Biology · Systems & Quantitative Biology
Models, assumptions, parameters and scales of explanation
An imagined pool contains twelve amount units. Its volume doubles from two to four volume units while its amount stays twelve. The concentration changes from six to three. A reader who sees only concentration could mistake dilution for loss. A model needs to say what its numbers represent before explaining why they change.
Research Design, Data & Scientific Practice introduces observations, comparisons and uncertainty. Gene Regulation & Cell Signalling develops biological feedback, while Population Ecology & Life-History Patterns develops rates and population boundaries. Their English chapters are available in this working manuscript; publication remains pending. Start with arithmetic, averages and ordinary graph reading. A derivative is introduced as an instantaneous rate of change. Exponentials and the circular sine–cosine example are explained where used; a first reading can follow their values and arrows, then return to their formulas with further mathematical preparation.
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Separate a model's state, parameters and observation rule, and recognise how boundaries and scale affect what it explains.
Describe the system and the record separately
A model is a deliberately specified representation used to answer a question. It may be a verbal account, a diagram, a set of equations or a probability rule. It keeps the features thought relevant to that question and leaves others out. A useful simplification makes its omissions and expected domain visible.
The state describes the quantities included at a time: for example, an amount in a bounded pool. A parameter specifies a feature of the rule, such as a rate coefficient. An initial condition supplies the starting state. An observation rule connects the model state to a possible record. Amount divided by volume gives a concentration; a detector signal might require a further calibration rule. These are different parts of the explanation.
The original question map separates question, state, parameter, observable and prediction. Its arrows express information relationships. They do not mean that writing a question physically creates a molecule or that a measured signal must cause the state it represents. Making a link explicit also makes it possible to question that link.
A state and its observable occupy different roles
Read the diagram in words
The question determines which state quantities matter. A rule and its parameters describe change. The observation rule maps included quantities to a record, and the prediction states the record expected under those assumptions. These roles are linked without being interchangeable.
| Role in an explanation |
|---|
| Question |
| Included state |
| Rule and parameters |
| Observation rule |
| Predicted record |
| Role in an explanation | Role in an explanation | Information relationship |
|---|---|---|
| Question | Included state | Quantity included by the model |
| Rule and parameters | Included state | Feature of the specified rule |
| Included state | Observation rule | Quantity a record would report |
| Observation rule | Predicted record | Consequence under stated assumptions |
Original conceptual map. Positions group explanatory roles, not biological locations or a time sequence. Each edge names an information relationship; no experimental or molecular causal effect is claimed.
A change in one quantity need not mean a change in another
In the original two-state example, amount is twelve in both cases. Concentration is amount per volume. Twelve divided by two is six; twelve divided by four is three. The concentration halves because the denominator doubles. The example contains no amount crossing the pool boundary.
Units help keep this distinction clear. An amount is measured in amount units. A concentration uses amount units per volume unit. Neither is automatically a reaction rate, which would also involve an interval of time. A numerical value such as three has no biological interpretation until the quantity, unit and boundary are specified.
A measurement can leave more than one explanation open. If only the concentration three were reported, a smaller amount in an unchanged volume and the same amount in a larger volume could both produce it. Observing concentration alone does not reveal which happened. A separate volume record, amount record or other justified evidence would be needed.
The same amount can have two concentrations
Read the diagram in words
The first state has concentration six because twelve amount units occupy two volume units. The second has concentration three because the same amount occupies four volume units. Amount changes by zero, volume doubles and concentration halves.
| Assigned state | Amount units | Volume units | Amount units per volume unit |
|---|---|---|---|
| First state | 12 | 2 | 6 |
| Second state | 12 | 4 | 3 |
| Difference between states | Δn | V₂/V₁ | c₂/c₁ |
|---|---|---|---|
| Same amount, different denominator | 0 | 2 | 0.5 |
c=n/V
Original arithmetic comparison of two stipulated pool states. Amount and volume use arbitrary compatible units. No actual cell, growth mechanism, loss, rate or intervening trajectory is supplied.
Choose the resolution required by the question
Every model has a boundary and a scale. A pool described by its total amount treats internal locations as interchangeable for that question. A spatial model may instead distinguish positions and transport between them. Neither is automatically more useful: extra detail matters when it changes the answer being sought and can be supported by suitable evidence.
A deterministic model assigns one future trajectory when its rule and initial state are fixed. A stochastic model assigns probabilities to possible events or trajectories. Treating an average population trajectory as though it were the exact future of each individual changes the claim. A population total also conceals how that total is distributed among cells or organisms.
A clear model therefore states its question, variables, boundaries, units, initial conditions, parameters and observation rule. It also states conditions under which its simplifications are expected to work. Biological evidence can challenge any of these choices. A good numerical fit supports a particular comparison with records; it does not automatically show that every hidden component of the model has been measured.
Worked example
Read the two volume states
State one has amount twelve and volume two, so its concentration is 12/2 = 6. State two has amount twelve and volume four, so its concentration is 12/4 = 3. The change in amount is zero and the concentration difference is minus three.
A different assigned state with amount six and volume two also has concentration three. That numerical match shows an ambiguity in a concentration-only observation. It does not make the two underlying states identical.
The question map helps locate the ambiguity: the state can include amount and volume, while the observable supplies only their ratio. More detailed language about the same concentration does not supply the missing information.
Check your understanding
Does a concentration decrease from six to three establish that half the amount disappeared?
Show reasoning
No. In the assigned example, amount remains twelve and volume doubles. A concentration is a ratio; a decrease can arise through a change in its numerator, denominator or both.
Why is a parameter different from an initial condition?
Show reasoning
A parameter specifies a feature of the model rule; an initial condition supplies the state from which that rule is followed. They may both be unknown, but they occupy different roles in a dynamical explanation.
Does including every molecular detail necessarily make a model more useful?
Show reasoning
No. Detail should serve the question and be supported by evidence. Unneeded or unconstrained detail can conceal the assumptions and make prediction harder to assess.
What to carry forward
Name the quantity and observation rule before interpreting a number, and match the model's scale to the question.
References
Resources supporting this lesson’s explanations and examples.
- Computational systems biology (Nature 420, 206–210 (2002); doi:10.1038/nature01254)Hiroaki KitanoResearch abstract
- Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood (Bioinformatics 25(15), 1923–1929 (2009); doi:10.1093/bioinformatics/btp358)A. Raue, C. Kreutz, T. Maiwald, J. Bachmann, M. Schilling, U. Klingmüller and J. TimmerOriginal methodological paper
- The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models (Bioinformatics 19(4), 524–531 (2003); doi:10.1093/bioinformatics/btg015)M. Hucka and colleagues, with the SBML ForumResearch abstract