Models and parameters

Every built-in model, with its inputs, parameters and default bounds. This page is generated from pk describe by docs/gen_models.py; pk describe <model> prints the same information as JSON.

Fixed by default parameters are held at their default unless you free them with --fit (or fit_params in Python). Bounds are hard limits the fitter never crosses.

Model	Tool	Name in the tool
empiricalA	photosynthesis	Empirical
fvcb	photosynthesis	Farquhar–von Caemmerer–Berry (C3)
fvcbC4	photosynthesis	von Caemmerer (C4)
vulnerability	hydraulics	Sigmoidal vulnerability curve
pvcurve	hydraulics	Pressure–volume curve
vanGenuchten	soil	van Genuchten
modifiedVanGenuchten	soil	modified van Genuchten
raats	soil	Raats superclass
brooksCorey	soil	Brooks–Corey
fractalSWRC	soil	fractal (Li et al. 2025)
bta	stomatal	Buckley–Turnbull–Adams 2012
medlyn	stomatal	Medlyn et al. 2011 (optimality)
lightResponse	Python only	—

empiricalA

Empirical — pk fit photosynthesis <file> --model empiricalA

\[ A = A_{max}(T)\,\frac{Q}{Q + K_q(T)}\cdot\frac{C_i - \Gamma^*(T)}{C_i + K_{co}(T)} - R_d(T) \]

Inputs: Ci (intercellular CO₂, µmol mol⁻¹), Q (absorbed light (PPFD), µmol m⁻² s⁻¹), T (leaf temperature, K)

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
Amax25	25	1	200		max assimilation at 25C
Rd25	2.5	0	10		dark respiration at 25C
GammaS25	42.75	10	120		CO2 compensation point at 25C
Kco25	700	50	3000		CO2 half-saturation at 25C
Kq25	50	5	2000		light half-saturation at 25C
Amax_dHa	50	1	200		Amax activation energy
Amax_dHd	200	100	500		Amax deactivation energy
Amax_Topt	308.12	288	330		Amax optimum temp (K)
Rd_dHa	46.39	1	200		Rd activation energy
GammaS_dHa	37.83	1	200
Kq_dHa	70	1	200
Kco_dHa	79.43	1	200
gm	0.25	0.01	99		mesophyll conductance

fvcb

Farquhar–von Caemmerer–Berry (C3) — pk fit photosynthesis <file> --model fvcb

\[ A = \left(1-\tfrac{\Gamma^*}{C_c}\right)\min\{\,w_c,\ w_j,\ w_p\,\} - R_d,\quad C_c = C_i - A/g_m \]

Inputs: Ci (intercellular CO₂, µmol mol⁻¹), Q (absorbed light (PPFD), µmol m⁻² s⁻¹), T (leaf temperature, K)

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
Vcmax25	100	5	400		max Rubisco carboxylation at 25C
Jmax25	200	5	700		max electron transport at 25C
TPU25	25	1	100		triose-phosphate utilization at 25C
Rd25	1.5	0	15		day respiration at 25C
gm	0.25	0.01	0.5		mesophyll conductance (Cc = Ci − A/gm)
alpha	0.5	0.1	0.9		quantum yield of electron transport
theta	0.7	0.1	0.99		curvature of J(Q)
alphaG	0	0	1		glycolate carbon returned (TPU)
Vcmax_dHa	65.33	1	200
Vcmax_dHd	200	100	500
Vcmax_Topt	311.15	288	333
Jmax_dHa	43.9	1	200
Jmax_dHd	200	100	500
Jmax_Topt	311.15	288	333
TPU_dHa	53.1	1	200
TPU_dHd	201.8	100	500
TPU_Topt	306.4	288	333
Rd_dHa	46.39	1	200
GammaS25	42.75	10	120	yes	CO2 compensation point (no Rd) at 25C
GammaS_dHa	37.83	1	200	yes
Kc25	404.9	100	1000	yes
Kc_dHa	79.43	1	200	yes
Ko25	278.4	100	600	yes
Ko_dHa	36.38	1	200	yes
O	213.5	100	300	yes	chloroplastic O2 (mmol/mol)

Soft constraints (penalties added to the fit; turn one off with --no-constraint <key>, all with --constraints none, reweight with --weight <key>=<w>):

Key	Constraint	Default weight	Per curve	What it does
ap_nonneg	Ap ≥ 0	1		TPU-limited rate non-negative at every point
aj_le_ac_mid	Aj ≤ Ac (mid Ci)	0.3	yes	Light-limited ≤ Rubisco-limited at Ci 500–700
rubisco_region	Rubisco region exists	1	yes	A low-Ci Rubisco-limited stretch must be present
ap_above_cross	Ap above Ac/Aj crossing	1	yes	TPU not limiting at the Ac–Aj transition
ap_limits_high	Ap limits at high Ci	1	yes	TPU is the limiter at the highest-Ci point
vcmax_window	Vcmax25 ∈ [20, 130]	0.05		Soft prior keeping Vcmax25 in a typical range (PhoTorch)
alphag_zero	αG → 0	2		Pull the glycolate-return fraction toward 0 (PhoTorch)
vj_ratio	Jmax:Vcmax ≈ 1.6–2.0	0.1		Soft prior on the Jmax25:Vcmax25 ratio (single-curve analog of PhoTorch's correlation term)

fvcbC4

von Caemmerer (C4) — pk fit photosynthesis <file> --model fvcbC4

\[ A = \min\{A_c,\ A_j\},\quad V_p = \min\!\left(\tfrac{C_m V_{pmax}}{C_m+K_p},\ V_{pr}\right),\quad C_m = C_i - A/g_m \]

