Team:GreatBay SCIE/Test Line Position and Sample Volume

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Introduction

The construction of this model aims to give a quantitative relationship between the concentrations of several different species participating in the conjugation reaction and several parameters involved.

We hope to solve the following problem:

What distance from the start of the whole test paper (where the sample and the aptamer solution enter) to the start of the test line containing immobilized toxins (which removes excess aptamers not bound to the toxin in the sample), should be designed in order for the aptamer and the possibly present toxin in the sample to descend fully, or to a certain extent, into chemical equilibrium, before reaching the test line, so the amount of unbound or excess aptamers left on the test line after the sample has flown through could be assessed to determine whether the sample contains toxins?

Abbreviations

• Apt: Aptamer
• ToxS: Toxin in sample

Parameters

• D: Diffusion rate constant
• U: Average flow speed of liquid on test paper
• $[\ce{Apt}]_0$: Original concentration of aptamer
• $[\ce{ToxS}]_0$ Original concentration of toxin in sample liquid

Model

Based on the following chemical equilibrium of association and disassociation, and the definitions of the equilibrium constant, the association and disassociation rate constants:

$$\ce{Apt + ToxS <=> Apt-ToxS}\\ K=\frac{k_{a1}}{k_{d1}}=\frac{[\ce{Apt-ToxS}]}{[\ce{Apt}][\ce{ToxS}]}$$

We can define the formation rates of Apt-ToxS to be the following:

$$F_\ce{Apt-ToxS}=k_{a1}[\ce{Apt}][\ce{ToxS}]-k_{d1}[\ce{Apt-ToxS}]$$

 

Then we can construct the following convection-diffusion-reaction equation:

$$\frac{\part[\ce{Apt-ToxS}]}{\part t}=D_\ce{ToxS}\frac{\part^2[\ce{Apt-ToxS}]}{\part x^2}-U\frac{\part [\ce{Apt-ToxS}]}{\part x}+F_\ce{Apt-ToxS}\\$$

The first term on the right-hand-side refers to diffusion. This takes in account of the second partial derivative of concentration of [Apt] with respect to distance. This term shares some similarity with the differential equations of heat conduction, in which the rate in which heat dissipates from a given point is proportional to the second partial derivative of the temperature at that point with respect to horizontal distance. For instance, if the concentration of Apt is very high at a certain point, and very low at other places, this will create a sharp peak on the [Apt]-x graph, hence the second derivative is largely negative at that point. This means, A will diffuse very quickly away from that point, with a diffusion rate coefficient D.

The second term takes in account of the lateral flow of the sample liquid. As A moves with the sample liquid, towards larger and larger positive values of x, we take the first derivative of the concentration of A with respect to distance.

• If the function is increasing at a position, meaning that the concentration peak of A is already ahead of that position, then its derivative will be positive, and hence, due to the negative sign in the equation, it contributes negatively to the concentration of A.
• If the function is decreasing, meaning that the concentration peak of A is in front of that position and will come shortly, then its derivative will be negative, and hence, it contributes positively to the concentration of A.

And all these contributions will be multiplied by a velocity parameter U.

Hence, substituting in the expression for the formation rate of Apt-ToxS:

$$\frac{\part[\ce{Apt-ToxS}]}{\part t}=D_\ce{ToxS}\frac{\part^2[\ce{Apt-ToxS}]}{\part x^2}-U\frac{\part [\ce{Apt-ToxS}]}{\part x}+k_{a1}[\ce{Apt}][\ce{ToxS}]-k_{d1}[\ce{Apt-ToxS}]$$

We want to get rid of the two variables [Apt] and [ToxS] by expressing them in terms of their initial concentrations (which can be a fixed parameter) and [Apt-ToxS] under equilibrium:

$$[\ce{Apt}]_\text{eq}=[\ce{Apt}]_0-[\ce{Apt-ToxS}]_\text{eq}\\ [\ce{ToxS}]_\text{eq}=[\ce{ToxS}]_0-[\ce{Apt-ToxS}]_\text{eq}$$

Through a series of algebraic rearrangements, we arrive at:

$$\frac{\part[\ce{Apt-ToxS}]}{\part t}=D_\ce{ToxS}\frac{\part^2[\ce{Apt-ToxS}]}{\part x^2}-U\frac{\part [\ce{Apt-ToxS}]}{\part x}+k_{a1}[\ce{Apt-ToxS}]^2-(k_{a1}([\ce{Apt}]_0+[\ce{ToxS}]_0)+k_{d1})[\ce{Apt-ToxS}]+k_{a1}[\ce{Apt}]_0[\ce{ToxS}]_0$$

We define the equilibrium concentrations to be functions of both displacement and time.

We state that:

• $x$: the distance from the start of the test paper
• t: the time elapsed

Hence we can state an initial condition, namely, the concentration of the conjugate everywhere on the test paper is 0 at t=0.

$$[\ce{Apt-ToxS}](x,0)=0\\$$

And we state the boundary conditions. The first one states that at the very edge at the starting end of the test paper, the concentration of Apt-ToxS is always 0.

