Exchange states and parameters

ChemEx uses letters such as A, B, C, and D as stable identifiers for exchange states. The letters do not rank states by population or energy. In many analyses, an analyst assigns A to the major or ground state and B to a minor or excited state, but ChemEx does not enforce that convention or reorder labels after a fit.

This page defines the common parameter conventions. Individual kinetic models can add constraints or use a different parameterization; see Kinetic Models for the available model families.

State labels are identifiers

Selecting the 2st model creates states A and B in that order. A three-state model similarly uses A, B, and C. These labels remain attached to the same populations, rates, shifts, relaxation rates, and magnetization components throughout an analysis, regardless of their fitted populations.

Consequently, a result with $P_B > P_A$ is allowed. It means that the state labeled B has the larger fitted population; it does not trigger a relabeling. Assign physical conformations to the labels deliberately and keep the assignment consistent across input files and experiments.

Populations in the two-state model

PB is the equilibrium population fraction of state B. For 2st, ChemEx fits PB by default and derives PA from normalization:

$$P_A = 1 - P_B.$$

Both populations are fractions. The default bounds on PB are 0 to 1, but nothing imposes $P_A > P_B$.

Two-state exchange rates

KEX_AB is the total A↔B exchange rate, in s⁻¹:

$$KEX_{AB} = K_{AB} + K_{BA}.$$

The directional names describe the initial and final labels:

• KAB means A → B;
• KBA means B → A.

For the 2st parameterization, ChemEx derives them as

$$K_{AB} = KEX_{AB} P_B$$

and

$$K_{BA} = KEX_{AB} P_A.$$

Thus KEX_AB and PB are optimizer-controlled by default, whereas KAB, KBA, and PA are derived. A Method file can change whether eligible independent parameters are fitted or fixed, but it cannot turn a derived relationship into an unrelated independent parameter.

Multi-state models preserve the same directional naming rule and stable state labels. Their allowed pathways and population/rate parameterizations vary by model, however, so do not extrapolate the two-state formulas to every multi-state topology.

Chemical-shift differences

At the user-facing parameter level, chemical shifts and their differences are in ppm. ChemEx defines

$$DW_{AB} = CS_B - CS_A.$$

Where the common shift construction applies, ChemEx therefore derives

$$CS_B = CS_A + DW_{AB}.$$

DW_AB is generally specific to a spin system or residue: different nuclei can report different shift changes for the same shared exchange process. Internal frequency conversions used by a pulse-sequence calculation do not change the user-facing ppm convention.

State-specific parameters

Names such as R1_A, R2_A, R1_B, and R2_B attach a quantity to a state. The suffix identifies the state; it does not say how the quantity is obtained. Depending on the experiment, kinetic model, and Method-file roles, a state-specific quantity may be fitted, fixed from an input value, or derived from another parameter or expression.

For example, many common parameterizations initially make state-B relaxation rates equal to the corresponding state-A rates. Other experiments or Method roles can require separate state-specific values. Do not infer that every state-specific parameter is independently fitted merely because it appears in an output file.

What changes when A and B are swapped

Relabeling a two-state description is a coordinated transformation, not a population sort. If old A becomes new B and old B becomes new A, then:

• the new PB is the old $P_A = 1 - P_B$;
• the new KAB represents the old B → A rate, and the new KBA represents the old A → B rate;
• the new CS_A is the old CS_B;
• DW_AB reverses sign.

All other state-specific quantities must be exchanged consistently. This simple two-state relabeling example does not imply that an arbitrary relabeling leaves every multi-state model or topology unchanged.

Practical interpretation checklist

Before interpreting fitted state parameters, check that:

• the physical assignment of each letter is documented outside ChemEx;
• all experiment and parameter files use that assignment consistently;
• directional rates are read from the first state to the second;
• DW_AB is interpreted as B minus A;
• fitted, fixed, and derived parameter roles are distinguished;
• no conclusion depends only on assuming that A must be the most populated state.

For the next part of the fitting convention, see Scaling, Uncertainties, and Residuals.