From Molecular Fluxes to State Equations
A coupled mRNA–protein model describes two changing states: the amount or concentration of mRNA and the amount or concentration of its protein product. The central task is to translate a biological account of production and loss into rates of change. Each equation follows the same accounting rule: rate of change equals total production flux minus total loss flux.
Here we construct the simplest coupled model: transcription supplies mRNA at a constant rate, translation produces protein in proportion to the available mRNA, and each molecular species is lost by a first-order process. This is a model of the specified fluxes, not a claim that all expression systems behave this way under all conditions.
The equations assume a well-mixed system whose volume is fixed or whose state variables are defined so that volume changes are already accounted for. Transcription and translation coefficients are constant over the modeled interval; translation is proportional to \(M\); and loss is proportional to the amount of the species currently present. There is no explicit delay between transcription and translation, and neither state feeds back on these fluxes. The states and parameters are nonnegative. These assumptions make the model linear in \(M\) and \(P\), with a constant mRNA source.
Deriving the Equations by Flux Balance
A flux is a quantity entering or leaving a state per unit time. For a sufficiently short interval \(\Delta t\), the change in a state is approximately its net flux at the start of the interval multiplied by \(\Delta t\). Taking the limit as the interval becomes arbitrarily short gives the ODE. The signs follow from whether a flux adds to or removes material from the state.
Constant transcription adds mRNA at rate \(r_M\). First-order loss removes mRNA at rate \(k_M M\). Thus, over a short interval, \(\Delta M=[r_M-k_M M(t)]\Delta t+o(\Delta t)\).
Divide the increment by \(\Delta t\) and let \(\Delta t\) approach zero. The remainder satisfies \(o(\Delta t)/\Delta t\to0\), so \(dM/dt=r_M-k_M M\).
Translation adds protein at rate \(k_T M\); first-order protein loss removes it at rate \(k_P P\). Therefore \(\Delta P=[k_T M(t)-k_P P(t)]\Delta t+o(\Delta t)\).
Divide by \(\Delta t\) and let \(\Delta t\) approach zero. The result is \(dP/dt=k_T M-k_P P\).
In the protein equation, \(M\) is the source state for translation, while \(P\) is the state being removed by protein loss. The equations are coupled because the protein rate depends on the mRNA state. In this basic model, the mRNA equation does not depend on protein. An equation that puts \(P\), rather than \(M\), in the translation term would represent a different mechanism and would not describe translation proportional to mRNA.
Checking Units and Signs
Suppose \(M\) is measured in \(\mathrm{nM}_{\mathrm{mRNA}}\), \(P\) in \(\mathrm{nM}_{\mathrm{protein}}\), and time in minutes. Then the parameters must have units that make each flux compatible with the derivative it enters:
| Quantity | Units | Role in the model |
|---|---|---|
| \(r_M\) | \(\mathrm{nM}_{\mathrm{mRNA}}/\mathrm{min}\) | mRNA production flux |
| \(k_M\) | \(\mathrm{min}^{-1}\) | mRNA loss rate constant |
| \(k_T\) | \(\mathrm{nM}_{\mathrm{protein}}/(\mathrm{nM}_{\mathrm{mRNA}}\cdot\mathrm{min})\) | translation rate coefficient |
| \(k_P\) | \(\mathrm{min}^{-1}\) | protein loss rate constant |
Multiplying \(k_M\) by \(M\) gives \(\mathrm{nM}_{\mathrm{mRNA}}/\mathrm{min}\), as required in the mRNA equation. Multiplying \(k_T\) by \(M\) gives \(\mathrm{nM}_{\mathrm{protein}}/\mathrm{min}\), as required in the protein equation. Likewise, \(k_P P\) has protein concentration per minute as its units. The translation coefficient need not have units of inverse time: it converts an mRNA state into a protein production flux. If the states are molecule counts rather than concentrations, the coefficient’s units must instead reflect protein molecules produced per mRNA molecule per unit time.
