TF24
model
This page builds on The big picture and the FF16 model. The Assimilation & hydraulics page gives the leaf-level equations in more detail.
The idea first
TF24 adds explicit plant hydraulics to the trait-based growth framework used by FF16. It was developed to ask how plant water use, tissue costs, and growth interact as water availability and atmospheric conditions change (Towers, O’Reilly-Nugent, Sabot, Vesk, & Falster, 2024).
The central idea is simple. Opening stomata allows carbon dioxide to enter a leaf, supporting photosynthesis, but it also allows water to escape through transpiration. Replacing that water puts the xylem under tension and increases the risk of losing hydraulic conductivity. TF24 therefore chooses an operating point that maximises photosynthetic carbon gain after accounting for a carbon-equivalent hydraulic cost.
The cost is expressed in carbon units so it can be subtracted from photosynthetic gain. In the original formulation, it represents the carbon needed to restore xylem conductivity lost through cavitation.
This operating point is not prescribed by an empirical stomatal-response curve. It emerges from the interaction among leaf biochemistry, atmospheric demand, soil water, plant size, and hydraulic traits. The calculation combines:
- the Farquhar model of C3 photosynthesis;
- a vulnerability curve describing how xylem conductivity declines as water potential becomes more negative;
- a water-supply calculation linking soil, xylem, and transpiration; and
- a numerical search for the point where photosynthetic benefit minus hydraulic cost is greatest.
The Assimilation & hydraulics page develops this leaf-level calculation equation by equation. The sections below explain why it was added to the model and how it connects to whole-plant growth.
How TF24 relates to FF16
| FF16 | TF24 | |
|---|---|---|
| Carbon income | prescribed saturating light-response curve | Farquhar photosynthesis coupled to stomatal conductance |
| Water | implicit | explicit supply–demand hydraulics with a vulnerability curve |
| Stomata | — | conductance set by profit maximisation |
| Allocation / growth | functional balance (shared) | functional balance (shared) |
TF24 changes how the plant calculates carbon income at the leaf. The subsequent steps are inherited from FF16: assimilated carbon is allocated to height, diameter, and tissue pools, and these individual-level rates feed into the size-structured PDE.
Model origin and scope
Earlier versions of the plant framework could explain how traits affect growth through tissue construction, allocation, respiration, and turnover, but water availability entered that framework only indirectly. Towers et al. connected those whole-plant growth rules to an explicit stomatal and hydraulic model. This made it possible to follow a continuous chain from the environment to plant performance:
- soil water and atmospheric conditions constrain water supply and demand;
- the leaf chooses a stomatal and hydraulic operating point;
- that operating point determines net carbon assimilation;
- respiration, tissue turnover, and hydraulic costs reduce the carbon available for growth; and
- the plant’s allocation and allometry determine how that growth changes its height and tissue pools (Towers et al., 2024).
The paper and its supporting information provide the scientific basis for this new carbon-income calculation. Biochemical demand for carbon dioxide is set by Rubisco activity and electron transport, while diffusive supply depends on stomatal conductance. TF24 solves for the intercellular carbon dioxide concentration at which supply and demand agree. Leaf respiration is also linked to structural and photosynthetic nitrogen, so photosynthetic machinery provides a benefit but also carries a maintenance cost (Towers et al., 2024).
Two different optimisations
The article uses the word optimisation at two different levels. Keeping them separate prevents a common misunderstanding.
- The leaf operating-point optimisation is part of TF24. The model varies water potential and the associated stomatal conductance to maximise instantaneous photosynthetic gain minus hydraulic cost. TF24 repeats this calculation as the plant and environment change.
- The trait-strategy optimisation was part of the paper’s analysis. The researchers varied leaf mass per area, Huber value, wood density, and sapwood conductivity to find the strategy with the fastest height growth in each environment. This outer search is not part of an ordinary TF24 run.
In other words, an individual in a standard TF24 run does not continually evolve or re-optimise its leaf mass per area, wood density, or other strategy traits. Those values define the strategy being simulated. The paper compared many possible strategies to ask which one would grow in height fastest under each set of conditions (Towers et al., 2024).
From carbon gain to height growth
TF24 does not treat photosynthesis as the final measure of plant performance. Carbon income is only the start of the growth calculation. A trait can increase photosynthesis while also making leaves, sapwood, bark, or roots more expensive to build and maintain.
