States and Fluxes Need Explicit Pools
An expression model is only interpretable if it is clear what its variables count. “mRNA” might mean all copies of a transcript in a cell, only copies in the cytoplasm, or copies measured in a particular assay. “Protein” might mean a free protein, a mature protein, or all forms of that protein in a chosen compartment. These are different pools, and a flux can be called an input or output only relative to a specified pool.
For a two-species expression model, the state variables describe the amounts or concentrations of one selected mRNA species and one selected protein species. The fluxes describe the rates at which molecular events add material to or remove material from those pools. At this stage, the important task is to define the bookkeeping consistently—not to select particular mathematical rate laws.
Define the Two State Variables
Let \(m(t)\) denote the concentration of one specified mRNA species and \(p(t)\) the concentration of its encoded protein species. The subscript-free symbols are useful only after the modeler has stated exactly which transcript and protein are represented, what cellular region is included, and what concentration denominator is used.
A molar concentration counts mRNA molecules or protein molecules per volume: one mole of the selected mRNA species contains Avogadro’s number of transcript molecules, just as one mole of the protein species contains Avogadro’s number of protein molecules. These are molecular concentrations, not concentrations of nucleotides, amino-acid residues, or mass. A mass concentration such as ng/mL can also be used, but it is not interchangeable with a molar concentration without molecular-mass information.
A common convention is to use nM for both \(m(t)\) and \(p(t)\), and hours for time. Under that convention, each concentration flux has units nM/h. Other choices are valid if they are applied consistently. For example, an amount state in molecules per cell requires fluxes in molecules per cell per hour. A transcription flux reported in molecules per cell per minute cannot be subtracted from a concentration flux in nM/h without converting both the amount basis and the time basis.
Worked Example: Naming the Pools Before Writing Fluxes
Consider a hypothetical model of one cytokine transcript and its encoded protein in a cultured cell. The researcher wants to describe transcript concentration in the cytoplasm and total mature protein concentration in the cell. State the variables and identify what the pool definitions imply.
mRNA state. Define \(m(t)\) as the cytoplasmic concentration of that specific cytokine transcript, in nM. Nuclear copies are outside this pool.
Protein state. Define \(p(t)\) as the concentration of total mature cytokine protein in the whole-cell pool, in nM. “Total” must be clarified operationally—for example, whether protein bound in complexes is included.
Implication for flux labels. Transcription initially produces RNA in the nucleus, so nuclear transcription is not automatically a direct input to this cytoplasmic mRNA pool. If the model contains only \(m(t)\), a flux called “transcription” entering \(m\) must be understood as an effective input to the modeled cytoplasmic pool, potentially combining production and delivery across the omitted nuclear boundary. Translation adds protein to the defined protein pool only to the extent that the newly made protein meets its inclusion criteria.
Fluxes and Their Bookkeeping
For the two selected species, the basic flux labels are transcription, translation, mRNA degradation, and protein degradation. A flux is named for the process, but its sign in a particular pool balance follows from the pool boundary. The table states the usual bookkeeping for a model in which transcription supplies the modeled mRNA pool and translation supplies the modeled protein pool.
| Flux | Pool affected | Usual bookkeeping effect | Units if states are in nM and time in h |
|---|---|---|---|
| Transcription, \(J_{\mathrm{tx}}\) | Selected mRNA pool | Input of mRNA | nM/h of mRNA |
| Translation, \(J_{\mathrm{tl}}\) | Selected protein pool | Input of protein; ordinarily no direct removal of mRNA | nM/h of protein |
| mRNA degradation, \(J_{\mathrm{deg,m}}\) | Selected mRNA pool | Removal of mRNA | nM/h of mRNA |
| Protein degradation, \(J_{\mathrm{deg,p}}\) | Selected protein pool | Removal of protein | nM/h of protein |
The fluxes are rates, not states. For example, \(m(t)\) is a concentration at a time, whereas \(J_{\mathrm{tx}}\) is a concentration increase per unit time due to transcription. Translation uses an mRNA molecule as a template and produces protein; in ordinary expression bookkeeping, translation does not consume the transcript. If the model instead includes transcript destruction coupled to translation or explicitly tracks ribosome-bound and free RNA pools, those processes and boundaries must be represented accordingly.
The general accounting principle is that the rate of change of a pool equals the sum of its inputs minus the sum of its outputs. This follows over a short interval: the amount at the end is the starting amount plus material entering, minus material leaving; dividing by the interval and taking the limit gives the rate balance. In a concentration model, this simple form assumes a fixed reference volume, or that volume-related effects have been handled separately. The flux names and pool definitions determine which terms belong in each balance; they do not by themselves specify the flux magnitudes or their dependence on state.
Keep the Unit Basis Consistent
A concentration flux can be calculated from a molecular production rate only when the cell or compartment volume and the amount basis are known. For a fixed volume \(V\), a rate of \(q\) molecules per cell per hour corresponds to \(q/(N_A V)\) mol/L/h, where \(N_A\) is Avogadro’s constant and \(V\) is in liters. If the volume changes, the conversion from amount to concentration can change even when the number of molecules does not.
Worked Example: Converting a Per-Cell Production Rate
Suppose, hypothetically, that 15,000 molecules of a specified transcript enter a modeled cellular pool per cell per hour. The reference cell volume is \(0.750\) pL, or \(7.50\times10^{-13}\) L. Convert this input rate to nM/h.
