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  1. #1

    Default Seeking help from people in Biology and related fields/majors

    To those in Biology and related fields/majors:

    I am looking for both the mathematical and historical background for the following sub-fields that lie at the intersection of mathematics and biology...

    • The work of R.A. Fischer on population biology. He invented the ANOVA, as well as the modern notion of "Design of Experiments." I would like to gain some insight into the thought processes that led to these inventions.
    • The inference of gene function through "homology." I would like to know what it means for things to be homologous in a mathematical sense, and how exactly, gene functions are inferred from homology. As well as, perhaps, a brief history of how this sub-field developed.
    • The study of gene sequences in relation to evolutionary trees. It is my understanding that the modeling here is mathematically intensive. I believe this area of genetics is called coalescent theory. Here I would like a good primer on the biology involved, and perhaps the (possibly mathematical) definitions of things like "gene", "allele", "genetic drift", "gene flow", "population structure," and other such things.
    • Cellular Electrophysiology. It is my understanding that this amounts to treating lines of electro-active (a.k.a. nerve?) cells as transmission lines like in electrical engineering (but with lots of discontinuities). But I would like to learn the actual physiology to make sure this conception is accurate.
    • Medical Imaging Technology. As I understand it, the signal processing done in this domain is among the trickiest forms of signal processing. For instance you get a bunch of resonances and you need to infer from those signals what was actually on the inside. Any sources on the algorithms used for MRIs and CTs would be very much appreciated.

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  2. #2
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    I'm not quite sure what you're looking for here, but will try to do the best that I can.

    With regards to ANOVA, I didn't do any research at all into its history and the only background that I have in it is statistics. I believe population geneticists probably use it more. As far as I remember, it's a test of significance. i.e. the probability with which the null hypothesis can be rejected. Calculating the Z-values/p-values using a t-test enables people to determine if values are significantly different from each other, and therefore draw conclusions. The analysis takes into account the sample size and therefore a smaller difference is required for significance to be reached if the sample size is larger. i.e. An average difference of 2 may be significant in a population of 1000000, but the average difference observed in a population of 100 may need to be much bigger to achieve statistical significance. There are various assumptions at play here, which I won't go into because I'm not a statistics major and would only embarrass myself by speaking in general terms.

    The term "homology", like "species" and "gene", carries several different definitions and emphasis according to which biological subfield you're from. I am not sure what you are asking. If you're talking about evolutionary biology, the definition of the terms above would be very different than if you were talking about genetics or ecology.

    Upon further thought (tell me if I'm wrong) you may be talking about conserved sequences of genes through evolution. Is that right?

    It's quite simple, in that case. People find that a particular gene, when cloned, transcribes a particular protein. They name the protein. They create knockouts in other species, e.g. mice. They find that a phenotype is observed, consistent with the speculated function of the protein. They do a search across genomes mapped in various different species and find that previously un-assigned stretches of sequences that originally had no known function are very similar in sequence. i.e. mathematically the chance that they were so similar by random chance is very very small. They call these genes "orthologs" of the original gene that was characterised, and assume that they produce a similar protein with similar function. "Homologs" are the same allele of the same gene in the same species, and when an individual carries 2 copies of the same allele, they are "homologous" at that particular gene locus.

    I didn't study evolutionary biology or genetics in detail (zero interest) but can give you a general outline of the terms as they are used in genetics.
    "gene" = DNA that encodes a protein, is transferred from generation to generation (how is outlined in the central dogma of molecular biology).
    "alleles" = different forms of the same gene that can cause the development of an abnormal/different phenotype.
    "genetic drift" = changes in allele composition in a population, i.e. not following mendelian genetics. This can be due to several reasons, including (but not limited to) selection.
    "gene flow" = transfer of alleles between populations.

    It's really difficult to explain all of this without context, or the assumptions/conditions of the above definitions. But if I was going to type it all out, it would be like conducting a basic genetics course over the Internet, and I'm not going to do that. Also note that the above definitions are actually highly artificial constructs... Whether the modeling has any relation to reality is highly debatable.