The second one means that the rate of change (increase) of the concentration of the conjugate is always 0 at the "solvent front" (liquid-phase chromatography analogy), namely, the very frontal edge in which the liquid has reached over a time period t, with flow speed U.

$$[\ce{Apt-ToxS}](0,t)=0\\ \frac{\part[\ce{Apt-ToxS}]}{\part x}\biggr\rvert_{x=Ut}=0$$

Next, we would like to figure out the equilibrium concentration of Apt-ToxS. We find the breakpoint by stating that at equilibrium, $F_\ce{Apt-ToxS}=0$ because any change in [Apt-ToxS] arose from the association-disassociation equilibrium is equal to 0. Hence we get a quadratic equation about [Apt-ToxS]:

Upon solving, we acquire:

$$k_{a1}[\ce{Apt-ToxS}]^2-(k_{a1}([\ce{Apt}]_0+[\ce{ToxS}]_0)+k_{d1})[\ce{Apt-ToxS}]+k_{a1}[\ce{Apt}]_0[\ce{ToxS}]_0=0$$

 

$$[\ce{Apt-ToxS}]=\frac{1}{2}\left([\ce{Apt}]_0+[\ce{ToxS}]_0+\frac{k_{d1}}{k_{a1}}-\sqrt{\left([\ce{Apt}]_0+[\ce{ToxS}]_0+\frac{k_{d1}}{k_{a1}}\right)^2-4[\ce{Apt}]_0[\ce{ToxS}]_0}\ \right)$$

And hence we can calculate the equilibrium concentration of the conjugate by only using four parameters.

Because we will use this value later on, we label this fixed parameter as $[\ce{Apt-ToxS}]_\text{eq}$.

An optimum Apt-ToxS concentration is needed because equilibrium takes long time for achieving that may result in very long length of reaction membrane before test line, which may not be practically feasible. Thus, positioning the test line at a location where the optimum Apt-ToxS concentration is reached assures that an adequate amount of aptamer is attached to any possibly-existing toxin molecules in the sample, hence the rest of the aptamers could be considered as unbound excess aptamers, and they could be confidently immobilized onto the test line for subsequent processes. This is a percentage of the expected equilibrium concentration of Apt-ToxS. In this model, it is taken as 90%.

The average velocity of liquid movement on a nitrocellulose membrane is taken as $10^{-2}$ centimeters per second. The diffusion rate constant, $D$, is taken the value $10^{-12}$. The association and disassociation rate constants $k_{a1}$ and $k_{d1}$ are taken the values $1\times 10^6$ $\text{mol}^{-1}\text{dm}^3\text{s}^{-1}$ and $1\times 10^{-3}$ $\text{s}^{-1}$, respectively. The initial concentration of the aptamer and the toxin in the sample ($[\ce{Apt}]_0$ and $[\ce{ToxS}]_0$) are both taken the value $1\times 10^{-8}$ $\text{mol dm}^{-3}$.

Through calculation using the formula we derived earlier, we can know that the equilibrium concentration of the aptamer-toxin conjugate ($[\ce{Apt-ToxS}]_\text{eq}$) is $7.30\times 10^{-9}$ $\text{mol dm}^{-3}$. Hence, because we take $[\ce{Apt-ToxS}]_\text{opt}=0.9[\ce{Apt-ToxS}]_\text{eq}$, the optimal concentration of the aptamer-toxin conjugate is $6.57\times 10^{-9}$.

We use these parameter values to solve the partial differential equation mentioned above (Using the NDSolve function in Wolfram Mathematica), and we obtain the following curved plane:

The vertical axis is the concentration of the aptamer-toxin conjugate concentration, and the horizontal axes are the position $x$ and time $t$, as labelled.

Taking the $[\ce{Apt-ToxS}]$-$x$ curve of multiple $t$ values, we can get the following collection of curves:

As shown in the legend, the blue horizontal dashed line represents $[\ce{Apt-ToxS}]_\text{opt}$, and the lines in red, orange, olive and green represent curves with $t$ values ranging from 100, 200, 300 and 400 seconds. As $t$ grows larger, these curves approach the black curve.

We examine the intersections of these curves with the optimal Apt-ToxS concentration line. For the curves $t=100$ and $t=200$, the concentration doesn't intersect with $[\ce{Apt-ToxS}]_\text{opt}$ at all. While for larger $t$ values, for $t=300,400,500$, there is an intersection point. This intersection point stays relatively the same when $t$ becomes large.

As mentioned earlier, we wish to place the test line at a position where the equilibrium concentration of the aptamer-toxin conjugate is greater than the optimal conjugate concentration. Hence, the test line should be placed after that intersection point, where it is certain that $[\ce{Apt-ToxS}]>[\ce{Apt-ToxS}]_\text{opt}$. However, due to cost and test efficiency concerns, we wish to reduce the length of the test strip as much as possible. Therefore, the optimal test line position should be at that very distance, where the black $[\ce{Apt-ToxS}]$-$x$ curve (maximum $t$) intersects with the $[\ce{Apt-ToxS}]_\text{opt}$ line.

We arrive at the conclusion that there exists an optimal test line position, where the aptamer-toxin equilibrium has completed to a certain extent ($[\ce{Apt-ToxS}]_\text{opt}=0.9[\ce{Apt-ToxS}]_\text{eq}$, for instance), and the time is minimized. We refer back to this conclusion when we are designing our test strip hardware, where an appropriate amount of space will be left between the starting position and the test line.

 

References

• Qian, S., & Bau, H. H. (2003). A mathematical model of lateral flow bioreactions applied to sandwich assays. Analytical Biochemistry, 322(1), 89-98
• Ragavendar, M. S., & Anmol, C. M. (2012). A mathematical model to predict the optimal test line location and sample volume for lateral flow immunoassays. In 2012 Annual International Conference of the IEEE Engineering in Medicine and Biology Society (pp. 2408-2411). IEEE.