Mathematics and theoretical biology
by Lara López
MATHEMATICIANS observe, arrange, and describe. They examine the complexity of the world and articulate it in the language of mathematics. Many scientists can read this language, and some use mathematical modeling to identify patterns, analyze structures, and simulate the phenomena of the physical world. With the help of computers, phenomena such as fingerprints, the designs on a zebra’s skin, genetic expression, the spread of epidemics, and traffic patterns are being mathematically modeled and analyzed. At the University of Puerto Rico, Río Piedras Campus, computational mathematician Mariano Marcano, Ph.D., creates models that simulate the complex urine concentration mechanism (UCM) of the kidney.
When water is scarce, mammals and some birds are able to produce urine that is more concentrated than blood plasma, allowing their bodies to dispose of impurities and filter blood while losing the least possible amount of water. This concentrated urine, produced in the nephron and collection tubules by the UCM, is one of the most efficient methods of water conservation in mammals. The UCM is an adaptive feature that helps a species to survive during times of drought. As may be expected, desert mammals—especially camels and certain rodents, such as gerbils and Australian mice—have very powerful UCMs, capable of producing urine that is up to 22 times more concentrated than blood plasma.
Biologists, physiologists, and nephrologists have worked simultaneously to devise a valid explanation of how the UCM works. To date, no hypothesis has been proved or disproved. Marcano himself has been working with other scientists for more than ten years to develop the mathematical tools needed to create computational models to test hypotheses about the UCM.
“The results of our research contribute more to biology than mathematics. In fact, many well-recognized universities place their biomathematics programs within their medical schools. Creating mathematical models with biomedical purposes may help us to understand how the human body works,” says Marcano. Computational mathematicians create mathematical models that simulate real life situations by starting with a series of equations that describe the phenomenon or physical conditions being studied. This description does not necessarily reflect the reality of what is being modeled because it must often be simplified. “Many experimental measurements taken by biologists and physiologists in laboratories do not fully describe what happens in the body, because they are done in vitro. For example, if one wishes to study the permeability of a kidney cell membrane, it must be removed from the kidney and placed on a pipette before any experiments can be performed. In the process, the membrane loses some characteristics, and some factors that affect its function in life may disappear. My work can show whether or not this in vitro technique causes any errors in the results.
“Perhaps with mathematical models we may introduce some characteristics that are difficult for researchers to observe and study in a laboratory. This is why I try to create comprehensive, all-embracing, complete models. Mathematicians who do this type of work attempt to look at the whole puzzle at once,” says Marcano.
Mathematical models of the kidney have been around for more than 30 years, but thus far they have not been able to generate urine that is as highly concentrated as is naturally produced by mammals. Marcano has been able to test his models with data from bird kidneys, which have just one internal medulla. The bird medulla, although much simpler, is very similar to the external medulla of a mammal, which possesses both internal and external medullae. Knowledge about the external medulla of mammals has allowed Marcano to test his models on the medullae of bird kidneys and compare the results with those obtained by biologists and physiologists who study bird kidneys in laboratories.
In his latest research, Marcano and his colleagues from Duke University’s Department of Mathematics, Harold E. Layton, Ph.D. and Anita T. Layton, Ph.D., solved a problem for a mathematical model that demonstrates how morphological characteristics and the permeability of tubules in bird kidneys affect the UCM. This study may shed some light on the relationships between the tubular and vascular elements that participate in the UCM and how they function.
“Creating a mathematical model is like preparing a recipe in a creative kitchen,” says Marcano. “The ingredients already exist; scientists just need to combine them in the right way.” The problems that Marcano solves usually consist of three phases. First, he proposes the model. Then he studies the equations that comprise the proposed UCM model. Lastly, he presents the model and shares his results with the scientific community, so they can apply and interpret it.
In the second phase, Marcano applies mathematical algorithms to solve the equations that describe the model. The values of the parameters that Marcano used for his most recent research came from experimental data obtained by biologists in laboratories. Because these results were reported in uncertainty ranges, Marcano decided to employ an alternative mathematical approach: the inverse problem. This method allows him to identify parameters needed for a desired solution and study errors that occur when forming the model.
In Marcano’s research, the inverse problem allowed him to systematically explore various parameters until finding the closest possible solution to the results reported by scientists. “Otherwise, finding this group of parameters would have been an exhaustive process, because we would have had to vary them little by little until reaching the solution reported by experimental scientists,” explains Marcano. He decided to formulate his inverse problem as a non-linear optimization problem, which maximizes (highest concentration of urine) or minimizes (lowest concentration of urine) the water retained by the organism during droughts.
Marcano’s research has technically advanced the application of optimization methods to more complex problems and has confirmed the validity of experimental data on birds. “Our models make general predictions that are consistent with previous physiological studies of the UCM of birds and with earlier studies that worked on modeling the UCM of quails. This means that the models and technique work, even when they have been simplified because of the organ’s complexity. “My goal is to use more sophisticated models, which include more of the details that are present in these processes and actually occur in the kidney; but when we increase complexity, we must also develop the mathematical technique and find more computational resources at the same time. The long-term goal of this project is to achieve a comprehensive three-dimensional mathematical model of the UCM of the mammalian kidney,” says Marcano.
Illustration used with permission from the American Physiology Association (Braun, E.J. and Dentzler, W.H. "Function of mammalian-type and reptilian-type nephrons in kidney of desert quail") 222, 617-629, 1972.
When results of Marcano’s research came to the attention of the organizers of the Renal Physiome Project, they asked Marcano to collaborate in exploring the viability of developing a series of Internet tools. These tools would facilitate the use of the inverse problem technique to help scientists analyze biophysical models of membrane cotransporters. The long-term goal of the RP Project—created by an international community of researchers interested in modeling renal physiology—is to create an interactive Web page that will allow renal physiologists and nephrologists to utilize mathematical models in experimental designs and analysis.
“If this effort is completed, the results of our research will be available to researchers who work with kidneys. Also, it will demonstrate the power of inverse methods for analyzing biological models,” says Marcano.
mmarcano@cnnet.upr.edu
http://www.necker.fr/kidneysim/index.php
