Kinetic Models

The kinetic model (specified with the -d or --model option) defines the type of exchange model to be used for data analysis. Available models include:

Model Name	Description
2st	2-state exchange model (default)
3st	Complete 3-state exchange model
3st_triangle	Historical explicit compatibility name for 3st
3st_linear	Linear A ↔ B ↔ C exchange model
3st_fork	A-centered B ↔ A ↔ C exchange model
4st	Complete 4-state exchange model
4st_linear	Linear 4-state exchange model
4st_fork	A-centered 4-state exchange model
5st	Complete 5-state exchange model
5st_linear	Linear 5-state exchange model
5st_fork	A-centered 5-state exchange model
6st	Complete 6-state exchange model
6st_linear	Linear 6-state exchange model
6st_fork	A-centered 6-state exchange model
2st_hd	2-state exchange model for H/D solvent exchange studies
2st_eyring	2-state exchange model for temperature-dependent studies
3st_eyring	Compatibility name for the linear 3-state Eyring model
3st_eyring_linear	Linear A ↔ B ↔ C Eyring model for temperature-dependent studies
3st_eyring_fork	Fork B ↔ A ↔ C Eyring model for temperature-dependent studies
4st_eyring	4-state exchange model for temperature-dependent studies
2st_binding	2-state exchange model for ligand binding studies
3st_binding_cs	3-state conformational-selection ligand binding model
3st_binding_if	3-state induced-fit ligand binding model
4st_hd	4-state exchange model for simultaneous normal and H/D solvent exchange studies

In these models, each state in the exchange process is represented with a unique parameter suffix (A, B, C, D, etc.). For example, R1_A and R2_B refer to the R₁ rate of state A and the R₂ rate of state B. In many analyses A is assigned to the major or ground state and B to a minor or excited state, but this is an analyst-defined convention rather than an ordering enforced by ChemEx. See Exchange States and Parameters for populations, directional rates, shift signs, and label swapping.

note

For any kinetic model, you can add the .rs suffix to make the kinetic parameters residue-specific (for example, 2st.rs or 3st_eyring.rs). Suffixes can be combined, such as 2st.rs.mf. The legacy 2st_rs name remains supported as an alias for 2st.rs.

note

For any kinetic model, you can add the .mf suffix to create a model that fits model-free parameters directly (e.g., TAUC_A, S2_A), rather than individual relaxation parameters (e.g., R1_A, R2_A). For an example, see CEST_15N_TR/ under Examples/Experiments/.

Generic N-state models

The generic family supports three through six states. Its public model name is the topology contract:

Model	Topology
3st, 4st, 5st, 6st	Complete graph: every unordered state pair exchanges
3st_linear, 4st_linear, 5st_linear, 6st_linear	Chain A ↔ B ↔ C ↔ D ↔ E ↔ F, truncated at the selected state count
3st_fork, 4st_fork, 5st_fork, 6st_fork	A-centered star: A exchanges directly with every other state
3st_triangle	Exact historical compatibility name for the canonical complete 3st model

PB, PC, and any higher non-A populations are independent fractions; PA is derived so that all populations sum to one. ChemEx enforces the complete closed simplex: every population must be finite and nonnegative and their sum must be exactly representable as no greater than one. Exact zero populations and PA = 0 are valid. Values are not clipped, renormalized, or replaced by a small positive floor.

For every structural edge i ↔ j, KEX_ij is the total exchange scale in s⁻¹:

$$KEX_{ij} = K_{ij} + K_{ji}.$$

ChemEx derives the directional rates from the prescribed populations:

$$K_{ij} = KEX_{ij}\frac{P_j}{P_i + P_j}, \qquad K_{ji} = KEX_{ij}\frac{P_i}{P_i + P_j}.$$

The first state in a directional name is its source, so KAB means A → B. KEX_ij = 0 produces exact zero in both directions and dynamically switches off an existing structural edge. If exactly one endpoint population is zero, the outgoing rate from that endpoint equals KEX_ij and the reciprocal rate is zero. If both endpoint populations are zero, KEX_ij = 0 remains valid, but a positive KEX_ij is rejected because its directional split is undefined. Positive rates too small to be represented as positive binary64 values are also rejected instead of being changed into structural zero.

An absent edge in a _linear or _fork model is different from an edge fixed to zero in a complete model: the absent edge has no KEX or directional-rate parameters and cannot be selected by a Method Step. In a complete model, setting selected structural KEX values to zero can create disconnected kinetic components. Positive prescribed populations may remain in multiple components; ChemEx treats them as static mixture weights for non-interconverting subensembles and does not require an irreducible generator or a unique global stationary distribution.