    I don't have any background at all in the last 2 fields, sorry.

  3. #3
    ish red no longer *sad* nightning's Avatar
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    On cellular electrophysiology... I have the basic class note taken during an undergrad course. Unfortunately the class did not use a textbook so I can't just refer you to something direct. The action potential is described by Hodgkin and Huxley's equation. If you can understand what goes into the equation, then everything else should follow.
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  4. #4

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    First off, thanks for the responses. I didn't really expect to get a "course" in this thread. However, if you have pointers to good sources on the subjects you mentioned, I would greatly appreciate it.

    Quote Originally Posted by nonsequitur View Post
    I'm not quite sure what you're looking for here, but will try to do the best that I can.

    With regards to ANOVA, I didn't do any research at all into its history and the only background that I have in it is statistics. I believe population geneticists probably use it more. As far as I remember, it's a test of significance. i.e. the probability with which the null hypothesis can be rejected. Calculating the Z-values/p-values using a t-test enables people to determine if values are significantly different from each other, and therefore draw conclusions. The analysis takes into account the sample size and therefore a smaller difference is required for significance to be reached if the sample size is larger. i.e. An average difference of 2 may be significant in a population of 1000000, but the average difference observed in a population of 100 may need to be much bigger to achieve statistical significance. There are various assumptions at play here, which I won't go into because I'm not a statistics major and would only embarrass myself by speaking in general terms.
    That's OK. I am well versed in the various ANOVAs and the Design of Experiments. I was just hoping to get a better idea of what Fisher was actually working on when he invented these things.

    BTW, t-tests work on means. ANOVAs work on variances.

    I find these types of history lessons very enlightening…like reading the Principia, The Origin of Species, Edison's Diary, Tesla's autobiography, Watson's and Crick's independent accounts of the events that led to the discovery of the structure of DNA, etc.

    Quote Originally Posted by nonsequitur View Post
    The term "homology", like "species" and "gene", carries several different definitions and emphasis according to which biological subfield you're from. I am not sure what you are asking. If you're talking about evolutionary biology, the definition of the terms above would be very different than if you were talking about genetics or ecology.

    Upon further thought (tell me if I'm wrong) you may be talking about conserved sequences of genes through evolution. Is that right?
    I believe this is pretty much the question I was asking. But I am such a n00b to these things that I have a hard time making the questions more crisp.

    Quote Originally Posted by nonsequitur View Post
    It's quite simple, in that case. People find that a particular gene, when cloned, transcribes a particular protein. They name the protein. They create knockouts in other species, e.g. mice. They find that a phenotype is observed, consistent with the speculated function of the protein. They do a search across genomes mapped in various different species and find that previously un-assigned stretches of sequences that originally had no known function are very similar in sequence. i.e. mathematically the chance that they were so similar by random chance is very very small.
    Ah, thanks. I believe that was the insight I was looking for. If you have pointers to the actual math/algorithms used, would really appreciate it.

    Quote Originally Posted by nonsequitur View Post
    They call these genes "orthologs" of the original gene that was characterised, and assume that they produce a similar protein with similar function. "Homologs" are the same allele of the same gene in the same species, and when an individual carries 2 copies of the same allele, they are "homologous" at that particular gene locus.
    This actually confused me a bit, because my own research turned up different meaning from what you listed. The meaning of Orthologs pretty much matches what I read, I think. But I thought Homologs referred to general similarity through common ancestry…and that what you were calling Homologs were called "paralogs". Perhaps it depends on the source defining the words?

