Temperature-dependent chemical shifts

The .tc model suffix gives chemical shifts a shared linear temperature dependence. It is independent of the kinetic model: 2st.tc, 4st.tc, 2st_eyring.tc, and compositions with .mf or .rs all use the same shift parameterization. Every Experiment in a .tc analysis must specify temperature under [conditions] in degrees Celsius.

Select it through the ordinary model option, for example:

chemex fit -e Experiments/*.toml -p Parameters.toml -m Method.toml -d 2st.tc

.tc changes chemical shifts only; 2st.tc leaves the two-state exchange kinetics and populations on their ordinary model. To combine shift-temperature coefficients with Eyring temperature-dependent kinetics, compose the features explicitly:

chemex fit -e Experiments/*.toml -p Parameters.toml -m Method.toml -d 2st_eyring.tc

Model and units

With dT = T - TREF, ChemEx resolves

$$CS_A(T) = CS0_A + CS1_A\,dT,$$ $$DW_{AX}(T) = DW0_{AX} + DW1_{AX}\,dT,$$ $$CS_X(T) = CS_A(T) + DW_{AX}(T).$$

CS0_A and DW0_AX are in ppm and are the respective values at TREF. CS1_A and DW1_AX are in ppm/°C. A Celsius difference is numerically equal to a kelvin difference, so .tc composes consistently with Eyring kinetic models: Eyring calculations convert absolute temperatures to kelvin, while the shift polynomial uses the Celsius difference.

The coefficient names are order-indexed, but .tc implements only the linear orders 0 and 1.

Parameter	Meaning	Default	Bounds	Default role
TREF	common reference temperature	25 °C	finite and above −273.15 °C	protected fixed constant
CS0_A	state-A shift at TREF	0 ppm	[−100, 300] ppm	Experiment Type authority
CS1_A	state-A linear coefficient	0 ppm/°C	[−1, 1] ppm/°C	Experiment Type authority
DW0_AX	A-to-X shift difference at TREF	0 ppm	[−100, 100] ppm	fitted
DW1_AX	A-to-X linear coefficient	0 ppm/°C	[−1, 1] ppm/°C	fitted

Direct chemical-shift Experiment Types already declare the absolute reference shift fit-capable. Under .tc, that declaration expands to both CS0_A and CS1_A. CEST and CPMG Experiment Types do not start fitting absolute shifts merely because .tc is selected; a Method Plan can explicitly select those coefficients. The shift-difference coefficients keep the established .tc default fitted behavior.

Configuring the reference temperature

An analysis has exactly one unqualified model constant:

[GLOBAL]
TREF = 20.0

TREF is shared across all residues, nuclei, fields, concentrations, Experiment Types, and temperatures in that analysis. It cannot be residue-, nucleus-, or condition-scoped, and it cannot have fitting bounds or a grid step. Method Plan FIT, FIX, CONSTRAIN, GRID, and DE-coordinate operations cannot target it.

The coefficients retain their normal spin/nucleus and state-pair scope but omit temperature and magnetic field. Consequently one coefficient line is shared by all temperatures and fields unless the user applies an explicit supported constraint or sharing rule.

Multistate semantics

ChemEx keeps state A as the chemical-shift reference. A multistate model has one CS0_A/CS1_A pair for state A and a DW0_AX/DW1_AX pair for each non-A state X. State X is always formed as CS_A + DW_AX; there are no independent CS0_X or CS1_X coordinates. This prevents the reference-state contribution from being counted twice.

The temperature-specific CS_A, DW_AX, and CS_X entries remain derived model quantities. Select, initialize, fix, fit, share, or constrain the order-indexed coefficients instead of overriding those derived values.

Reference-temperature invariance

Changing the reference from r to r' preserves the physical line when each coefficient pair is transformed as

$$Q0' = Q0 + Q1(r' - r), \qquad Q1' = Q1,$$

where Q is either CS or DW. ChemEx does not automatically rewrite arbitrary fitted parameter files between reference temperatures. Independent coefficient bounds, priors, grid ranges, and constraints generally do not retain the same meaning under this coordinate transformation. Change TREF only together with an explicit scientific review of those settings.

Fitting limitation

At one temperature, a free order-0 coefficient and its free order-1 coefficient enter observables only through one linear combination and cannot be estimated independently without an additional constraint. Use data at multiple temperatures, or explicitly fix or constrain one coefficient. ChemEx does not silently change the fitting roles for a single-temperature analysis.

Breaking change from the previous .tc interface

The released .tc implementation used an uncentered line, DW_AX(T) = DWP_AX + DWM_AX T, with T in °C. This release intentionally replaces DWP_AX and DWM_AX with DW0_AX and DW1_AX; the old names are not aliases and are not migrated automatically. Existing .tc parameter and Method files must be rewritten in the centered coordinate system:

$$DW0_{AX} = DWP_{AX} + TREF\,DWM_{AX}, \qquad DW1_{AX} = DWM_{AX}.$$

Review bounds and constraints when converting because changing the reference temperature changes their coordinate meaning.

Output, provenance, and restart

New parameter output contains TREF, CS0/CS1, and DW0/DW1 only. TREF is written as a fixed global value whenever its associated shift polynomial is active. Temperature-specific resolved shifts may appear as derived output, but they are not independent restart coordinates. Run provenance archives the input files as supplied; restart from ChemEx's canonical output so the reference and coefficients remain together.