What is complex system modeling

Modeling complex worlds

Computers and new calculation methods make a significant contribution to the progress of the biological and environmental sciences and their technical application. They not only make it possible to record and process huge amounts of data, but also to calculate mathematical models that describe the physical, chemical and biological processes that occur. At the Interdisciplinary Center for Scientific Computing (IWR), research groups deal with the modeling and analysis of complex model equations as well as the development of computational methods and the simulation of biological systems and environmentally relevant processes. Willi Jäger and Gabriel Wittum report on the ongoing research.

Processes in biological systems and in the environment are usually complex, as are their mathematical models. The model equations are mostly non-linear, high-dimensional dynamic systems, which describe the temporal-spatial development and the interactions in the real systems. Despite this complexity, modeling and simulation have proven to be useful and necessary, as in physics and chemistry, for example in order to elucidate the processes in a cell, the interactions between cells, the coding and regulation of biological processes. They are just as useful for recording the development of biological structures and systems, the mode of operation of organs, growth and death, the interaction of biological populations and the dynamics of complex ecological systems. With methods of mathematical modeling it is possible to incorporate experimental observations into a consistent theory and to use them for prognoses, planning, optimization and control of processes.

Computers have now become experimental devices in which molecules react with one another or virtual organisms grow like in a fermenter. They simulate ecosystems in which species survive or become extinct; they determine the development of CO2 in the atmosphere and the effects of pollutants in the soil. Mathematical methods are used to analyze sequences of genes and to compare them with one another - a task which, because of the optimization problems to be solved, places great demands on computing power and computational processes. In order to elucidate the function of genes and their role, for example in the development of biological structures, it is not enough to statistically establish a cause-effect correlation - for example the interaction between a genetic deviation and a disease - with extensive data analyzes. Rather, it is necessary to grasp the biophysical-biochemical processes, to understand the action of a biomolecule or the chemical action of an enzyme in order to design compounds that have predetermined properties.

The modeling records the interaction of the atoms in a molecular chain with one another and with the surrounding medium. Modeling and simulation not only provide insights into the microscopic building blocks and processes of life or the interrelationships of complex biological systems, but also difficult tasks in bio, medical and environmental technology can be mastered. The research groups at the IWR develop suitable models, solution processes and software tools.

Mathematical methods are characterized by "portability", that is, they are structural and independent of the specific application. The modeling of biological populations and processes follows the ideas and schemes that are successfully used in molecules and physico-chemical processes. Equations for the spread of pathogens mathematically lead to the same questions as they have to be solved in chemical reactors.

In order to study the interaction of microorganisms with one another and with the environment, they are examined in a controlled biological reactor, in a chemostat. A column of chemostats, a gradostat, is ideal for analyzing biological growth in a variable environment. Models for biological processes in this apparatus have the same shape as a column of chemical reactors connected by pumps. In the interplay between modeling, simulation and experiment, the growth of microorganisms in chemical gradients was investigated. In particular, the question of the conditions under which different species coexist in such an environment was studied.

Organisms can exist in different phases in a variable environment. They adapt their metabolism to the environment, they change their growth behavior. Bacterial cultures in gradients of chemical substances can develop growth patterns that are very specific and sensitive to the substance concentrations. The concentration of bacteria that grow in a radial gradient of a chemical substance shows growth rings in the experiment, which are reproduced in the simulation. The growth behavior can also serve as a bio-indicator for pollutants. A well-known example is the Ames test, which is used to identify carcinogenic substances. With the computer it is possible to determine the critical dose of a pollutant from the observed growth patterns.

Simulation of micro-ecological systems

The mathematical modeling and computer simulation of such "microecological" systems of organisms are of both theoretical and practical interest. In order to master bioreactors, it is necessary to understand and control the ongoing biochemical processes of the microorganisms involved. Biological populations are now used in chemical or pharmaceutical production, in pest control or to dispose of pollutants. The modeling and simulation of the running processes is an important aid for the planning and control of the procedures. They require the full range of scientific computing methods.

This is demonstrated below using the example of biological cleaning of pollutants in the soil. The investigation of this biotechnological process is being carried out at the IWR in cooperation with IBL Umwelt- und Biotechnik, Heidelberg. It uses the data from an incident in the Sandhausen area, where chemicals from a factory got into the groundwater. Soil bacteria are able to break down chlorinated hydrocarbons in a reaction chain. Anaerobic and aerobic reaction steps alternate, that is, the oxygen content of the soil influences the degradation. The depth-dependent structure of the pollutant plume, which was observed during drilling, can be traced back to the anaerobic and aerobic growth of organisms in the oxygen gradient in a model by means of simulation. For the planning and controlled implementation of a biotechnological soil remediation, groundwater flow, material transport, reaction with the soil and the microorganisms are coupled and combined in a model. The model is adapted to the real data and simulated. Software tools are being developed for this purpose. The process must be controlled in such a way that the degradation of the pollutants by the bacteria is optimized. Through the simulation and optimization, the number and position of the required boreholes and pumps as well as the supply of activators for biological growth can be determined in order to create optimal conditions for the remediation. Even when planning measures, it is necessary to estimate the prospects of success and the costs that will arise. The industrial partner expects simulation tools that he can generally use for his renovation projects.