    Quote Originally Posted by nonsequitur View Post
    I didn't study evolutionary biology or genetics in detail (zero interest) but can give you a general outline of the terms as they are used in genetics.
    "gene" = DNA that encodes a protein, is transferred from generation to generation (how is outlined in the central dogma of molecular biology).
    "alleles" = different forms of the same gene that can cause the development of an abnormal/different phenotype.
    "genetic drift" = changes in allele composition in a population, i.e. not following mendelian genetics. This can be due to several reasons, including (but not limited to) selection.
    "gene flow" = transfer of alleles between populations.
    I did take a basic Biology course, and most of what you listed seems like review (I had just forgotten).

    "Gene flow," however, is new.

    When you say: "transfer of alleles between populations," do you mean transfer of alleles from one generation to the next, or a population at one instance in time, and the population at another instance in time (that is much of the 1st population may be re-represented in the second.)?


    Quote Originally Posted by nonsequitur View Post
    It's really difficult to explain all of this without context, or the assumptions/conditions of the above definitions. But if I was going to type it all out, it would be like conducting a basic genetics course over the Internet, and I'm not going to do that. Also note that the above definitions are actually highly artificial constructs... Whether the modeling has any relation to reality is highly debatable.
    Thanks for your effort nonetheless. If you know of good sources, please let me know.

    As far as the reliability of the models, I figured as much considering how much the models leave out. Wikipedia says the simple models leave out:
    recombination, natural selection, and gene flow or population structure. Even though I don't know what recombination and population structure really are, that list seems like a lot to leave out.

    It may be a pipe dream, but part of the reason I am asking these questions is to see if I can use my applied math skills and programming skills (and perhaps some physics) to develop new and useful models.

    Quote Originally Posted by nightning View Post
    On cellular electrophysiology... I have the basic class note taken during an undergrad course. Unfortunately the class did not use a textbook so I can't just refer you to something direct. The action potential is described by Hodgkin and Huxley's equation. If you can understand what goes into the equation, then everything else should follow.
    Ah. It is certainly not a traditional transmission line like model since it has built in voltage and current sources. And no inductive or capacitive components (though the gating mechanisms seem like they creates implicit capacitances).

    It also seems like this model does a good job since it is (according to Wikipedia at least) "widely regarded as one of the great achievements of 20th-century biophysics."

    Well, the only thing not touched on are the Medical Imaging Algorithms. Hopefully, someone else can address that.

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  5. #5
    insert random title here Randomnity's Avatar
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    I graduated as a bio major (BSc), but I try to stay away from math as much as possible (I am more into the cell/tissue/whole animal level than the molecular/genetic level). It seems like what you're asking is beyond the scope of what I've learned in those areas.

    Honestly I hated all that stuff when I had to take it, too. I'm far more interested in more tangible things. Hence, the bio major.

    You can probably learn the basics for biostatistics/ecology and neurophysiology from various textbooks or even online (wiki is great for understanding fundamentals). For more in-depth topics (esp the MRI stuff - though it's possible that the newer stuff is proprietary?) I would research online journals (ie pubmed) or even look up profs in the area at local colleges; I'm sure many would be happy to talk briefly with you about their research, especially if you've researched the area previously.
    -end of thread-

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    I was tought when an allele pair are the same it's homozygous and when they're different it's heterozygous.
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    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    FYI, the specialization for my MS was in Math Ecology, which deals with population models (mostly predator-prey models is what I studied). That doesn't sound much like what you are interested in here though, and I make no claims to having any exceptional knowledge in biology whatsoever.
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    Senior Member Scott N Denver's Avatar
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    Isn't MRI and other stuff all based on Radon transforms, which are a type of INtegral transform? IN particular the Radon trasnform of f(x) is the integral of f(x)?

    [runs off to grab copy of Lokenath Debnaths Integral Transforms text...]

    Yes..starting on page 539 of ITaTA 2nd edition, X-ray tomography, x-rays,. CAT...

    Yes, radon transform is integral [over path length] of f(x,y) "In other words, the totality of all these line integrals constitutes the radon transform of f(x,y) and each line integral is called a sample of the Radon transform of f(x,y).