Inputs: Ci (intercellular CO₂, µmol mol⁻¹), Q (absorbed light (PPFD), µmol m⁻² s⁻¹), T (leaf temperature, K)

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
Vpmax25	200	1	600		max PEP carboxylation at 25C
Vcmax25	40	1	200		max Rubisco carboxylation at 25C
Jmax25	247.69	5	700		max linear electron transport at 25C
Rd25	1	0	15		day respiration at 25C
Vpmax_dHa	50.1	1	200
Vcmax_dHa	78	1	200
Jmax_dHa	77.9	1	200
Jmax_dHd	260	100	500
Jmax_Topt	316.15	288	333
Rd_dHa	66.4	1	200
gm	1	0.05	5	yes	mesophyll conductance (Cm = Ci − A/gm)
gbs	0.003	0.0001	0.05	yes	bundle-sheath conductance
Vpr	80	20	200	yes	PEP regeneration cap
alpha_psII	0	0	1	yes	fraction of PSII activity in bundle sheath
x_etr	0.4	0.05	0.95	yes	fraction of linear ETR to mesophyll
fcyc	0.45	0	0.85	yes	cyclic electron flow fraction
theta_etr	0.7	0.1	0.99	yes	curvature of J(Q)
Kc25	1210	500	3000	yes	Rubisco Kc at 25C (μbar)
Kc_dHa	64.2	1	200	yes
Ko25	292000	100000	500000	yes	Rubisco Ko at 25C (μbar)
Ko_dHa	10.5	1	200	yes
Kp25	82	20	300	yes	PEPC Kp at 25C (μbar)
Kp_dHa	38.3	1	200	yes
GammaStar25	0.000381679	0.0001	0.001	yes	half Rubisco O2/CO2 specificity (dimensionless)
GammaStar_dHa	31.1	1	200	yes
Om25	210000	100000	300000	yes	bundle-sheath O2 partial pressure (μbar)

Soft constraints (penalties added to the fit; turn one off with --no-constraint <key>, all with --constraints none, reweight with --weight <key>=<w>):

Key	Constraint	Default weight	Per curve	What it does
vpmax_gt_vpr	Vpmax > Vpr	1		PEP kinetic curve must reach its regeneration cap
vp_breakpoint	Vpr breakpoint in Ci range	0.02	yes	Keep the Vpmax→Vpr transition inside the data (Vpmax/Vpr identifiability)
acaj_regime	Ac/Aj regime at high Ci	0.1	yes	High light → Ac binds; low light → Aj binds, at saturating Ci
rd_tether	Rd ≈ 0.02·Vcmax	10		Soft tether of day respiration to Vcmax (poorly identified in C4)
rate_windows	Rate windows	0.1		Soft priors: Vpmax∈[20,400], Vcmax∈[5,200], Jmax∈[50,600]

vulnerability

Sigmoidal vulnerability curve — pk fit hydraulics <file> --model vulnerability

\[ K = \dfrac{K_{max}}{1+\exp\!\left[-\tfrac{S}{25}\,(\Psi - P_{50})\right]} \]

Inputs: P (water potential, MPa (negative))

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
Kmax	1	1e-09	—		maximum hydraulic conductance at full hydration
P50	-2	-15	-0.05		water potential at 50% loss of conductivity (MPa)
S	60	1	400		slope at P50 (% MPa⁻¹)

pvcurve

Pressure–volume curve — pk fit hydraulics <file> --model pvcurve

\[ \Psi = \pi_o\left[\max\!\left(0,\tfrac{w-w_{tlp}}{1-w_{tlp}}\right)\right]^{\varepsilon} - \dfrac{\pi_o}{w} \]

Inputs: w (relative water content, 0–1)

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
πo	2	0.1	5		osmotic pressure at full turgor (MPa)
wtlp	0.85	0.5	0.99		relative water content at turgor loss point
ε	1	1	3		wall-elasticity exponent on turgor

vanGenuchten

van Genuchten — pk fit soil <file> --model vanGenuchten

\[ \theta = \theta_r + (\theta_s-\theta_r)\left[1+(\alpha|\psi|)^{n}\right]^{-m},\; m=1-\tfrac{1}{n} \]

Inputs: psi (matric potential, MPa (negative))