Direct TRF and its Monte Carlo/bootstrap reruns use closed, bounded simplex coordinates internally while continuation and output retain public PB, PC, and higher-state coordinates. MCMC remains in those public coordinates and rejects proposals outside the simplex through its normal zero-density path. Qualified interior deterministic covariance is propagated analytically to PA and the directional rates. At a population boundary, symmetric errors are reported only with ChemEx's boundary warning; the positive-KEX zero/zero split has no valid central rate or derivative.

Breaking default change for complete 4/5/6-state models

Bare 4st, 5st, and 6st are now genuinely complete by default. Every structural KEX, including KEX_AD and KEX_BD, starts at 200 s⁻¹, is bounded to [0, 1e6] s⁻¹, and is fitted by default. Earlier releases inherited historical KEX_AD = KEX_BD = 0 fixed defaults. This is an intentional breaking change.

To reproduce the old sparse behavior for any of 4st, 5st, or 6st, use a normal parameter file:

legacy-complete-parameters.toml

[GLOBAL]
KEX_AD = 0.0
KEX_BD = 0.0

and explicitly preserve the roles in a version 2 Method Plan:

legacy-complete-method.toml

FORMAT_VERSION = 2

[FIT]
ROLES = [{ FIX = ["KEX_AD", "KEX_BD"] }]

Select the required state count normally:

chemex fit ... -d 4st -p legacy-complete-parameters.toml -m legacy-complete-method.toml
chemex fit ... -d 5st -p legacy-complete-parameters.toml -m legacy-complete-method.toml
chemex fit ... -d 6st -p legacy-complete-parameters.toml -m legacy-complete-method.toml

Old generated run_info/restart.toml files preserve the numerical zero values and their bounds, but do not preserve the historical fitted/fixed roles. When such a restart is loaded under the new complete model, KEX_AD and KEX_BD therefore become ordinary fitted coordinates unless the Method Step above fixes them. Plain historical parameter files that omitted the two values carry no reliable version or intent signal; ChemEx applies the new complete defaults rather than guessing their age.

Three-State Association Models

The 3st_binding_cs and 3st_binding_if models separate equilibrium composition from tagged-magnetization exchange speed. In both models, KD_APP and the equilibrium ratio determine PA, PB, and PC; changing a KEX or KOFF value does not change those populations.

All independent association parameters are scoped by temperature. KD_APP is in M, KOFF_* and KEX_* are in s⁻¹, and KEQ_* is dimensionless. The upper bounds shown below are broad safety defaults rather than scientific priors; parameter files can override them using the normal explicit-bound syntax.

Conformational selection: 3st_binding_cs

The reaction scheme is

A ⇌ B
    B + L ⇌ C

where A is apo conformer 1, B is the apo binding-competent conformer, and C is the bound complex. The independent parameters are:

Parameter	Meaning	Default	Default bounds
KD_APP	apparent dissociation constant against total unbound protein	1e-6 M	(0, 1.0] M
KOFF_BC	tagged B←C dissociation-rate scale	100.0 s⁻¹	[0, 1e6] s⁻¹
KEQ_AB	apo equilibrium ratio B / A	1.0	(0, 1e6]
KEX_AB	total tagged A↔B exchange scale	200.0 s⁻¹	[0, 1e6] s⁻¹

Writing q = KEQ_AB and U = A + B, the apo distribution is A = U / (1 + q) and B = U q / (1 + q). The apparent equilibrium definition is KD_APP = U L / C, so the intrinsic binding constant is KD_BC = KD_APP q / (1 + q). Directional conformational rates are KAB = KEX_AB q / (1 + q) and KBA = KEX_AB / (1 + q). The tagged binding edge uses KCB = KOFF_BC and detailed balance, PB KBC = PC KCB.

KEX_AB = 0 sets both A↔B tagged rates to zero without changing the apo equilibrium. KOFF_BC = 0 freezes both tagged binding directions without changing composition. L_TOTAL = 0 retains the A/B apo distribution and makes the tagged association rate zero. KD_APP = 0, KEQ_AB = 0, and P_TOTAL = 0 are rejected; exact-zero KEQ_AB is incompatible with this finite reversible apparent-binding parameterization.

KAB, KBA, KD_BC, KON_BC, C_L, KBC, KCB, PA, PB, and PC remain public derived outputs. KD_BC and KON_BC are report-only, so an unrepresentable intrinsic value cannot block otherwise finite tagged dynamics. When representable, these report-only values are included in normal parameter output with propagated deterministic uncertainty when its registered derivative qualifies; otherwise the output gives the explicit uncertainty-unavailable reason.