    See, now don't say that PDE's and their solutions methods never did anything for you...

  9. #9
    ish red no longer *sad* nightning's Avatar
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    Quote Originally Posted by ygolo View Post
    Ah, thanks. I believe that was the insight I was looking for. If you have pointers to the actual math/algorithms used, would really appreciate it.
    Search for related sequences in genome database (BLAST is a commonly used search tool. Which just comparisons how similar your sequence is compared to stretches of sequences in its database. That's the only part that I can think of requiring algorithms. The site might tell you more about the algorithms they use...

    When you say: "transfer of alleles between populations," do you mean transfer of alleles from one generation to the next, or a population at one instance in time, and the population at another instance in time (that is much of the 1st population may be re-represented in the second.)?
    From my understanding... populations here specifically means 2 groups of organism who normally do not interact with one another (aka does not interbreed). However since you imply allele transfer can happen, likely some geological barrier prevented interbreeding. So transfer of alleles is most likely to occur when one member from population A, somehow ends up in the territory of population B, thereby carrying the new allele over and passing it to its offsprings. If the geological barrier is breached (members can freely cross over), then the alleles mixing is more rapid until you get a single homologous population.

    Ah. It is certainly not a traditional transmission line like model since it has built in voltage and current sources. And no inductive or capacitive components (though the gating mechanisms seem like they creates implicit capacitances).
    My bad. The membrane is inherently a capacitor. It's a phospholipid bilayer. You can also further increase capacitance by increasing the thickness of the myelin wrapping (which is just many layers of membrane) around the axon.

    The resistor capacitor "cable" model is used for passive conduction, which occurs in parts of myelinated membranes. The well insulated section carries signal passively until you hit the Node of Ranvier where action potentials are used to boost up the signal then you get another section of myelinated axon etc.

    It also seems like this model does a good job since it is (according to Wikipedia at least) "widely regarded as one of the great achievements of 20th-century biophysics."
    *nods* Pretty much everything in neurophysiology is based on the H-H model. Current work in the field seems to be more focused on channel physiology (looking at the properties of individual channel subtype. There's mathematical modelling involved there, but mostly focused on modelling channel/protein structure in predicting channel current behaviour.) or effect of drugs on electrophysiology (through what mechanism are they altering electrophysiology). You might want to look into local anaesthetics if you have time... they block sodium channels to prevent action potentials. The charged quaternary local anaesthetics are a hot topic right now. Under the conventional theories, they should not work... but in practice, they do.
    My stuff (design & other junk) lives here: http://nnbox.ca

  10. #10

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    Quote Originally Posted by Scott N Denver View Post
    Isn't MRI and other stuff all based on Radon transforms, which are a type of INtegral transform? IN particular the Radon trasnform of f(x) is the integral of f(x)?

    [runs off to grab copy of Lokenath Debnaths Integral Transforms text...]

    Yes..starting on page 539 of ITaTA 2nd edition, X-ray tomography, x-rays,. CAT...

    Yes, radon transform is integral [over path length] of f(x,y) "In other words, the totality of all these line integrals constitutes the radon transform of f(x,y) and each line integral is called a sample of the Radon transform of f(x,y).

    See, now don't say that PDE's and their solutions methods never did anything for you...
    Thanks. I just ordered that book. Never worked with Radon Transforms before.

    Also, I have found PDE's and their solution methods to among the most useful of things in mathematics. Thinking in the Laplace domain is second nature for most EEs. Most of the math I have learned has been useful.

    Quote Originally Posted by nightning View Post
    Search for related sequences in genome database (BLAST is a commonly used search tool. Which just comparisons how similar your sequence is compared to stretches of sequences in its database. That's the only part that I can think of requiring algorithms. The site might tell you more about the algorithms they use...
    Thanks nightning. Their interface isn't intuitive at first glance. Maybe when I get time, I'll try and decipher it. Work has been kicking my butt, lately.

