The chemistry in the flow chambers.

Lagrée P.-Y.
L.M.M. UMR CNRS 7607, Université PARIS VI, FRANCE
www.lmm.jussieu.fr/~lagree/TEXTES/ BIACORE/BIA-Jsb2001.html
c 4/09/01, à jour sept 04; mai 06

What is a flow chamber and what is a BIACORE?

In this page we present some results on the chemical reactor used in biological analysis. Those apparatus are called "flow chamber" or "flux chamber", one of them is the BIACORE 2000 (now 3000!!!). This machine has been designed by the company Biacore.

The flow chambers are commonly used to measure binding rates of macromolecular interactions with a large area of biological applications (ADN/ADN, ADN/proteins, polymerisation...). The measurement is done with reflexion of polarized light ("SPR" technique see explanation in french). Those chambers (such as the BIACORE device) have been designed to allow the use of simple kinetic theories (for example, the experimental conditions are chosen in order to have nearly a constant spatial concentration on the chip). There are recent studies which show the limits of those theories: they present more complex theories considering the coupling between the reaction kinetics and the mass transport. They permit the understanding of the influence of the mass flux increase upon the reactions taking place on the wall (the "chip").

biacore
h=5 10-3cm, L=0.24 cm. The channel is very small! 		biacore
We prefer to look upside down to the flow... the chip (the reacting part) is now on the bottom:
...

The reactant "C" is in the flow (green spheres) and reacts with "D" (looks like a red "Y") on the wall (there is association and disssociation). C comes to the wall by diffusion, when no C is no more present in the flow, the dissociation process gives finaly D alone.

reaction at the wall

The classical simple kinetic theory as an exact solution: C reacts at the wall with D to give B (with constant of adsorption ka et and desorbtion kd):

kd

C + D <=> B

ka

The differential equation associated is:

B'(t) = ka CT (RT -B(t)) - kb B(t) and B(0)=0

where CT is the concentration provided by the flow, RT is the initial concentration of D at the wall. This must be solved for t<tc the injection time, when t>tc, we have dissociation

B'(t) = - kb B(t).

Examples of solutions B(t), here tc=60.

 

The question is:

what is the influence of the flow on the chemistry?

The following links are pointed to PDF files giving the response:

  • Those problems and a new way to solve the equations in the BIACORE are found in the paper:
    P.-Y. Lagrée & A. Ivan-Fernolendt (2004)," Direct comparison of asymptotic models of surface reacting flows in flow chambers", Eur. Phys. J./AP vol 26, pp 133- 143), note the Erratum page 10: the figure 8 displays "B" and not "C".

  • The abstract of a contribution at the "Dispositifs d'écoulement pour l'étude du comportement de populations cellulaires sous contraintes" is :
    "The chemistry in the BIACORE cell".
    The text of the slides is:
    "La chimie dans les chambres à flux, comparaison de modèles simplifiés."

  • congrès SFB de CRETEIL 2004 abstract, (P.-Y. Lagrée & A. Ivan-Fernolendt (2004), "the chemistry in a flow cell", Arch Phys Bio, vol 112, p 81).

  • congrès SFB de CRETEIL 2004 (P.-Y. Lagrée & A. Ivan-Fernolendt (2004), "the chemistry in a flow cell", slides).

  • The exam (2001) of the ENSTA Engineers gives some details in the computations: "Transferts Thermiques dans les fluides."

  • References on the subject
    Myszka DG, He X, Dembo M, Morton TA, Goldstein B (1998): "Extending the Range of Rate Constants Available from BIACORE: Interpreting Mass Transport-Influenced Binding Data", Biophysical Journal 75:8:583-594,
    mais il faut aussi consulter les articles de Edwards dont:
    Edwards D. A. (1999): "Estimating rate constants in a convection diffusion system with a boundary reaction", IMA J. Appl. Math., 63 , pp89 -112.
    http://www.math.udel.edu/%7Eedwards/download/pubdir/pubhome.html#pubj14






    • Hint :
    the flow is a Poiseuille one, one has to solve the diffusion/convection equation for C in the flow.


    with boundary conditions

    but the channel is very thin so that the equations may be simplified as:

    Using those equations, we may compute the concentration,


    Figure: plot of the concentration of the formed species B(x,t) on the chip (x between 0 and 1) for various times. We see that there is a little non uniformity in x. Pe=372, K=1, Da=0.7, DaPe-1/3=0.1
    Conclusion for our model

    It is a simplified model, but it remains precise enough.



    Link with previous studies "Edwards solution"

    Near the chip it is a linear shear flow, so that we may consider tht the velocity is in fact linear:


    Figure: près de la paroi le cisaillement est constant.


    this allows to reobtain the "Lévêque" problem (see Edwards)

    Figure: tracé de B fonction du temps pour différentes valeurs de Da et Pe, ainsi que la courbe prédite par Edwards.

    Les résultats sont confondus pour DaPe-1/3 <.1 (donc la description "Lévêque" marche bien)

    Pour DaPe-1/3~.5, l'erreur est de 10%

    La solution d'Edwards tombe en défaut pour DaPe-1/3~1
    Link with previous studies "Integral solution"


    At first, when the Biacore was designed, the flow was neglected. The next approximation was to do a "integral" or "mean" resolution for the flow. So an "exchange coefficient" was introduced. We show in our paper that one has to be carefull with the choice of this coefficient.

    Figure: Réponse de la concentration de ligand formé B(t) en fonction du temps pour K=1, DaPe-1/3=0, 1, 10 et100. Points: résolution numérique du système couplé PDE et ODE, pointillés le modèle intégral avec le coefficient 0.87 (correct pour DaPe-1/3 très petit) et traits: le modèle de Myszka avec le coefficient 0.807 (correct pour DaPe-1/3 très grand, en pratique supérieur à 0.5)

    `



    Numerical integration

    Ici les fichiers source en C pour calculer l'image ci dessus. Le fichier pdf détaille la suite des calculs.