(72c) On Parabolic and Hyperbolic Diffusion Effects during Hybridization Kinetics with Convection in Microfluidic Chamber

The hybridization reaction between target molecules that is going to be studied with the probe molecules that are used for detection can be seen to be reversible reaction. Thus,

k_f

                                                          P + T      P-T                                                   (4.27)

k_r

where P is the probe molecules, T – target molecules and P-T the probe-target hybridized duplex that is formed on the substrate. 

                                                                    (4.28)

Where [P] is the concentration of the probe molecules, [T] is the concentration  of the target molecules, [P-T] is the concentration of the probe-target duplex molecules, k_f is the rate constant of the forward reaction and k_r is the rate constant of the reverse reaction. The ratio of [P-T] to [P]₀ provides a measure of the efficiency of the hybridization.  [P]₀ is the concentration of the fresh probe molecules available for tagging in the beginning of the operation. This also can be seen to be the maximum concentration of probe molecules during the reaction process. When the target molecules are assumed to be in excess the rat expression in Eq. () can be seen to obey pseudo-first order kinetics. The rate would only vary with the concentration of the probe molecules and probe-target duplex molecules and the concentration of the target molecules can be lumped along with the rate constant, k_f and can be expected to remain constant during the reaction process. The rate expression can be simplified as;

    (4.29)

Assuming that at t = 0, no probe-target duplex molecules were formed, i.e., [P-T]₀  = 0, Eq. () can be integrated as follows;

                          (4.29)

Where 

Integrating Eq. ();

                        (4.30)

or

                                    (4.31)

                                (4.32)

Where K is the equilibrium rate constant given by; . Eq. () is for the ideal case. This represents the highest rate possible for hybridization. Diffusion effects can result in a lower overall rate of hybridization.

      The relative effects of intrinsic hybridization reaction rate and diffusion rate were evaluated using a Damkohler number. Damkohler number was defined by Pascault et. Al. [2008] as the ratio of maximal forword reaction rate to the maximum normal diffusion rate. When Da is greater than 1 they found the process to be diffusion limited.

      Sharma [2005] has shown that when the dimensions of the chamber is small and the time of transport are low damped wave diffusion effects may become important.  The relaxation times of the oligonucleotides may also have to be considered.  The relaxation times can be expected to increase with the increase in size of the oligonucleotide. When the relaxation  times are large and/or the microfluidic chamber size is small the concentration of the probe molecules can be expected to undergo subcritical damped oscillations. The probe molecules will have to meet the target molecules and form the duplex. The duplex molecules can be attached to the microarray substrate using linker molecules. Optimal probe concentration is discussed in Sharma [2009]. Quality of surface needed is also discussed.  Single stranded DNA molecule will be 12 A⁰ in diameter. So the P-T duplex molecules will be 24 A⁰ in diameter.  Due to major and minor grooves the effective diameter will be 20 A⁰.  The optimal target concentration is achieved when the spacing of target molecules are at least 20 A⁰ each.  Further packing will cause steric hindrance. 

Consider the microfluidic chamber to be made of a finite slab with thickness of 2a (Figure 4.3).  At one end of the surface the probe concentration is at the maximum and the constant wall concentration boundary condition can be assumed. The other end can be assumed to be impervious to the probe concentration. As the probe molecules diffuse through the target solution simultaneous reaction and damped wave diffusion effects may become important.

Figure 4.3 Microfluidic Chamber for P-T Duplex Formation

The governing equation for simultaneous damped wave diffusion, relaxation and reaction that obeys the irreversible first order kinetics, can be written as follows;

                  (4.33)

Where C_p is the concentration of the probe molecules, t_r is the relaxation time of the probe molecules, D_pt is the binary diffusion coefficient of the probe molecules in target solution, k is the rate constant of the probe-target duplex formation assuming that the probe-target reaction is first order and irreversible and k* = kt_r. The governing equation can be made dimensionless by the following substitutions;

                                                          (4.34)

The governing equation becomes;

                                 (4.35)

The boundary conditions can be written as follows;

X = 0, u = 0                                                     (4.36)

                                            (4.37)

The governing equation needs to be made homogenous. The solution can be assumed to comprise of two parts, i.e., steady state part and a transient state part. Thus,

                                                                                                                             (4.38)

Pluggind Eq. (4.38) in Eq. (4,35) the governing equation can be written as two equations;

                                                          (4.39)

                           (4.40)

The governing equation to the steady state part is a second order differential equation with constant coefficients. The k* term will result in a integrating factor of another constant a₃.  The solution to the  second order differential equation with constant coefficients  can be written as a sum of sinh and cosh hyperbolic functions. The integrating factor a₃ can be seen to be -1 by a g = (1+ u^ss) substitution.  Thus;

                        (4.41)

Applying the boundary conditions to the solution a₂ can be seen to be 1 from the  X = 0, u = 0 condition. Applying the impervious boundary condition at x = 2a;

                             (4.42)

                               (4.43)

         (4.44)

The solution to the transient part of the dimensionless concentration can be obtained by the method of separation of variables. The dimensionless transient concentration can be expressed as a superposed product of decaying exponential and wave concentration. This can also be arrived at by multiplying Eq. () by e^nt. By choice of n and grouping the terms the wave concentration can be seen to be;

                                                                                                                                (4.45)

For , the wave concentration can be seen to obey;

                                                     (4.46)

The hyperbolic second order partial differential equation for wave concentration can be solved for by the method of separation of variables. Let W = g(t)f(X).  Then Eq. () becomes;

                               (4.47)

The space domain solution can be written as follows;

                             (4.48)

From the boundary condition at X = 0, c₂ can be seen to be 0. From the impervious boundary condition at x = 2a;

                                                   (4.49)

                             (4.50)

Or, ,  n = 0, 1, 2, 3, 4,…….                            (4.51)

The time domain solution can be seen to be;

                                     (4.52)

It was shown by Sharma [2005] that final condition in time can be applied and solutions within the frameword of second law of thermodynamics can be obtained. When t ®∞, what happens to the wave concentration ?

                                              (4.53)

The dimensionless concentration u becomes 0 and the exponential becomes infinity. Although the product of 0 and infinity is of the indetermintate form of the fourth kind [Piskunov, 1965] it can be  assumed that W will be finite or zero at infinite time and will not be infinity.  Applying this condition to the time domain solution it can be seen that c₃ is 0. The generation solution for the transient concentration can be written as sum of infinite modified Fourier series and;

                                        (4.54)

The initial condition can be used in order to obtain the c_n by invoking the principle of orthogonolity. C_n can be seen to be . When the relaxation times are large the concentration of the probe can be expected to undergo subcritical damped oscillations.  This is when;

                                           (4.55)

This can happen with more probe molecular types especially when a is small. This is the case for the microfluidic chamber in Figure 4.3. This is about 100 µm. k* also contains a contribution from relaxation time, t_r.   For systems with slow reaction and higher diffusivity Eq. () will yield a lower threshold value for the relaxation time. For systems with large relaxation times the concentration profile can be written as follows;