    Quote Originally Posted by nightning View Post
    From my understanding... populations here specifically means 2 groups of organism who normally do not interact with one another (aka does not interbreed). However since you imply allele transfer can happen, likely some geological barrier prevented interbreeding. So transfer of alleles is most likely to occur when one member from population A, somehow ends up in the territory of population B, thereby carrying the new allele over and passing it to its offsprings. If the geological barrier is breached (members can freely cross over), then the alleles mixing is more rapid until you get a single homologous population.
    Ah, so is it that the null hypothesis is that the two populations have no inbreeding, and we know roughly what to expect in terms of genetic similarity in that case. But if there is significantly more genetic similarity than expected, then we say there is gene flow?

    Quote Originally Posted by nightning View Post
    My bad. The membrane is inherently a capacitor. It's a phospholipid bilayer. You can also further increase capacitance by increasing the thickness of the myelin wrapping (which is just many layers of membrane) around the axon.
    Actually, there is a membrane capacitance in the H-H model. I just missed it the first time I looked.

    However, it is really strange for capacitance to increase when the space between the two sides increases. Capacitance is usually proportional to area and inversely proportional to the distance between the two "plates."

    Quote Originally Posted by nightning View Post
    The resistor capacitor "cable" model is used for passive conduction, which occurs in parts of myelinated membranes. The well insulated section carries signal passively until you hit the Node of Ranvier where action potentials are used to boost up the signal then you get another section of myelinated axon etc.
    So there is passive conduction along the myelin sheath of axons, that can be represented as a lossy transmission line at low frequency transmission (I am assuming low frequency because there is no inductance in the model).

    The H-H model, in contrast, seems to model the flow of current from outside the cell membrane to inside it. It has a capacitance (lipid bilayer itself), several variable conductances (ion channels), several current sources (ion transporter), and a fixed conductance (caused by nodescript "leakage")

    I solved the (1st order non-homogeneous) differential equations for activation and inactivation. They were simple exponential decays. But solving for the voltage across the membrane as a function of time (I suppose this is the "action potential") seems quite a bit trickier. I think if I sat down and did it using Laplace or Fourier Analysis it could be straightforward….but those alpha and beta coefficients look scary.

    I presume the hard part of this form of modeling is empirically fitting the model to various types of nerve cells.

    Also, I am not sure how the two models fit together. Is it that the H-H model for one region is connected by a small loss (resistance) to the H-H model of another region?

    Also, what about inside the membrane? Is the inside of the membrane supposed to be thought of as a conductor (i.e. as being at the same potential), or do you model the inside as a cable with losses as well?

    Do you know of any good sources to flesh out the electrophysiology?

    Quote Originally Posted by nightning View Post
    *nods* Pretty much everything in neurophysiology is based on the H-H model. Current work in the field seems to be more focused on channel physiology (looking at the properties of individual channel subtype. There's mathematical modelling involved there, but mostly focused on modelling channel/protein structure in predicting channel current behaviour.) or effect of drugs on electrophysiology (through what mechanism are they altering electrophysiology). You might want to look into local anaesthetics if you have time... they block sodium channels to prevent action potentials. The charged quaternary local anaesthetics are a hot topic right now. Under the conventional theories, they should not work... but in practice, they do.
    Interesting that so much research goes into ion channels. Do you mean ion channels of ion pumps…because you mentioned the action of proteins.

    The gating potentials of the ion channels seem like they would follow the Nernst equation from electrochemistry. Aren't they basically concentration cells?

    Beyond that it seems like modeling the variable conductances of the ion channels would be challenging because there are six parameters to fit.

    I suppose the anesthetics actually block the action potential by disrupting (the previously constant?) concentrations of ions. So one of the fixed parameters of the H-H model now becomes variable? IDK. I am stretching my limited understanding at this point.

    As far as the specific type of anesthetic you mentioned, I wasn't able to find anything useful on the web. Could you point me to an introductory source?

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