As in FF16, TF24 expresses height growth as a chain of four relationships:
\[ \frac{dH}{dt} = \frac{dH}{dA_l} \frac{dA_l}{dM_a} \frac{dM_a}{dB} \frac{dB}{dt}. \tag{1}\]
Read from right to left, new biomass production (\(dB/dt\)) is allocated to living tissue (\(dM_a/dB\)); that investment supports additional leaf area (\(dA_l/dM_a\)); and the plant’s allometry translates added leaf area into height growth (\(dH/dA_l\)). This is the same kind of functional-balance logic used by FF16. The hydraulic leaf model changes the carbon income entering the right-hand side, while traits can also change the construction and maintenance terms further along the chain (Towers et al., 2024).
This complete chain matters because traits can affect more than carbon income. They can also change tissue construction, maintenance, turnover, and allometry. TF24 therefore evaluates their combined consequences for whole-plant growth; their effects cannot be inferred from photosynthesis or biomass production alone (Towers et al., 2024).
Traits that connect hydraulics to growth
Towers et al. focused on four measurable traits that connect water transport to the wider carbon budget. In TF24, each trait can help one part of the plant’s carbon-water economy while imposing a cost elsewhere.
- Leaf mass per area, \(\phi\), is the dry leaf mass needed to deploy a unit of leaf area. Higher values make each unit of leaf area more expensive to build, but the leaves turn over more slowly. Leaf respiration also changes with the nitrogen investment associated with this trait.
- Huber value, \(\theta\), is the cross-sectional sapwood area supplying a unit of leaf area. More sapwood improves water supply, but the extra sapwood and bark must be built and maintained. A larger sapwood volume can also incur a larger absolute hydraulic repair cost.
- Wood density, \(\rho\), is sapwood mass per unit volume. It makes a given volume of sapwood more expensive to construct. In the current default parameterisation, it does not change cavitation resistance: the density-to-
d_I, density-to-g1_TF24, and density-to-k_sexponents are all zero. Those links can be enabled by changing their hyperparameters, but they are inactive by default. - Maximum sapwood-specific conductivity, \(K_{s,\max}\), describes how quickly a unit of sapwood can transport water at maximum conductivity. High conductivity improves water transport. In the current parameterisation it also determines \(p_{50}\) and, through it, the vulnerability-curve parameters \(c\), \(b\), and \(\psi_{crit}\); this is the model’s hydraulic safety–efficiency trade-off.
These are not four independent “good” or “bad” settings. For example, increasing Huber value improves water supply per unit leaf area but also diverts more biomass into supporting tissue. Whether that change improves height growth depends on plant size and on the current soil, light, carbon dioxide, and atmospheric conditions (Towers et al., 2024).
How the paper relates to current TF24
Towers et al. introduced the physiological module and used it in an outer trait-optimisation analysis. That analysis is best read as a worked application of TF24: it shows how the model can be used to investigate trait-environment trade-offs, but its predicted trait responses are not rules built into the strategy. A normal TF24 simulation uses fixed strategy traits unless the user places it inside a separate evolutionary or optimisation analysis (Towers et al., 2024).
The distinction between the published experiment and the current plant implementation is also important. Towers et al. supplied a fixed soil water potential and did not simulate competition, reproduction, mortality, or a changing soil water balance. The current TF24 strategy can instead receive water from the dynamic, multi-layer soil-water and root-uptake model described below. The paper is therefore the source for TF24’s physiological logic and original tests, while the linked theory pages describe the broader model in which that physiology now operates.
Soil water and root uptake
The leaf model needs to know how much water the roots can supply. In TF24, the plant’s hydraulic calculation extends down to the root collar. A separate soil water model tracks the water balance and calculates uptake from multiple soil layers, providing the water supply at that boundary. These calculations are described on their own core theory page:
- Soil water & root uptake — the multi-layer bucket model, infiltration and drainage, the retention curve, and the root-collar solve that couples soil moisture into the leaf model.
The notebooks below record the original prototyping and benchmarking work:
This page explains the scientific logic of TF24 and how the published model connects leaf hydraulics to the FF16 growth framework. It is not a parameter reference. The exact current parameter set and its links to the whole-plant implementation still need a separate, version-specific account.
