Evolution and Mathematics: Help Needed!

Factual Questions
1

The matrix I need help with has 20 entries and because I can’t include a gif here it can only be seen at this url:

http://www.geocities.com/jorolat/matrix.html

Below is a copy of the info I’ve put on the page, hopefully it will be sufficiently interesting for someone to look at it and give me advice! :slight_smile:

“Cytochrome C is a protein about 100 amino acids long thats occurs in organisms ranging from bacteria to man. The matrix below is obtained by comparing the number of differences there are in this amino acid sequence for a number of different organisms.”

(matrix appears here)

"Mathematically I need to know:

Finding the average value for bacteria (dark green) seems an obvious case of adding the values up and dividing by 20. To find the average value for birds (light blue) do I add up the values and divide by 9? Is this mathematically valid?. If so what do I do about the values for mammals (white) where organisms are compared against one another? Do I include the zeroes and divide by nine or exclude them and divide by 6?

Biologically I need to know:

The average value for bacteria (as I’m currently calculating it) is about 65, that of yeasts 42, flowers 43, insects 25, fish 16, reptiles 10, and mammals either 4 or 6.

I’m looking for a general trend rather than mathematical exactness and for this exercise I’ve included the lamprey with the other fish and the snapping turtle with birds. How great a heresy is this? (after all, it’s only a small database).

The reason I’m doing this exercise is because the values for the groups, after factorizing, are (bacteria first, and very approximately cos I don’t have the actual results to hand):

13, 8.2. 8.1, 5.2, 3.4, 2, 0.8 (note how similar yeasts and plants are)

This may be an occurence of the fibonacci series which is potentially significant for a project I am working on (the possible existence of a testable internal evolutionary mechanism) and could explain, as an example, why the proto-whale evolved into the whale rather than devolving to any earlier evolutionary stage.

The matrix is taken from M. Denton’s book “Evolution: A Theory In Crisis” (Secker & Warburg, London) and originally appears in “Dayhoff: Atlas of Protein Sequence and Structure”. It is interesting to note that Denton emphatically states the matrix doesn’t support evolution because there isn’t any linear relationship between the groups.

While I’m at it, as long as a reference is given, is it ok to reprint a single table from a book without seeking permissions?.

I would be very grateful for help in any of the above areas!"

Regards,

Jorolat

2

Well this is pathetic. Can’t even get an answer at SDMB/GQ.

Jorolat, you shouldn’t include the diagonal, so for mammals, the average is 20/3.

But remember, what you are calling “birds” (light blue) is really birds compared with mammals. What you are calling mammals is mammals compared with mammals. And the yellow rectangle is snapping turtles compared with birds and mammals together.

Not sure what he means by this. maybe someone else will clarify, once this is back to the top.

3

I am not sure what would be accomplished by this. Why are you comparing R. rubrum to everything else, but comparing mammals only to other mammals and birds only to other birds and mammals? Also, you are assuming that the pre-existing classifications are in fact true.

What evolutionary biologists generally do with these kinds of matrices is to do a cluster analysis using all pairwise comparisons. For example, compare the horse not only to the dog and the kangaroo, but also to everything else in the matrix. As ZenBeam said, you shouldn’t include the diagonal in the comparison. I won’t bore everyone with the details, but the final result can be portrayed as a dendrogram, which generally a rough approximation of the evolutionary tree.

I am also confused by the quote

Why would you necessarily assume that there is a relationship between the groups in the first place?

4

Oops! Please insert “linear” into my last statement:

Why would you necessarily assume that there is a linear relationship between the groups in the first place?

5

Hi Zenbeam,

Thank you for replying. Though this subject matter is unfamiliar to me I do realize that these are comparitive values.

Denton states:

“However, the most striking feature of the matrix is that each identifiable subclass of sequences is isolated and distinct. Every sequence can be unambiguously assigned to a particular subclass. No sequence or group of sequences can be designated as intermediate with respect to the other groups. All the sequences of each subclass are equally isolated from the members of another group. Transitional or intermediate classes are completely absent from the matrix.”

Equally isolated (allowing for a slight spread) means, as an example, that:

silkworm and horse = 27 (mammal)
pigeon = 25 (bird)
turtle = 26 (reptile)
carp = 25 (fish)
lamprey = 30 (jawless vertebrate)

He further states:

“Each class is isolated and unique… The classification system that is derived from these comparative molecular studies is a highly ordered non-overlapping system composed entirely of groups within groups, of classes which were inclusive or exclusive of other classes. There is a total absense of partially inclusive or intermediate classes, and therefore none of the groups traditionally cited by evolutionary biologists as intermediate gives even the slightest hint of a supposedly transitional character.”

I hope that makes his argument clearer.

Jorolat

6

Hi Terminus Est.

Thank you for your post. I am interested in the hierarchy apparent in the matrix and in any pattern that may or may not exist between the groups.