Induced fit: 3st_binding_if

The reaction scheme is

A + L ⇌ B ⇌ C

where A is free protein, B is the first bound complex, and C is the rearranged bound complex. The independent parameters are:

Parameter	Meaning	Default	Default bounds
KD_APP	apparent dissociation constant against total bound protein	1e-3 M	(0, 1.0] M
KOFF_AB	tagged A←B dissociation-rate scale	100.0 s⁻¹	[0, 1e6] s⁻¹
KEQ_BC	bound-state equilibrium ratio C / B	1.0	[0, 1e6]
KEX_BC	total tagged B↔C exchange scale	200.0 s⁻¹	[0, 1e6] s⁻¹

Writing q = KEQ_BC, the apparent equilibrium definition is KD_APP = A L / (B + C), with B:C = 1:q; therefore KD_AB = KD_APP (1 + q). Directional conformational rates are KBC = KEX_BC q / (1 + q) and KCB = KEX_BC / (1 + q). The tagged binding edge uses KBA = KOFF_AB and detailed balance, PA KAB = PB KBA.

KEX_BC = 0 and KOFF_AB = 0 freeze their respective tagged edges without changing composition. KEQ_BC = 0 is supported exactly: C has zero equilibrium weight, KBC = 0, KCB = KEX_BC, and KD_AB = KD_APP. L_TOTAL = 0 puts all protein in free A and makes the tagged association rate zero. KD_APP = 0 and P_TOTAL = 0 are rejected.

KBC, KCB, KD_AB, KON_AB, C_L, KAB, KBA, PA, PB, and PC remain public derived outputs. KD_AB and KON_AB are report-only, with the same representability and uncertainty-reporting policy as KD_BC and KON_BC above.

Migrating directional-rate inputs

Directional rates remain available as outputs but can no longer be supplied as independent parameter values, Method Plan roles, constraints, or grid targets. For 3st_binding_cs, replace a positive legacy pair using KEQ_AB = KAB / KBA and KEX_AB = KAB + KBA. For 3st_binding_if, use KEQ_BC = KBC / KCB and KEX_BC = KBC + KCB.

A legacy (0, 0) pair determines only KEX = 0; its equilibrium ratio is ambiguous and must be chosen explicitly. For induced fit, KBC = 0 with KCB > 0 maps exactly to KEQ_BC = 0 and KEX_BC = KCB. A positive forward rate with a zero reverse rate would require an infinite equilibrium ratio and cannot be migrated to the finite parameter domain. Conformational selection also rejects a legacy KAB = 0 mapping because KEQ_AB must be positive.

The migration diagnostic prints a numerical replacement only when the exact positive ratio and sum are representable in binary64. If either result is outside that domain, it asks for manual model/parameter reconsideration instead of printing a false zero or infinity. A representable replacement can still exceed the new broad default upper bound of 1e6. In that case keep the exact migrated value and override the bound explicitly with the normal three-value parameter syntax, for example KEX_BC = [1200000.0, 0.0, 1200000.0]; for positive CS KEQ_AB, use a positive lower bound such as 5e-324.

Temperature-Dependent Eyring Models

The 2st_eyring, three-state Eyring variants, and 4st_eyring models implement exchange systems with temperature-dependent rate constants calculated using Eyring transition state theory. These models are particularly useful for studying exchange processes where thermodynamic parameters govern the temperature dependence of exchange rates.

Thermodynamic convention and temperature

ChemEx input temperatures are in degrees Celsius. Eyring calculations convert them to Kelvin internally with T = temperature + 273.15. The public domain is every finite temperature strictly above -273.15 °C; absolute zero, lower temperatures, NaN, and infinities are rejected. There is no artificial upper temperature limit.

State A is the thermodynamic reference, so H_A = S_A = 0. Every DH_I and DS_I parameter is the coordinate of state I relative to A. Likewise, DH_IJ and DS_IJ are the shared IJ transition-state coordinates relative to A, not separate forward activation parameters. Enthalpies use J mol⁻¹ and entropies use J mol⁻¹ K⁻¹.

For the directional transition i → j, ChemEx therefore uses

Delta H double dagger (i -> j) = H_TS_ij - H_i
Delta S double dagger (i -> j) = S_TS_ij - S_i

log(k_ij) = log(k_B / h) + log(T)
            + (S_TS_ij - S_i) / R
            - (H_TS_ij - H_i) / (R * T)

Here T is in Kelvin, k_ij is in s⁻¹, and the transmission coefficient is one. A shared transition state makes the forward/reverse pair obey

log(k_ij / k_ji) = -((H_j - H_i) - T * (S_j - S_i)) / (R * T)

The equilibrium populations are calculated directly from state coordinates, not from rate magnitudes:

log(w_i) = S_i / R - H_i / (R * T)
p_i = w_i / sum(w)

The topology still selects the kinetic edges. This population authority ensures detailed balance for every present edge and thermodynamic cycle consistency in 4st_eyring, even when all transition states are shifted to make kinetics much slower.