The signs encode the direction of the flux relative to the state. Transcription and translation add material, so their terms are positive. Degradation or other modeled loss removes material, so its term is negative. With nonnegative parameters and states, a loss flux cannot be negative. If a calculation gives a negative loss flux, check the parameter sign and the definition of the flux before interpreting the result.
Worked Example: Deriving and Evaluating Both Rates
Suppose a hypothetical cell model uses \(M\) in \(\mathrm{nM}_{\mathrm{mRNA}}\), \(P\) in \(\mathrm{nM}_{\mathrm{protein}}\), and time in minutes. Let \(r_M=0.12\ \mathrm{nM}_{\mathrm{mRNA}}/\mathrm{min}\), \(k_M=0.040\ \mathrm{min}^{-1}\), \(k_T=0.30\ \mathrm{nM}_{\mathrm{protein}}/(\mathrm{nM}_{\mathrm{mRNA}}\cdot\mathrm{min})\), and \(k_P=0.060\ \mathrm{min}^{-1}\). At the time of interest, \(M=2.0\ \mathrm{nM}_{\mathrm{mRNA}}\) and \(P=5.0\ \mathrm{nM}_{\mathrm{protein}}\). Write the equations and calculate the instantaneous rates.
The assumed fluxes give
For mRNA, transcription produces \(0.12\ \mathrm{nM}_{\mathrm{mRNA}}/\mathrm{min}\), while loss is \(k_M M=(0.040\ \mathrm{min}^{-1})(2.0\ \mathrm{nM}_{\mathrm{mRNA}})=0.080\ \mathrm{nM}_{\mathrm{mRNA}}/\mathrm{min}\). Therefore,
For protein, translation produces \((0.30)(2.0)=0.60\ \mathrm{nM}_{\mathrm{protein}}/\mathrm{min}\), and protein loss is \((0.060)(5.0)=0.30\ \mathrm{nM}_{\mathrm{protein}}/\mathrm{min}\). Hence
At this instant both states are increasing, because production exceeds loss for each. These are instantaneous rates, not amounts accumulated over an arbitrary time interval.
Worked Flux Checks
Worked Example: Translating Units into the Translation Term
Consider a hypothetical model with \(M=0.15\ \mathrm{\mu M}_{\mathrm{mRNA}}\). A translation coefficient is reported as \(k_T=0.20\ \mathrm{\mu M}_{\mathrm{protein}}/(\mathrm{\mu M}_{\mathrm{mRNA}}\cdot\mathrm{h})\). Calculate the translation flux and determine whether it can appear in the protein equation.
The flux is the coefficient multiplied by the mRNA state:
The mRNA units cancel, leaving protein concentration per hour. Thus this term is dimensionally suitable as the positive translation term in \(dP/dt\), provided the protein state is measured in \(\mathrm{\mu M}_{\mathrm{protein}}\) and time in hours. It would not be suitable in an equation using minutes unless the time units were converted consistently.
Worked Example: Recovering the Net Rates from Fluxes
At one instant in a hypothetical system, transcription is \(0.050\ \mathrm{nM}_{\mathrm{mRNA}}/\mathrm{min}\) and mRNA loss is \(0.030\ \mathrm{nM}_{\mathrm{mRNA}}/\mathrm{min}\). Translation is \(0.24\ \mathrm{nM}_{\mathrm{protein}}/\mathrm{min}\) and protein loss is \(0.19\ \mathrm{nM}_{\mathrm{protein}}/\mathrm{min}\). Find both state rates and state their directions of change.
Each net rate is production minus loss. For mRNA,
For protein,
Both rates are positive, so both states are increasing at that instant. The separate fluxes have the same units as their respective net rate. The flux values alone do not identify the state values or parameter values; they only determine the instantaneous net rates at the stated time.
Assumptions, Diagnostics, and Alternatives
The equations are useful when the stated fluxes are plausible over the time and state range being modeled. They should not be treated as a default that overrides data or biological knowledge. Several checks can reveal a mismatch between the model and its intended interpretation:
- Check dimensions term by term. Every term in one equation must have the same units as that state per unit time. A protein loss term cannot be added to an mRNA production term.
