Understand the relationship
The reasoning behind the result
Three rate constants play different roles
dθ/dt=konLfree(1−θ)−koffθ
kon is a second-order association constant; multiplying it by free ligand in mol/L gives an association term in s⁻¹. koff is a first-order dissociation constant. At fixed free ligand, their sum kobs determines the speed of relaxation toward equilibrium occupancy θeq=konLfree/(konLfree+koff).
The initial occupancy need not be zero. The exponential solution moves from θ0 toward θeq; it may rise or fall depending on the starting state. A dissociation phase without further association instead approaches zero at koff. This assumes an independently justified one-site mechanism and adequate mixing rather than a transport-limited sensor response.
Concentration dependence separates association from dissociation
kobs versus Lfree: slope=kon; intercept=koff
A linear fit across distinct free-ligand concentrations can estimate both parameters when each observed rate was obtained under the same one-site conditions. The implementation performs unweighted least squares, then converts the fitted slope and intercept to molar and second units.
A positive slope and positive intercept are required for the stated reversible model. Replicates remain individual rows. No parameter uncertainty, automatic outlier removal or mechanism acceptance threshold is inferred. A single observed rate at one ligand concentration is insufficient to identify both constants without other evidence.
A baseline-subtracted decay can be inspected on a log scale
koff=−slope of ln(y−baseline) versus time
The dissociation mode holds a visitor-entered baseline fixed, selects the inclusive time window and fits the logarithm of positive residual signal. Its intercept defines the fitted exponential amplitude at the entered phase time origin. The full raw-signal ledger shows fitted values and original residuals.
This is a log-linear error model, not unweighted nonlinear least squares in raw signal. Near-baseline noise is magnified by the logarithm; a zero or negative baseline-subtracted included measurement is rejected rather than clipped. Association kon and equilibrium Kd are not inferred from a dissociation-only trace.
Half-life and mean lifetime are different summaries
Dissociation half-life=ln(2)/koff; mean bound lifetime=1/koff
For the single exponential with no rebinding, half-life is the time for half the initially bound population to remain. Mean bound lifetime is the expected residence duration under that same exponential model. The approach-to-equilibrium half-time during association is instead ln(2)/kobs.
These laboratory model times are not durations of a clinical effect or evidence of in-vivo exposure. Rebinding, changing ligand concentration, transport and multistep binding can change observed behavior.
Follow the numbers
Relaxation at a half-occupancy ligand concentration
- Enter kon=1,000,000 L/(mol·s), koff=0.01 s⁻¹ and free ligand=10 nM=10⁻⁸ mol/L.
- kobs=1,000,000×10⁻⁸+0.01=0.02 s⁻¹ and equilibrium occupancy=0.01/0.02=0.5.
- Starting from zero, occupancy at 100 s is 0.5(1−e⁻²)=43.2332%. The approach half-time is ln(2)/0.02=34.6574 s; the dissociation half-life is ln(2)/0.01=69.3147 s.
Equilibrium level and speed of approach are separate outputs, even at the same free-ligand concentration.
Quick guide
How to use this calculator
- Choose an entered-constant scenario, an observed-rate dataset or a dissociation trace.
- Keep time units explicit. kon is entered in L/(mol·s), while observed-rate data use inverse units of the selected recorded time.
- Association assumes fixed free ligand. Dissociation assumes that new binding and rebinding are excluded.
- Inspect the predicted time ledger or measured fit residuals. A fitted relaxation rate alone does not identify both kon and koff at one concentration.
Calculation method
Calculation and interpretation
Separate a second-order association constant, a first-order dissociation constant and a ligand-dependent observed relaxation rate.
kobs=konLfree+koff; θ(t)=θeq+(θ0−θeq)e^(−kobst); without rebinding θ(t)=θ0e^(−kofft); ln(signal−baseline)=ln(amplitude)−kofft.
Worked example
Relaxation at a half-occupancy ligand concentration
Equilibrium level and speed of approach are separate outputs, even at the same free-ligand concentration.
kobs=konLfree+koff; θ(t)=θeq+(θ0−θeq)e^(−kobst); without rebinding θ(t)=θ0e^(−kofft); ln(signal−baseline)=ln(amplitude)−kofft.
Supported inputs
Precision and limits
A one-to-one experimental model
Assumes one class of independent equivalent sites, reversible one-to-one binding and stated assay conditions. It does not establish affinity in an organism, immune protection, antibody performance, clinical effect or a treatment dose.
Specific binding and free concentration
Multivalent avidity, cooperativity, nonspecific binding, multiple site classes, transport limitation and rebinding require other models. An antigen–antibody system can use this arithmetic only when the stated one-to-one independent-site approximation is justified.
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