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
θr	0.05	0	0.5		residual volumetric water content
θs	0.45	0.1	1		saturated volumetric water content
α	1	0.0001	100000		inverse air-entry pressure (ψ⁻¹)
n	1.5	1.01	12		pore-size distribution parameter

modifiedVanGenuchten

modified van Genuchten — pk fit soil <file> --model modifiedVanGenuchten

\[ \theta = \theta_r + (\theta_s-\theta_r)\left[1+(\alpha_1|\psi|)^{n_1}\right]^{-(1+1/n_1)} \]

Inputs: psi (matric potential, MPa (negative))

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
θr	0.05	0	0.5		residual volumetric water content
θs	0.45	0.1	1		saturated volumetric water content
α1	1	0.0001	100000		MVG retention scale parameter (ψ⁻¹)
n1	1.5	0.2	12		MVG retention shape parameter
α2	1	0.0001	100000	yes	MVG-RHC conductivity scale parameter (ψ⁻¹)
n2	1.5	0.2	12	yes	MVG-RHC conductivity shape parameter

raats

Raats superclass — pk fit soil <file> --model raats

\[ |\psi| = b^{*}\left[\dfrac{1-S^{\alpha}}{S^{\beta}}\right]^{\gamma},\quad S=\dfrac{\theta-\theta_r}{\theta_s-\theta_r} \]

Inputs: psi (matric potential, MPa (negative))

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
θr	0.05	0	0.5		residual volumetric water content
θs	0.45	0.1	1		saturated volumetric water content
b*	0.05	1e-05	100000		characteristic pressure head (ψ units)
α	5	0.01	10		Raats shape parameter α
β	5	0.1	40		Raats shape parameter β (dry end)
γ	0.8	0.01	0.999		Raats shape parameter γ (wet end, 0–1)
λ	0.5	-5	5	yes	Mualem tortuosity exponent (conductivity)

brooksCorey

Brooks–Corey — pk fit soil <file> --model brooksCorey

\[ S_e = \left(\dfrac{\psi_b}{|\psi|}\right)^{\lambda}\ (|\psi|>\psi_b),\quad \theta=\theta_r+(\theta_s-\theta_r)S_e \]

Inputs: psi (matric potential, MPa (negative))

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
θr	0.05	0	0.5		residual volumetric water content
θs	0.45	0.1	1		saturated volumetric water content
ψb	0.03	1e-05	10000		air-entry (bubbling) potential (ψ units)
λ	0.5	0.05	10		pore-size distribution index
L	0.5	-5	5	yes	Mualem tortuosity exponent (conductivity)

fractalSWRC

fractal (Li et al. 2025) — pk fit soil <file> --model fractalSWRC

\[ s = \dfrac{4\lambda}{\pi}\dfrac{2-D}{1-D}\dfrac{1-\delta}{r_{max}}\dfrac{1-\left(1-\tfrac{1-S_e}{(1-\delta)^2}\right)^{\frac{1-D}{2-D}}}{S_e} \]

Inputs: psi (matric potential, MPa (negative))

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
θr	0.05	0	0.5		residual volumetric water content
θs	0.45	0.1	1		saturated volumetric water content
D	1.5	1.001	1.999		fractal dimension of the pore-size distribution
r_max	5	0.001	10000		maximum equivalent pore radius (µm)
δ	0.02	0	0.3		relative adsorptive water-film thickness
L	0.5	-5	5	yes	Mualem tortuosity exponent (conductivity)

bta

Buckley–Turnbull–Adams 2012 — pk fit stomatal <file> --model bta

\[ g_s = \frac{E_m\,(Q + i_0)}{k + b\,Q + (Q + i_0)\,D_s} \]

Inputs: Q (absorbed light (PPFD), µmol m⁻² s⁻¹), Ds (leaf-to-air vapour pressure deficit as a mole fraction, mmol mol⁻¹)

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
Em	0.2	0.001	20		conductance scale (mol m⁻² s⁻¹)
i0	20	0	500		light compensation offset (µmol m⁻² s⁻¹)
k	100000	1	1e+07		denominator constant
b	100	0	20000		VPD/light sensitivity

medlyn

Medlyn et al. 2011 (optimality) — pk fit stomatal <file> --model medlyn

\[ g_s = g_{s,0} + 1.6\left(1 + \frac{g_1}{\sqrt{D}}\right)\frac{A}{C_s} \]

Inputs: A (net CO₂ assimilation, µmol m⁻² s⁻¹), Cs (CO₂ at the leaf surface, µmol mol⁻¹), D (leaf-to-air vapour pressure deficit, kPa)

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
gs0	0.01	0	1		residual conductance (mol m⁻² s⁻¹)
g1	3	0.01	30		slope, sensitivity to VPD (√kPa)

lightResponse

Available from Python (fit_arrays, fit_many, predict); no pk fit tool uses it.

\[ \frac{ Amax\cdot Q}{\left( Q+ Kq\right)}- Rd \]

Inputs: Q (absorbed light (PPFD), µmol m⁻² s⁻¹)

Parameter	Default	Lower bound	Upper bound	Fixed by default	Description
Amax	25	1	200
Kq	300	5	2000
Rd	1.5	0	10