. These two checks test the equation and the starting value separately.
For a test case, choose X0 = 18 nM and k = 0.3 min^-1. The initial rate is -5.4 nM/min. A positive starting concentration must decrease toward zero in this model. Run the code with the same parameters, time units and production setting, then compare several output times with the formula.
A numerical solver may have small approximation errors. Tightening its error settings or reducing its time step, where applicable, should reduce those errors for a well-behaved calculation. A persistent mismatch needs investigation of the implemented equation, units, initial conditions and numerical method.
Passing this check supports the implementation in this particular limit. It does not test the production terms that were disabled or establish that the full model describes the biological system.
2 · Watch
Take it off screen
Where it sits
Optional connections
This opens up
Nothing builds on it yet.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.