- Check the state each flux acts on. First-order mRNA loss is proportional to \(M\); first-order protein loss is proportional to \(P\). Translation depends on \(M\) in this model.
- Check directions at boundary states. At \(M=0\), the model gives \(dM/dt=r_M\), which is nonnegative. At \(P=0\), it gives \(dP/dt=k_T M\), also nonnegative for nonnegative \(M\). A negative rate at a zero state would be a warning that a loss term or sign has been entered incorrectly.
- Check whether the assumed flux laws are supported. If measured rates do not scale approximately with the relevant state, a first-order loss or proportional translation term may not describe the observed range. A constant source is likewise inappropriate if transcription changes over the interval.
- Check units and time bases before comparing rates. A value per hour cannot be combined directly with a value per minute. Convert one consistently rather than changing only one term.
A flux-balance derivation is preferable to memorizing the final equations: it makes the source, destination, and sign of every term explicit. Writing separate production and loss fluxes and then subtracting them is mathematically equivalent, but is often easier to audit. Amount-based and concentration-based models can also express the same accounting, but their state and parameter units must match. For example, a change in volume requires care when converting an amount balance to a concentration balance; the equations above assume a fixed volume or a state definition that absorbs that issue.
The model is limited by its assumptions. It does not represent a time delay, nonlinear or saturating translation, changing transcription, feedback, or a loss process that is not first order. Those are different flux specifications and require corresponding terms. Adding such a term should follow from a stated mechanism and must preserve dimensional consistency; it should not be added merely to improve the appearance of a fit.
Common Mistake: Treating Every Rate Constant as Inverse Time
A frequent error is to assign \(k_T\) units of \(\mathrm{min}^{-1}\) automatically because \(k_M\) and \(k_P\) have inverse-time units. That assignment is not generally valid when \(M\) and \(P\) are distinct concentration states: \(k_T M\) must have protein concentration per time, so \(k_T\) must include the conversion from mRNA concentration to protein concentration. The correction is to derive parameter units from the term in which each parameter appears, not from its name.
A second sign error is to write \(+k_P P\) for protein loss. With nonnegative \(k_P\) and \(P\), that term increases protein and describes production proportional to protein, not loss. The flux-balance statement resolves the issue: protein loss is subtracted, giving \(-k_P P\).
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- If mRNA production is \(0.08\ \mathrm{nM}_{\mathrm{mRNA}}/\mathrm{min}\), and mRNA loss is \(0.05\ \mathrm{nM}_{\mathrm{mRNA}}/\mathrm{min}\), what is \(dM/dt\) at that instant?
- For \(dP/dt=k_T M-k_P P\), which state determines the translation flux, and which state determines first-order protein loss?
- If \(M\) is in \(\mathrm{\mu M}_{\mathrm{mRNA}}\), \(P\) in \(\mathrm{\mu M}_{\mathrm{protein}}\), and time in hours, what units must \(k_T\) have?
- At \(P=0\), with \(M\geq0\), \(k_T\geq0\), and \(k_P\geq0\), can this model give a negative \(dP/dt\)? Show why.
- A proposed mRNA equation is \(dM/dt=r_M+k_M M\), where \(k_M>0\) is described as a degradation rate constant. Identify the inconsistency.
Solutions. (1) \(dM/dt=0.08-0.05=0.03\ \mathrm{nM}_{\mathrm{mRNA}}/\mathrm{min}\), so mRNA is increasing. (2) Translation depends on \(M\); first-order protein loss depends on \(P\). (3) \(k_T\) must have units \(\mathrm{\mu M}_{\mathrm{protein}}/(\mathrm{\mu M}_{\mathrm{mRNA}}\cdot\mathrm{h})\), so \(k_T M\) is protein concentration per hour. (4) No: at \(P=0\), \(dP/dt=k_T M\geq0\). (5) The positive \(k_M M\) term adds mRNA; degradation must be subtracted, giving \(dM/dt=r_M-k_M M\).