Improvements for Dissociation Curve Error Propagation

Separate Prompt to Validate Previous Work:

Your derivation is on target—the algebra matches Tong Ye Wang’s logic and the signs in the partials check out. Below are the main points I would tighten or add, plus a couple of practical nuances Tong hints at but doesn’t spell out.

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1 · Algebraic details worth tweaking

Spot	Comment
Relative‑error prefactor	In Eq (2) the factor in front of the brackets should be $1/|\ln [(R_t-R_{\infty })/R_a]|$. Because you already substituted $k_d=-\ln [(R_t-R_{\infty })/R_a]/t$, no extra $t$ remains.
Variance path (Eq 3)	Good. If you later want covariances between $R_a$ and $R_{\infty }$ (they anti‑correlate in a single‑exponential fit) insert cross‑terms $2\mathrm{cov}(R_a,R_{\infty })/(t^2{R}_a(R_t-R_{\infty }))$. Tong’s Eq 14 keeps them when he fits all variables simultaneously.
Timing jitter	With modern SPR clocks ${\sigma }_t/t<10^{-4}$; you can drop the last term in Eq (4) unless you sample at sub‑ms resolution for very fast off‑rates.

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2 · How Tong’s two paths map onto a dissociation trace

Tong’s symbol	Your analogue	Physical meaning in dissociation kinetics
$a$ (correlated gain)	$c=1/|\ln [(R_t-R_{\infty })/R_a]|$	Time‑point‑dependent amplification of any offset in amplitude or baseline.
$\alpha$ (uncorrelated noise in R)	$({\sigma }_{R_t}^2+{\sigma }_{R_{\infty }}^2)/(R_t-R_{\infty }{)}^2$	Shot noise & drift in the detector at the chosen $t$.
$\lambda$ (uncorrelated noise in amplitude)	${\sigma }_{R_a}^2/R_a^2$	Fit uncertainty of the starting signal.
$\beta$ (timing noise)	${\sigma }_t^2/t^2$	ADC sampling jitter or flow synchronisation.

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3 · Practical tips Tong omits but matter for SPR/BLI dissociation

1. Pick two evaluation times, not one.
   Early (e.g. 0.5 s) highlights $R_a$; late (e.g. 5 × t½) stresses $R_{\infty }$. Quoting both lends intuition—Tong does this implicitly by plotting the biphasic curve in his Fig 3.

2. Exploit the fit covariance matrix.
   Most software returns the full Hessian: extract ${\sigma }_{R_a},{\sigma }_{R_{\infty }},\mathrm{cov}(R_a,R_{\infty })$ directly rather than estimating them from replicate runs. Plug those numbers into the variance path (with the covariance term) for an in‑run precision estimate rather than a run‑to‑run one.

3. Guard against hidden correlation between $t$ and $R_t$.
   In pixel‑clocked imagers or in parallel‑channel SPR, time stamping and signal quantisation share hardware. A global gain drift shows up in both $R_t$ and the stopwatch, breaking the “fully uncorrelated” assumption. If you suspect that, bias‑correct first (e.g. normalise with a reference channel) and then apply Eq (4).

4. Report accuracy + precision side‑by‑side.
   Tong’s main point is that a single “± σ” is incomplete. The tidy way is

   $$k_d=(3.2{\text{s}}^{-1})\pm 0.11{\text{s}}^{-1}(\text{precision}){}_{-0.09}^{+0.25}{\text{s}}^{-1}(\text{accuracy})$$

   which directly mirrors his Table 1 layout. Downstream modelers can then propagate whichever limit matters for their simulation.

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4 · Next analytical step if you need a time‑averaged $k_d$

If you plan to fit the whole dissociation segment rather than a single point, integrate your error expressions over the sampled times $t_i$ (weights $w_i$). The correlated path becomes

$$\Delta k_d^{\text{(fit)}}=\sum _i{w}_i\Delta k_d(t_i),$$

and the uncorrelated path becomes the usual weighted RSS. Doing so typically cuts the random component (${\sigma }_{k_d}$) by $\sqrt{N_{\text{points}}}$, but it hardly changes the systematic term unless the baseline bias drifts with time.

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5 · Literature cross‑check

The prefactor $c$ and RSS structure above reproduce Tong’s Eqs 12 & 14 (for $K_d$) if you swap his $R$, $T_0$, $L_0$ with your $R_t$, $R_a$, $R_{\infty }$ and note that his “fraction $R$” plays the same logarithmic role as $\ln [(R_t-R_{\infty })/R_a]$ in a first‑order decay. (ChemRxiv)

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Bottom line: Your framework is solid; just (i) insert the covariance term if you use simultaneous fitting, (ii) keep the $1/\ln []$ prefactor outside the brackets, and (iii) label the two results explicitly as accuracy vs precision in Tong’s style. That will give you a fully comparable dissociation‑rate error budget.