The composition of soils is obviously very heterogeneous. In specific cases, their data situation is poor. Soils are "random" media. Processes in such complex porous media with only stochastically describable geometric and physico-chemical properties require new methods. We are dependent on describing such media with "effective" properties and also introducing stochastic effects in the simulation. Further examples are biological tissues and technical materials. The visualization of the outermost layer of the skin with the help of an imaging process (computed tomography) is graphically similar to a section through a soil layer. It is therefore to be expected that material flows through tissue can be modeled and simulated using similar methods. Over the years, the Heidelberg research group has developed know-how to analyze such systems on different scales and to process them numerically. The technical terms "homogenization" and "multigrid process" give a rough indication of the procedure. Appropriate averaging provides macroscopic descriptions that approximate the behavior except for controllable and acceptable errors. Numerical methods are based on computational grids in space and time, with which structures on different scales are calculated. It is obvious that scales in a system can be taken into account by appropriately splitting them up into sub-tasks on grids of corresponding scales.

It requires very efficient methods to calculate the flow through a stochastically generated porous medium, as shown by the cube shown at the top right on page 39, by solving highly non-linear partial differential equations. Water flows into this cube from the base like in unsaturated soil. The result is shown immediately after the start of the process in a three-dimensional representation of the water content (increasing from blue to red).

The heterogeneity of the soil leads to effects in the flow of several phases (e.g. water and solvent), for the control of which computer simulations are necessary. Soils often consist of layers of very different properties. As shown in the picture on the left on page 40, lentils of fine-grain sand (red in the picture) may be embedded in a coarse-grained environment. Due to the non-linear coupling of the phases and the material jump, it happens that, for example, the solvent (green in the picture) does not penetrate the lenses at all because the pressure required is insufficient, and sometimes even in the opposite direction to the direction of flow of the water (in the picture from right) flows. The results obtained in the simulation reflect the phenomena observed in the field and confirmed in experiments. The degradation of a pollutant in a porous medium by a chemical substance transported in the flow was simulated on the computer and visualized using a method that was specially developed for the three-dimensional representation of flows. Spatial densities are translated into optical properties and made visible by tracking light rays in the optical medium created in this way. For the investigation of multidimensional processes, the visualization of measured or calculated data sets has become an important means of information.

A current research focus at the IWR is to model and calculate the dynamics of three-dimensional multiphase flows with chemical or biological reactions. Important advances have already been made. The software system UG, which the research group Scientific Computing and Technical Simulation developed, contains tools for the calculation of such multiphase flows. UG can be used in many areas of activity and contains the latest, mathematically based methods, especially for solving complex systems, especially non-linear partial differential equations. Because of the high demands on the computing power, it uses parallel processes and computing grids (unstructured grids) adapted to the individual computing steps.

Modeling of physiological processes

Membranes, filters, paper, biological or textile tissue, bones and porous catalysts provide a matrix for processes that can be processed using the appropriate methods. The modeling of physiological processes, for example in the skin, is important for pharmaceutical, medical or cosmetic applications. The penetration of substances through the skin was simulated for the case of regular tissue structures. The simulation of a pure diffusion without interaction with the cells shows the effect of the different permeability of the walls and the interior of the cells as well as the intercellular space. The necessary numerical processes have been developed to couple the transport of substances with chemical reactions.

To summarize the physiological processes in cells, in tissues and bones, in organs, the flow of substances, the flow of blood in the circulatory system and the processing of information in nerve networks in mathematical relations and equations, is a decisive step that the task on the development of calculation methods and the adaptation to real data is reduced. The chemical reactions in a test strip used to diagnose a disease are comparatively easier to model than the reactions in living tissues.

Diagnostic paper strips were simulated by the research group for the pharmaceutical industry in order to better quantify the test reactions and thus the diagnoses. In the future, medical technology will increasingly use computer simulations as an instrument to improve diagnostics, record the effects of therapeutic measures and plan interventions. There is still a mountain of research work to be done, for example to calculate the pressure distribution and the increase in volume of a brain damaged by an infarction, to plan the placement of a heart valve optimally on the computer or to estimate the long-term effect of an implant on bones and tissue.

Models always only capture partial aspects of the running processes and these only approximately. This applies in particular to biological systems and processes in the environment. Experience shows, however, that complexity can be reduced to essential processes depending on the respective questions. Computer simulations are used to determine the decisive factors.

In a huge network of chemical reactions, what are the essential components for the production of a substance? What are the decisive factors for the survival or extinction of a species in an ecological system? Which parameters determine the stability of a growth process, which changes lead to the formation of structures? Computer simulations help determine how a system will respond to changes in system data. With them it is possible to calculate controls and control processes. The fact that this is possible with comparable methods in apparently very different situations is due to the comparable structure of their mathematical models.

Research in this area is funded at the IWR by the German Research Foundation (in particular through the special research area "Reactive Flows, Diffusion and Transport"), the Federal Ministry of Education, Science, Research and Technology in the program "Mathematics in Industry" and through direct industrial cooperation . Research at the IWR is interdisciplinary and particularly includes experiment and observation data. The IWR contributes to the training of qualified young people in biotechnology and environmental technology, in which computer-aided methods have become indispensable.

Authors:
Prof. Dr. Willi Jäger and Prof. Dr. Gabriel Wittum,
Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg,
Telephone (0 62 21) 54 57 80