Posted by: khuang01 | October 8, 2008

Math and the Origins of Life

As a follow up to Elena’s post on symmetry in evolution, I saw this interesting article in the most recent issue of the Scientific American:

http://www.sciam.com/article.cfm?id=can-math-solve-origin-of-life

The article talks about how Prof. Martin Nowak of Harvard is using mathematical equations to model theoretical chemical systems. The hope is that by playing around with the math enough, we can understand the essentially elements of a “primordial soup” that would need to give rise to basic life.

So far, the results are very promising, essentially establishing that basic, well-understood chemical reactions can lead to such behavior as competitive selection, cooperation between molecules to increase “survival” chances, genetic diversity, and mutation. But the caveat to all of this is that all the work is theoretical, chemists have been unable to construct such a system in real life to study.

Prof. Nowak’s research represents a very important part of mathematics in science: Math can give us certain knowledge beyond what we can learn from experiments. Because mathematics operates on the fundamental principles of logic, any mathematical deductions one makes from a given set of assumptions are logically infallible as long as your assumptions hold.

But this does not mean that math is a replacement for empirical evidence by any means. Using Nowak’s research as an example, he must have made certain assumptions about the way molecules interact in order to produce his models. So if those assumptions hold, Nowak’s conclusions cannot be wrong (in the same way that 2 cannot equal 1). But even though we have Nowak’s model, empirical evidence is still crucial in both validating those assumptions and possibly revealing new knowledge about the origins of life (new knowledge which might be logically consistent with the model but not predicted by it).

It is this powerful application of mathematics in science that allows for great discoveries before we have the technology to empirically test them (Einstein’s theory of relativity is a great example). And it is another example of why math is not just for mathematicians.

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Responses

  1. Okay, I agree with you Mr.khuang01 and also of course to Prof.Nowak for his statement about the importance of mathematics in treating physical phenomenons. We must remember again that the models of mathematics which have been commonly used to the purpose are in differential equations (DE’s), including both ODE and/or PDE.

    I believe, although there are several establish methods for solving the DE’s, but maybe we still need to develop alternative method that perhaps it provides another solution form that giving more complete explanation about a phenomenon. Here I take two examples that reasonable to be discussed further. The first DE so-called arctangent DE that is of form:

    dy/dt=1+y^2.

    We know that the usual solution for t(0)=0 and y(0)=0 is y(t)=tan(t). But, by using a new method so-called SMT (shortened from Stable Modulation Technique) that recently posted in
    http://eqworld.ipmnet.ru/forum/viewtopic.php?f=2&t=34&start=20,
    I found it’s solution in the following form :

    y(t)=sin(2t)/[1+cos(2t)]

    You can verify that both tangent function are equal, but at t=pi/2 you will meet the equalty of 1/0=0/0. Of course, suppose L’Hospital Theory doesn’t exist until now, so we will agree that mathematics inexacly, isn’t?

    The second DE so-called BDE (Bernoulli DE) of constant coefficients that is of form:

    dy/dx+Py=Qy^n, for n not equal 1

    If you solve the BDE by the usual way, you will find it’s solution for x(0)=x0 and y(0)=y0 is in the form :

    y(x) = 1/[(1/y0^(n-1 )-Q/P )e^{P(n-1)(x-x0)} + Q/P]^(1/(n-1))

    But SMT gives in another form, that is

    y(x)=[(P/2Q)^1/(n-1)][1+tanh(-P(n-1)(x-x0)/2 +atanh((2Q/P)y0^(n-1) -1) ) ]^1/(n-1)

    Next, if we use both formula to solve mathematics model of the energy internal of black-body radiation as explained in this address: http://rohedi.com/content/view/31/26/ , the first solution of BDE gives the usual Planck’s Formula for the black-body radiation:

    U(T) = ћω / [exp(ћω/(kT) – 1]

    but, the second solution of BED gives a New Planck’s Formula in the form :

    U(T) = 0.5ћω[ tanh{0.5ћω/(kT) – iП/2} – 1]

    You know, when Einstein verified the phenomenon of black body radiation using harmonic oscillators, the usual Planck’s formula can only explain us that the energy difference between the sequence of two energy levels of harmonic oscillator is ћω. But the new Planck’s Formula will also verify that the minimum energy of the harmonic oscillator is 0.5ћω, and justifying that there are so many harmonic oscillators involved that known from П/2 as the phase differences representing between the two energy lavels.

    Of course the above of both U(T) is the internal energy of black body radiation in thermal equilibrium. Next question what ’s the form for non thermal equilibrium? Here, I give more information, that there is a possibility to include external perturbation to the differential equation representation of the above New Planck’s Formula.
    By this special feature, I believe the New Planck’s Formula can be used to explain not only disclose confidential global warming, but also in treating the experiment of regulated speed waves.

    Hopefully the above explanation useful for you all here.

    Thank you for your attention. Please correct me if there is a mistake.

    Sincerely,
    Rohedi.

  2. @rohedi

    special regard for you mr.Rohedi, first time
    for me know your eq. like:

    y(t)=sin(2t)/[1+cos(2t)]

    for tangent function

    tan(t)=sin(t)/cos(t)

    are quite equall numerically,
    but remembering about 1/0=0/0,
    hence should be a advance convention again.
    lets discuss and stay in this topic!!!!


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