--- title: "Modelling a biological system: rate equations, parameters and fit" description: "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 conclusio" canonical: https://lightmysky.com/learn/science/modelling-a-biological-system-rate-equations-parameters-and-fit-mt_Al6sNjUOW5 source: https://lightmysky.com/learn/science/modelling-a-biological-system-rate-equations-parameters-and-fit-mt_Al6sNjUOW5.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Science · Area: Genetics & Evolution · Ages 23 to 24 Page: https://lightmysky.com/learn/science/modelling-a-biological-system-rate-equations-parameters-and-fit-mt_Al6sNjUOW5 ## Ready when they can - 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 ## Lesson: Rate models you can defend with data 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. *(drawing: For pool A, input minus loss is 9 - 3 = 6 µM/h. These bars compare rates, not concentrations.)* 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. 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. 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. **Recap.** 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.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Single-cell analysis: normalisation, embedding and naming cell states](https://lightmysky.com/learn/science/single-cell-analysis-normalisation-embedding-and-naming-cell-states-mt_diYHLaLmij) ## Optional connections - [Deriving the Michaelis-Menten Equation from the Steady State](https://lightmysky.com/learn/science/deriving-the-michaelis-menten-equation-from-the-steady-state-mt_BFp0fwZ8_i) - [Phase Portraits, Equilibria and Stability](https://lightmysky.com/learn/mathematics/phase-portraits-equilibria-and-stability-mt_ULuZk4lmGr)