Rates are evaluated in the log domain and are not clipped or saturated. If a mathematically positive rate lies below the minimum positive binary64 value or above the maximum finite binary64 value, parameter evaluation fails explicitly instead of producing zero, infinity, or a capped rate.

Finite Eyring state coordinates likewise imply strictly positive populations. If a normalized state population lies below the minimum positive binary64 value, ChemEx rejects the evaluation instead of converting that state to structural zero.

2st_eyring Model Parameters

The 2st_eyring model uses the following thermodynamic parameters:

State Energies (relative to state A):

• DH_B: Enthalpy difference (J/mol) for state B relative to A
• DS_B: Entropy difference (J/mol/K) for state B relative to A

Transition-state coordinates (relative to state A):

• DH_AB: Enthalpy coordinate (J/mol) of the shared AB transition state
• DS_AB: Entropy coordinate (J/mol/K) of the shared AB transition state

The model automatically calculates both forward (k_AB) and reverse (k_BA) rate constants from these parameters.

Three-State Eyring Model Parameters

3st_eyring_linear has the linear topology A ↔ B ↔ C. 3st_eyring_fork has the fork topology B ↔ A ↔ C. The historical 3st_eyring name remains supported as a compatibility name for 3st_eyring_linear; it does not select a triangular topology.

Both topologies use the following state parameters:

State Energies (relative to state A):

• DH_B, DH_C: Enthalpy differences (J/mol) for states B, C
• DS_B, DS_C: Entropy differences (J/mol/K) for states B, C

Linear transition-state coordinates (3st_eyring, 3st_eyring_linear):

• DH_AB, DH_BC: Enthalpy coordinates (J/mol) of the AB and BC transition states
• DS_AB, DS_BC: Entropy coordinates (J/mol/K) of the AB and BC transition states

The linear models calculate k_AB, k_BA, k_BC, and k_CB. There is no direct A ↔ C pathway.

Fork transition-state coordinates (3st_eyring_fork):

• DH_AB, DH_AC: Enthalpy coordinates (J/mol) of the AB and AC transition states
• DS_AB, DS_AC: Entropy coordinates (J/mol/K) of the AB and AC transition states

The fork model calculates k_AB, k_BA, k_AC, and k_CA. There is no direct B ↔ C pathway.

Parameter files used with 3st_eyring or 3st_eyring_linear should not contain DH_AC or DS_AC; those parameters are not part of the linear topology.

4st_eyring Model Parameters

The 4st_eyring model implements a full 4-state system:

State Energies (relative to state A):

• DH_B, DH_C, DH_D: Enthalpy differences (J/mol) for states B, C, D
• DS_B, DS_C, DS_D: Entropy differences (J/mol/K) for states B, C, D

Transition-state coordinates (relative to state A):

• DH_AB, DH_AC, DH_AD: Enthalpy coordinates (J/mol) of the AB, AC, and AD transition states
• DH_BC, DH_BD, DH_CD: Enthalpy coordinates (J/mol) of the BC, BD, and CD transition states
• DS_AB, DS_AC, DS_AD: Entropy coordinates (J/mol/K) of the AB, AC, and AD transition states
• DS_BC, DS_BD, DS_CD: Entropy coordinates (J/mol/K) of the BC, BD, and CD transition states

The model automatically calculates all 12 rate constants (k_AB, k_BA, k_AC, k_CA, k_AD, k_DA, k_BC, k_CB, k_BD, k_DB, k_CD, k_DC).

All Eyring state and transition enthalpy parameters have default bounds of [-2×10⁵, 2×10⁵] J/mol. The corresponding entropy parameters have default bounds of [-5×10², 5×10²] J/mol/K. Explicit bounds in parameter files override these defaults.

Identifiability, uncertainty, and parameter scope

At one temperature, varying both H and S for the same state or transition state is non-identifiable: the data determine the corresponding free-energy combination, not H and S separately. Different H/S pairs can therefore give the same rate or population, and deterministic uncertainty can correctly be unavailable because the covariance is rank deficient. Measurements at multiple temperatures can restore mathematical rank, but H/S correlation remains strong over a narrow temperature interval. Use a meaningful temperature span when H and S are both fitted.

When the covariance is qualified, ChemEx propagates deterministic uncertainty to derived directional rates and to the coupled, normalized populations. Eyring models use the generic MCMC and resampling workflows; each proposal or replicate recalculates rates and populations through the same thermodynamic authority.

The current ChemEx scoping contract shares Eyring H/S parameters across temperature, magnetic field, and spin system, but splits them when p_total or l_total changes. This concentration scoping is a compatibility contract, not a universal thermodynamic requirement. Derived rates and populations are scoped by temperature, p_total, and l_total; changing magnetic field alone does not create another chemical kinetic rate. Public parameter names, units, defaults, and TOML syntax are unchanged.