The molar concentration rate is the molecular rate divided by the number of molecules in one mole and by the cell volume:
The denominator is \(4.5165\times10^{11}\ \mathrm{molecules\cdot L/(mol\cdot cell)}\) when the cell basis is canceled consistently. Thus,
The result is a concentration flux, not an mRNA concentration. It applies to the stated reference volume and pool. Reporting only “15,000 per hour” would leave both the per-cell basis and the concentration conversion unspecified.
Worked Example: A Translation Flux Has Protein Units
In a hypothetical system, translation produces 2,400 protein molecules per cell per minute. The defined cell volume is \(1.20\) pL. Express the translation flux as nM/h, assuming the produced protein enters the modeled protein pool.
First convert the time basis: \(2{,}400\ \mathrm{molecules/(cell\cdot min)}\times60\ \mathrm{min/h}=144{,}000\ \mathrm{molecules/(cell\cdot h)}\). The cell volume is \(1.20\times10^{-12}\) L, so the molar rate is
This is a protein concentration flux. It is not directly comparable to a transcript flux in molecules per cell per minute until both are expressed on a common amount and time basis. Nor does a high translation flux alone establish that protein concentration will rise at that rate over an extended period; removal and other modeled processes also affect the pool.
Pool Boundaries, Volume, and Diagnostics
A model boundary determines whether movement between compartments is an input, an output, or an internal transfer. If nuclear and cytoplasmic mRNA are modeled separately, export removes RNA from one pool and adds it to another. If only cytoplasmic RNA is represented, export is an input to that state. Likewise, if a protein state includes only extracellular protein, secretion is an output from the intracellular pool and an input to the extracellular pool. Choosing one boundary does not make another choice wrong; it makes the model’s interpretation specific.
Concentration states also require a reference volume. If a pool amount is \(A(t)\) and the volume is \(V(t)\), its concentration is \(C(t)=A(t)/V(t)\). A changing volume can therefore change concentration even when the amount is constant. A model may track amounts, assume a fixed volume, or include the consequences of volume change; mixing these conventions without accounting for volume can create apparent sources or sinks.
Worked Example: Whole-Cell and Cytosolic Concentrations Differ
Suppose a hypothetical cell contains 60,000 molecules of a protein. Its whole-cell volume is \(2.00\) pL, while the cytosolic volume used for a separate cytosolic pool is \(1.00\) pL. Calculate the concentration under each volume convention.
For the whole-cell volume:
If all 60,000 molecules are instead assigned to the cytosolic pool, the denominator uses \(1.00\times10^{-12}\) L:
The two values describe different concentration conventions. The second is appropriate only if the molecule count actually belongs to the cytosolic pool; one cannot assign a whole-cell count to the cytosolic volume merely to obtain a different concentration.
Several checks help reveal a bookkeeping problem before fitting or simulating a model:
- Write the pool name and compartment next to every state variable; “protein concentration” alone is not a complete definition.
- Check dimensions: a flux entering a concentration balance must be concentration per time, with the correct species as well as compatible units.
- Verify the direction of each process relative to the selected pool. Degradation is an output from the degraded species’ pool; translation is ordinarily an input to the protein pool.
- Check whether rates are per cell, per volume, or total amount, and whether time is in seconds, minutes, or hours.
- Ask whether assay measurements and modeled states refer to the same molecular form, compartment, and normalization volume.
Approaches and Limits
Two common state conventions are amount per cell and concentration per volume. Amounts make per-cell molecular counts and production rates direct to interpret, while concentrations are often convenient for comparing conditions or linking to concentration-dependent processes. Neither convention is universally superior. A conversion requires a volume, and concentration comparisons can be misleading if cell size or the relevant compartment volume differs across conditions.
A second choice is whether to represent compartments separately or combine them into a lumped pool. Separate pools make transport and localization explicit, but require additional states and measurements. A lumped pool is simpler, but any flux crossing an omitted boundary may need to be represented as an effective input or output. The lumped model cannot, by itself, distinguish mechanisms hidden inside that effective flux.
These definitions do not determine whether transcription or degradation is constant, proportional to state, saturable, regulated, or otherwise time-varying. They also do not guarantee that an observed assay signal equals the modeled state. Measurement units, normalization, detection of different molecular forms, and unmodeled compartments can all limit interpretation. Specify the state and flux meanings first; then choose kinetic assumptions that match the scientific question and available evidence.
Common Mistake
A frequent error is to treat translation as a loss of mRNA because the transcript is “used” to make protein, or to subtract a protein degradation rate directly from the mRNA balance. That reasoning confuses the process name with the pool being counted. Translation ordinarily adds protein while leaving the transcript available for further translation; mRNA degradation removes mRNA, and protein degradation removes protein. Each balance must include only fluxes that cross that particular pool’s boundary.
Check Your Understanding
Answer each question using explicit pool and unit definitions.
- A model tracks cytoplasmic mRNA concentration in nM and time in hours. What units should a transcription input to that pool have?
- Does translation ordinarily remove mRNA from the mRNA pool? Which pool does translation supply?
- A production rate is reported as 900 molecules per cell per minute. Name two pieces of information needed to convert it to nM/h.
- Why can a concentration change when the molecule count stays constant?
Solutions. (1) The input must be in nM/h of the specified mRNA. (2) No; ordinary translation supplies the modeled protein pool and does not itself consume the transcript. (3) The relevant reference volume and a conversion from minutes to hours are needed, along with Avogadro’s constant for converting molecules to moles. (4) Concentration is amount divided by volume, so a change in the pool’s volume changes concentration even at constant amount.