Modeling Of Enzymatic Processes In Thin Layers

It is possible that the progressive formation of tissue with distance from the active cell layer is due to slow diffusion of reactive species. This could result, for instance, in a decreasing concentration of an inhibitor away from the cell surface. Given tissue growth rates of a few millimeters per day, control mechanisms should operate on the scale of about a few hundred seconds if the active zone is 10 microns wide. Diffusion coefficients for molecules in aqueous solution range from about 10-5 to 10-7 cm2/sec as the molecular weight rises from 50 to 500,000. For a film of thickness 10 microns, these give relaxation times for equilibration of concentration of about 0.5 to 5 seconds. Thus a diffusion-based control system would require diffusion coefficients reduced by two orders of magnitude. This could be the case in a gel matrix, if either the diffusing species bound to the gel or were large compared to the pore size of the gel. Either of these may be true for proteins diffusing in extracellular matrix.

As an illustration of this model, figure 11 shows the formation of a product in a 100 micron layer of matrix against a surface (cell layer), which acts as a constant source. Also shown is the same system with a slowly diffusing inhibitor, which results in a reduced level product adjacent to the source. In order to produce a distribution similar to bone growth, with product formation at a distance of a few microns from the cell layer, the inhibitor can be produced slightly before the reagents so that a zone of no reaction is formed, as shown in figure 12. Thus, given a suitable combination of reaction and diffusion rates and release times, it is possible to control deposition in depth as seems to occur in tissue formation.

Figure 11. Product and reagent distribution in a 100 micron layer with co-diffusion for 6 seconds of reagents (D= 10-6cm2/sec) and a slowly diffusing inhibitor (D= 10-9cm2/sec).
Figure 12. Product and reagent distribution in a 10 micron layer after 3 seconds of diffusion of an inhibitor (D= 10-9cm2/sec) followed by diffusion of two reagents (D= 10-6 cm2/sec), with rapid reaction A+B->C.

5. Conclusions

We have shown that inkjet printing can be used to deposit multiple layers of reacting materials and so could be used to mimic biological growth. The form of the product will depend on the interaction of the printed materials with the substrate and with each other. Effects of surface tension and of timing are important. The product will depend on a combination of diffusion and chemical reaction which may need to be modeled in order to achieve detailed understanding. This approach may also shed light on the role of time, space and diffusional processes on biological tissue formation.

References

1. Calvert, P. (2001) Inkjet printing for materials and devices, Chem. Mater. 13, 3299-3305.

2. Landis, W.J. and Silver, F.H. (2002) The structure and function of normally mineralizing avian tendons, Comparative Biochemistry and Physiology Part A 133, 1135-1157.

3. Bianco, P. (1992) Structure and mineralization of bone, in E. Bonucci (eds.), Calcification in biological systems, CRC Press, Boca Raton.

4. Diekwisch, T.G.H., Berman, B.J., Anderton, X., Gurinsky, B., Ortega, A.J., Satchell, P.G., Williams, M., Arumughan, C., Luan, X., Mcintosh, J.E., Yamane, A., Carlson, D.S., Sire, J.-Y. and Shuler, C.F. (2002) Membranes, minerals and proteins of developing vertebrate enamel, Micros. Res. Tech. 59, 373-395.

5. Creagh, L.T. and McDonald, M. (2003) Design and performance of inkjet print heads for non-graphic arts applications, MRS Bulletin 28, 807-811.

6. Derby, B. and Reis, N. (2003) Inkjet printing of highly loaded particulate suspensions, MRS Bulletin 28, 815818.

7. Deegan, R. (2000) Pattern formation in drying drops, Physical Review E 61, 475-485.

8. Shmuylovich, L., Shen, A. and Stone, H. (2002) Surface morphology of drying latex films: Multiple ring formation, Langmuir 18, 3441-3445.

9. Fischer, B.J. (2002) Particle convection in an evaporating colloidal droplet, Langmuir 18, 60-67.

10. Deegan, R.D., Bakajin, O., Dupont, T.F., Huber, G., Nagel, S.R. and Witten, T.A. (1997) Capillary flow as the cause of ring stains from dried liquid drops, Nature 389, 827-829.

11. Shimoda, T., Mori, K., Seki, S. and Kiguchi, H. (2003) Inkjet printing of light-emitting polymer displays, MRS Bulletin 28, 821-827.

12. Calvert, P. and Liu, Z. (1998) Freeform fabrication of hydrogels, Acta Materialia 46, 2565-2571.

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