TF24f: acclimation lag instead of instantaneous re-optimisation
TF24 assumes that stomata and hydraulics adjust immediately when conditions change. At every ODE step, the model solves the profit-maximisation problem again and places the plant directly at the new optimum. This quasi-steady-state assumption is useful, but it cannot represent a plant that is still adjusting to a change in its environment.
TF24f introduces that adjustment time. It uses the same physiology as TF24: the same leaf model, xylem vulnerability curve, and soil-water coupling. The implementation subclasses TF24_Strategy, so the carbon-income and water-transport calculations described on Assimilation & hydraulics are inherited unchanged. The only difference is how the plant moves towards the most profitable hydraulic operating point.
In TF24, a golden-section search over root-collar water potential is used at every step to find the \(\psi\) that maximises the profit \(P(\psi_l)\) defined on the assimilation & hydraulics page. TF24f instead adds a sixth ODE state, opt_root_psi_state, which records the plant’s current root-collar water potential. Gradient ascent then moves this state towards the profit-maximising value rather than allowing it to jump there immediately:
\[ \frac{d\psi}{dt} = k_{\text{acclim}} \cdot \frac{dP}{d\psi} \]
Here, \(k_{\text{acclim}}\) controls how quickly the operating point changes, and \(dP/d\psi\) indicates which direction increases profit. At each leaf-solve step, TF24f evaluates profit at the current tracked value of \(\psi\). It then passes the local gradient \(dP/d\psi\) to the ODE system, which advances the state through time.
At establishment, the tracked state begins at the true optimum. There is therefore no artificial adjustment period at birth. A lag appears only after the environment changes and shifts the optimum away from the plant’s current operating point.
The use_ad_gradient setting selects one of two ways to calculate the gradient:
use_ad_gradient = true(the default) uses the exact gradientLeaf::dprofit_droot_collar_psi. It combines forward-mode automatic differentiation of the photosynthesis and cost calculations, the implicit-function theorem at the \(\psi_l \to C_i\) root-find, and analytic spline derivatives for the transport terms.use_ad_gradient = falseuses a finite-difference approximation. It steps \(\psi\) bypsi_fd_step(MPa) and compares the profit on either side.
The parameter k_acclim controls the acclimation rate. Changing it produces a range of behaviours rather than one fixed alternative to TF24:
Small k_acclim |
Large k_acclim |
|
|---|---|---|
| Behaviour | slow, lagged tracking of the optimum | tight tracking — approaches TF24’s instantaneous optimum |
| Interpretation | a real acclimation delay in stomatal/hydraulic adjustment | quasi-steady-state recovered as a limiting case |
| Calculation at each evaluation | one gradient evaluation | one gradient evaluation; a stiff state may require more ODE steps |
| Risk | plant can be caught away from the optimum by fast environmental change | stiff ODE state; may demand a smaller integration step |
TF24 vs TF24f
| TF24 | TF24f | |
|---|---|---|
| Operating-point solve | golden-section search for the profit optimum, every step | gradient-ascent relaxation toward the optimum, tracked as an ODE state |
| Assumption | instantaneous (quasi-steady-state) acclimation | finite acclimation lag, set by k_acclim |
| Extra state | — | opt_root_psi_state (6th ODE state) |
| Gradient source | not needed (search only evaluates profit) | Leaf::dprofit_droot_collar_psi (AD) or finite difference (psi_fd_step) |
| Physiology, hydraulics, soil coupling | shared | shared (inherited unchanged from TF24) |
| Bit-for-bit match to TF24 | — | no — by design, it tracks rather than reproduces the optimum |
Use TF24f when the timing of acclimation matters—for example, when asking how quickly stomatal or hydraulic behaviour can respond to a changing environment. It can also be useful when TF24’s repeated optimisation becomes a computational bottleneck across many individuals or time steps, provided that a large k_acclim is an acceptable approximation to instantaneous adjustment.
Use TF24 when the quasi-steady-state assumption is appropriate and you do not want to introduce an extra state or choose an acclimation-rate parameter. TF24 is also the right choice when results must match TF24 exactly: TF24f approaches its behaviour as \(k_{\text{acclim}} \to \infty\), but does not reproduce it bit for bit.