## Lateral Flow Assay Model

We proposed to use a lateral flow device to detect the labelled RNA, as stated in design page. We can reasonably assume that liquids will do one-dimensional diffusion in our lateral flow device since the width (y) and the height (z) of the device can be ignored compared to its length, as Fig.1 shows:

Fig.1 The schematic diagram for lateral flow assay deviceNow let’s consider the antibody-protein binding kinetics. There are two processes at work: the formation of protein-antibody complex (PA) from protein (P) and antibody (A), and the dissociation of the complex into protein and antibody. We can therefore use equation 2.1 to depict the binding kinetics:

where r is the net rate for protein-antibody complex formation, k

_{a}, k_{b}are the reaction kinetic constant for complex formation and dissociation. C_{p,A,pA}is the concentration of corresponding protein. When the reaction is at equilibrium, note that the reaction rate r can be depicts in another way:Now let’s consider the protein concentration inside the binding area. Assume the length of the binding region is x, then we will have the volume of the binding region (V):

Assume the flow rate of the liquid is v, then we have the volume of the flow into the region per unit time (D):

According to law of conservation of mass, the difference of the protein amount between inflow and outflow equals to the absorbed protein amount by the antibody:

Combine equation 2.1~2.4 with 2.5, we have:

where C

_{A,0}is the antibody amount before protein-antibody binding.We can derive the expression of protein concentration in outflow, C

_{p,out}, from equation 2.6:Let v/x = t, we can calculate the derivate of C

_{p,out}versus t:Apparently the denominator or the right side of equation 2.8 is no less than 0. So we can discuss the numerator alone to decide the sign of the derivate of C

_{p,out}versus tRewrite the numerator part:

Note that r = k

_{a}C_{p}C_{A}- k_{b}C_{PA}≥ 0, and C_{p,in}≥ C_{p}, therefore:therefore we have:

Combiner inequation 2.9 and 2.12, we have:

We therefore have proven that:

Two conclusions can be reached by analyzing inequation 2.14:

Conclusion a): the higher the flow rate v, the less efficient the protein can bind to the antibody, and the more unbinding protein will present in outflow.

Conclusion b): the wider the binding region is (the larger x is), the more efficient the protein can bind to the antibody.

# Lateral Flow

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