Mathematical modeling of nanotechnology

One of the challenges – and advantages – of nanotechnology is that many aspects of its use can, but also ought to be, tested by simulation. If minuscule applications are to carry active components with extreme precision to their target within the human body, be it to mark cells for diagnostic procedures or to support the buildup of tissues and structures or to deliver therapeutic agents, their functionality needs to be simulated by mathematical models and methods.

One such application is the development of biosensors capable of identifying tumor markers in blood that holds significant potential for cancer therapy. Nanowires composed of semiconductor material enable recognition of proteins that indicate the presence of existing tumors. Physical simulation of such an electronic system with biological application presents novel mathematical problems. Nanowires used as sensors are approximately 1 μm in length (one-hundredth of the diameter of a human hair) and 50 μm in diameter, i.e., one-twentieth of their length. DNA molecules examined by these nanowires are only 2 μm in diameter and tied to receptors on the wire permeated by a certain electric current. This changes conductivity and current flow of the sensor. A mathematical model suitable for simulation needs to reflect the relevant subsystems of this process by equations that describe the distribution of the charges, coupled with equations reflecting the movement of charges. This creates a system of partial differential equations reflective of the transport of charges that can be connected with equations descriptive of the motion of molecules on the outside of the sensor. Currently, Clemens Heitzinger at Vienna University of Technology’s Institute of Analysis and Scientific Computing is developing such models.

Certain systems of equations can also use multiscalar analysis problems that are common in nanotechnology. Both the behavior of fine structures (for example during bonding of a molecule) and the properties of the sensor itself are of interest. If one were to describe it all through a numerical simulation, it would require computing time beyond realizable limits. But partial differential equations combined with a solution of multiscalar problems permit simulation of both relatively large and very small structures in a single system. This can save extremely expensive custom-built items for lab experiments. It also permits conclusions about data and connections that could not be deducted by physical measurement techniques. The technology can target any molecule identifiable through antibodies, not only biological molecules but also poisonous gases.

However, the smaller the examined systems, the greater the importance of random movements and fluctuations. To account for such events, probability theories need to be integrated into systems of differential equations, transforming partial differential equations into stochastic partial differential equations that have random forcing terms and coefficients, can be exceedingly difficult to solve, and have strong connections with quantum field theory and statistical mechanics. This also has numerous applications outside of medicine, for example in information technology. One example is microchips that contain billions of transistors approximately 20 μm in size that cannot all be perfectly identical but need to function despite their fluctuation range. The same mathematical modeling approach permits optimal numerical simulation of such systems.

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