I know very little about this subject and am just exploring the possibility that it may be relevant to an area of interest that I have. The information you have given about what evolutionary biologists do is very useful and I shall be looking into it further.

I have included quotes from Denton in my reply to Zenbeam which hopefully will answer your query about the first quote. It’s quite likely I may have misrepresented Denton’s position though I must admit I was surprised by some of his statements concerning the data.

I am interested in the possibility of a homeostatic internal evolutionary mechanism and hierarchical relationships between the groups would be consistent with this.

Jorolat

7

There is a branch of statistics called cluster analysis that allows one to classify and group together different objects, starting from the kind of matrix that you already have. I don’t have a lot of time right now to get into the details. Here’s a web page that introduces some of the methods used:

http://149.170.199.144/multivar/ca.htm

A more technical description can be found here:

http://www.statsoftinc.com/textbook/stcluan.html

I have not actually read the book in question, and I’m afraid that this could easily become a Great Debate, so I’m only going to post a link to a critique of Denton’s book without further comment:

http://www.talkorigins.org/faqs/denton.html

8

Hi Terminus Est,

Thank you for the links, I’ve read them once and will do so again, along with further reading, later in the week.

I take your point about the “debate” aspect and have thought of posting to that forum - perhaps I’ll meet you there sometime! :slight_smile:

Jorolat

9

OK, I can see what he’s getting at. The flaw in his argument is that all of these data are for currently living species. The intermediary forms would all have existed in the past, and necessarily are not included in the data.

Further, an evolutionist would expect that the farther in the past a pair of groups shared a common ancestor, the greater the number of differences. So, for example, all the animals are more closely related to each other than any of them are to the plants, indicating the animals shared a common ancestor with one-another more recently than they and the plants shared a common ancestor. The data in the matrix are consistent with evolution.

10

Hi Jorolat, I understand your position for Lamarkian evolution. Does the argument against transitional species support Lamarkian evolution?

11

Hi ZenBeam,

I agree that the data supports evolution though in fairness must say that there is more depth to Denton’s argument than I can convey without extensive quotes. For example he argues that the degree of divergence is not consistent for different proteins etc., etc…

Jorolat

12

Hi PosterChild,

I’m not sure if I’ve said the first part of the following before so bear with me if I have:

My interest in evolution arose out of certain “anomalies” found in psychology. These anomalies, two characteristics of which are equilibrium and interaction with “DNA”, were occuring below the level at which trauma occurs and therefore not related to psychology as such.

The apparent connection with “DNA”, however, led me to look at evolution to see if there was any interest in, or evidence for, an internal evolutionary mechanism based on an extension to homeostasis.

From my perspective, rightly or wrongly, it was like having an answer but not knowing exactly what the question was.

The first thing I saw a possible connection with was the phenomena of stationary phase mutations followed by the Baldwin Effect. The results of this are what you have seen on my website.

More recently someone suggested that my “take” on evolution was remarkably similar to that of Jean Piaget except that I use different terminology. I looked him up on a search engine and was immediately intrigued by references to equilibrium and a little more digging revealed his interest in the Baldwin Effect. I’m trying to get hold of some of his books at the moment.

I’m interested in a possible and testable homeostatic internal evolutionary mechanism. Such a mechanism is indirect, as is the Baldwin Effect itself, and so I don’t “support” Lamarckian evolution at all. I entitled an earlier post “Evolution: Two Intriguing Lamarckian Experiments” simply because they had previously been described that way and people recognize "Lamarckian.

On the other hand I am interested in what Lamarck actually did say and have a copy of:

*"ZOOLOGICAL PHILOSOPHY

OR

EXPOSITION

Of the considerations relevant to the natural history of animals; to the diversity of their organic structure and of the faculties which they derive from it; to the physical causes sustaining life in them and producing the movements which they carry out; finally, to those causes which produce feeling in some and intelligence in others endowed with it.

By J.-B.-P.-A. LAMARCK"*

which I will reluctantly plough through soon.

The data in the matrix is entirely consistent with the mechanism I am interested in and I was delighted to find it. Over evolutionary time such a mechanism is bound to be hierarchical at the molecular level though until I came across amino acid sequencing I was not sure how it would manifest itself.

My specific interest in a possible occurence of the fibonacci series is because, in it’s “pure” form, one way a homeostatic mechanism can work is by making “jumps”:

Take organism x in equilibrium. Evolutionary changes occur, some are quickly lost (a la fruit fly subjected to heat shock), but generally they become accumulative.

When the degree of changes nears, or excedes, phi (“golden angle”) equilibrium can no longer be maintained so the changes are integrated into the start state and the whole process begins again.

Externally this may result in the fibonacci series being expressed but internally there has been no change over (evolutionary) time and homeostasis = “staying the same”.

I hope that makes sense though, of course, you’re bound to disagree with it ‘sigh’ [ :slight_smile: ]

Jorolat