Error Propagation of Dissociation Curve

Dissociation Curve Equation

$$R_t=R_a{e}^{-k_{d}t}+R_i$$

Where $R_t$ is the signal produced for any time $t$ Where $R_a$ is the signal at the start of the dissociation phase Where $R_i$ is the baseline as time $t$ turns to infinity Where $k_d$ is the dissociation rate constant

Dissociation Curve Equation isolated for $k_d$

$$k_d=-\frac{1}{t}\ln \left(\frac{R_t-R_i}{R_a}\right)$$

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Partial derivatives

Variable held	Expression
(t)	$\frac{\partial k_d}{\partial t}=\frac{\ln \left(\frac{R_t-R_i}{R_a}\right)}{t^2}=\frac{-k_d}{t}$
(R_t)	$\frac{\partial k_d}{\partial R_t}=-\frac{1}{t(R_t-R_i)}$
(R_i)	$\frac{\partial k_d}{\partial R_i}=\frac{1}{t(R_t-R_i)}$
(R_a)	$\frac{\partial k_d}{\partial R_a}=\frac{1}{t{R}_a}$

Domain requirement: $(R_t>R_i>0)$ and $(R_a>0)$ to keep the logarithm defined.

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Mapping Tong’s two “error‑propagation paths” onto the dissociation‑rate constant $k_d$

Tong Ye Wang splits accuracy analysis of $K_d$ into (i) fully correlated systematic biases and (ii) fully uncorrelated (independent) uncertainties, letting each path produce a compact formula that shows which experimental terms dominate.
Below the same logic is applied to

$$k_d=-\frac{1}{t}\ln [(R_t-R_{\infty })/R_a]$$

with the four underlying variables

symbol	nature	how obtained
$R_t$	instantaneous response	experimental data point
$R_{\infty }$	long‑time baseline	nonlinear fit
$R_a$	initial amplitude	nonlinear fit
$t$	elapsed dissociation time	instrument clock

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1 · Fully correlated (strongly coupled) systematic biases

Exactly as Tong’s eq 11 → 12 step, set all biases to the same‑sign source and sum differentials:

$$\Delta k_d=\frac{\partial k_d}{\partial R_t}\Delta R_t+\frac{\partial k_d}{\partial R_{\infty }}\Delta R_{\infty }+\frac{\partial k_d}{\partial R_a}\Delta R_a+\frac{\partial k_d}{\partial t}\Delta t$$

Combining like terms:

$$\Delta k_d=\frac{\Delta R_{\infty }-\Delta R_t}{t(R_t-R_{\infty })}+\frac{\Delta R_a}{t{R}_a}-\frac{k_d}{t}\Delta t\text{\hspace{1em}(1)}$$

Divide by the fitted value to expose the relative accuracy term:

$$\left|\frac{\Delta k_d}{k_d}\right|=\underset{\text{constant }c}{\underbrace{\frac{1}{|\ln [(R_t-R_{\infty })/R_a]|}}}\left[\frac{|\Delta R_{\infty }-\Delta R_t|}{|R_t-R_{\infty }|}+\frac{|\Delta R_a|}{|R_a|}\right]+\frac{|\Delta t|}{t}\text{\hspace{1em}(2)}$$

The prefactor $c$ (a pure number set by your chosen time point) plays the same role as Tong’s “$a$” in his correlated expression; it amplifies or damps how baseline/amplitude bias reaches $k_d$.

Practical reading

• Late times (small $R_t-R_{\infty }$): the first bracket explodes—baseline mis‑estimation dominates.
• Early times: amplitude bias rules.
• Timing bias matters only for very fast dissociation (large $k_d$).

Use worst‑case estimates for each $\Delta$ to quote an accuracy window:

$$k_d^{\text{fit}}\pm |\Delta k_d{|}_{\text{max}}$$

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2 · Fully uncorrelated (independent) uncertainties

Mirroring Tong’s switch to a root‑sum‑of‑squares rule (eq 13 → 14), propagate variances:

$${\sigma }_{k_d}^2=\frac{{\sigma }_{R_t}^2+{\sigma }_{R_{\infty }}^2}{t^2(R_t-R_{\infty }{)}^2}+\frac{{\sigma }_{R_a}^2}{t^2{R}_a^2}+\frac{k_d^2}{t^2}{\sigma }_t^2\text{\hspace{1em}(3)}$$

and the relative precision becomes

$$\frac{{\sigma }_{k_d}}{k_d}=\sqrt{\frac{{\sigma }_{R_t}^2+{\sigma }_{R_{\infty }}^2}{(R_t-R_{\infty }{)}^2{\ln }^2[(R_t-R_{\infty })/R_a]}+\frac{{\sigma }_{R_a}^2}{R_a^2{\ln }^2[(R_t-R_{\infty })/R_a]}+\frac{{\sigma }_t^2}{t^2}}\text{\hspace{1em}(4)}$$

The three square‑root terms parallel Tong’s “$\alpha ,\lambda ,\beta$” constants: two depend only on baseline‑to‑amplitude variance, the third folds in the (usually tiny) clock jitter.

Practical reading

• Identify which variance term dominates equation (3); that is your precision limit.
• Combine with the correlated accuracy bound if both bias and random noise exist:

$$\text{Total CI}\approx \pm \sqrt{{\sigma }_{k_d}^2+(\Delta k_d{)}^2}$$

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How to use these two paths in practice

Step	Correlated path (Eq 1–2)	Uncorrelated path (Eq 3–4)
1. Gather inputs	Realistic worst‑case signs for each systematic bias (e.g. sensor drift, offset in global fit, clock lag).	Standard deviations from replicate traces or fit covariance matrix.
2. Compute influence	Plug into Eq (1) and sum.	Plug variances into Eq (3) and take square root.
3. Report	Accuracy window: $k_d^{\text{fit}}\pm \Delta k_d$.	Precision (repeatability): $k_d^{\text{fit}}\pm z{\sigma }_{k_d}$ (choose $z=1.96$ for 95 %).
4. Improve assay	Extend dissociation to sharpen $R_{\infty }$; multi‑cycle global fitting to tighten $R_a$; verify clock calibration.	Increase replicate count; average adjacent data points; refine weighting in nonlinear regression.

Using Tong’s two‑path mindset ensures every $k_d$ you publish carries both a bias‑aware accuracy limit and a statistically rigorous precision, letting downstream modelers judge fitness for purpose just as Tong advocates for equilibrium $K_d$.

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