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The relationship between mathematics and biology

  1. Aug 4, 2007 #1

    I have been interested in the relationship between Mathematics and Biology for a while. I recently read Richard Feynman's "The character of physical law", which features a segment about this relationship:

    "In biology, for example, the action of a virus on a bacterium is unmathematical. If you watch it under a microscope, a jiggling little virus finds some spot on the odd shaped bacterium - they are all different shapes - and maybe it pushes its DNA in and maybe it does not. Yet if we do the experiment with millions and millions of bacteria and viruses, then we can learn a great deal about the viruses by taking averages. We can use mathematics in the averaging, to see whether the viruses develop in the bacteria, what new strands and what percentage; and so we can study the genetics, the mutations and so forth."

    I have some questions about this; any input is much appreciated!

    1. Feynman says that the action of one virus is not mathematical. Is this to say that the behavior of the individual virus is chaotic, or could it be understood by other, non-mathematical principles?

    2. Feynman says that mathematics is important in understanding genetics and mutations. Would it be correct to say that these fields have been reduced to information sciences best described by computers crunching numbers (I guess Ray Kurzweil would hold that view) or are they too complex to be described by computational models?

    Thank you!

  2. jcsd
  3. Aug 4, 2007 #2
    Feynman is talking about the importance of statistical analysis in biology. For example, forget the virus and take a single man. This man may find a woman to reproduce with and have kids, and then again he might not (and chances are high if he is a scientist that he has trouble finding a mate). The point is that you can't study the behavior of the human population as a whole by looking strictly at the behavior of any one man. You can only make generalizations by considering the behavior of a great many men, and then speak of the subject of reproduction and population growth (it should be noted that Feynman doesn't limit this view to strictly biology, he speaks about statistical mathematics at great length as a necessary ingredient of modern physics as well).

    It is important, but NOT all encompassing. The subject of genetics for example was understood in a primitive state in the early 1900's by breeding lots of animals together and statistically analyzing the observable characteristics on the children to be sure (the classical "Mendelian" approach). However, this doesn't necessarily lead you to the conclusion that a molecule inside the nucleus is responsible for this nor does it help describe how such a molecule behaves. There is much to genetics that has yet to be described in a satisfactory manner by math.
  4. Aug 8, 2007 #3
    To respond to your first question:

    Feynman's point should be taken along the lines that: individual situations have an immense amount of variation and as such individual behavior and observed activity does not generate enough data to form a model for virus behavior; however, over a large enough span of time enough instances occur that statistical methods can be used to create a model of generalized behavior in biology.

    It isn't that virus behavior is chaotic, it is that there are too many variations to deal with each virus's individual behavior mathematically. It is much akin to saying we could in principle attempt to determine something about the behavior of an ideal gas by computing all of the forces and motions of all of the atoms in a mole of gas, or we could treat the gas as an ideal gas and be done with it.
  5. Aug 13, 2007 #4
    Mathematics and Biology - on various levels

    The relationships may be found on various levels.

    The environment is extremely complex - we can neither know nor even hope to know all the details of microscopic structure of the cell, all the chemical compounds moving within it and interacting. Yet, there is little doubt today that there is no special elan vitale, and that chemical interactions are governed by `normal' chemical - which is physical - laws, and these are well described by mathematics. (Which is pretty surprising in itself: why does mathematics seem to describe our world so well...)
    So, I guess that to say that the virus' actions could be better described by other non-mathematical principles is a bit far fetched. Unless by description we mean just a general, well, description.

    The situation changes (?) when we move to higher organisms, where the language of purpose, planning and eventually thinking seems to be much better that attempts at mathematizations.

    Since Feynman's time the advances in understanding just how complex is the heredity, regulation and mutation/selection process have been enormous, taking into account development mechanisms, aspects supplementing DNA--> proteins information transfer (such as regulation by existing proteins) etc. etc.

    As a result it is not that genetics has been reduced to information sciences. On the contrary: information sciences in XXI Century will benefit from our progress in understanding biological processes. Eventually, yes, mathematics will play a crucial role, but my guess is that we shall discover similar surprises of complexity and beauty as were found when people first `discovered' fractals.

    (BTW: did fractals exist before people tried to run their little computers to see Mandelbrot set or Julia Set?)

    At the lowest level: there are quite interesting theories (I do not say I believe them all, but who knows), which state that at the basic level all Universe is `computational' The most widely known example is the New Kind of Science of Wolfram, where he postulates that all we find in the world is somewhere a cellular automaton. Far below quarks and Planck scale. I do not believe it, but if this would be true that all the world would be mathematical to the core.

    See me on http://countryofblindfolded.blogspot.com/
  6. Aug 13, 2007 #5
    I hold this to be false. If the initial state of the bacteria is x, and the transformation on the bacteria as a single system (e.g., the set of transitions) by the action of the virus is T, then one can then deduce the successive state values (the trajectory) of x as a series process, which is mathematically called integrating the transformation. When the transformation is a set of differential equations the process is called solving the equations. Nothing more than basic cybernetic theory. Clearly I have no idea what Feynman was taking about, all biological behavior (from the simple system such as a bacteria cell to the ecosystem) can be reduced to the mathematics of cybernetic science.
  7. Aug 14, 2007 #6
    Well, the problem is to know the initial state x. And although in theory we could write a more or less complete set of equations (say classical, nonrelativistic Schroedinger equations for all the protons, neutrons and electrons forming molecules of the virus, the bacterium and some of the nearest surroundings, the size of the dataset (dimension of the Hilbert space) would be enormous. Yest, we can deal with 10^23 particles in, say, solid state physics, but only by using Bloch theorem, which reduces the size of the problem by assuming translational symmetry. In the living cell there are no such props. If we talk about 10^19 atoms, 10^20+ particles, and if we want to solve it through rigorous equations, from state x through transformations T, we have to take it all into account. In this sense, biology is `theoretically' physical and thus mathematical.

    Of course one can construct simpler equations, for statistical description of processes or for simplified description of some biological processes (famous example: equations describing propagation of electric spikes in neurons). But these simplified equations are, as I said, CONSTRUCTED: one picks some properties and tries to describe them mathematically. And thanks to enormous wealth of mathematics and its tools (differential equations, PDE, integral equations, Markov processess, well... too many to think of) some sort of `fitting' would be found.

    But as to whether a particular virus behaviour versus particular cell can be PREDICTED by mathematical tools - deducing subsequent steps from the initial state, this would require not only knowing this state with arbitrary precision but also knowing how to solve QM equations for a `dirty, hot and wet system'.

    best regards,
    see me on http://countryofblindfolded.blogspot.com/
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