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Questions about simulating diffusion gradients
The following group of questions require that you understand the video of "diffusing" numbers which was shown in the lecture on March 30th. There are videos of two runs of this program on the web site:
number_gradients_1.mov
number_gradients_1.mov
Although this may seem abstract, it is very important and questions about it will definitely be included on the next hour exam and also on the final exam.
The two key points are (CONCEPT NUMBER ONE) that in order to simulate a diffusion gradient, computer programs assign numbers to points in space, then repeatedly compare each of these numbers with the average of the numbers of the immediate neighboring points, and change each number to make it closer to the average of its neighbors.
111
242
333
Sum of the 8 neighboring points: 1+1+1+2+2+3+3+3 = 16
Average of neighboring points = 16/8 = 2.00
Which is two less than the 4 in the middle,
Therefore decrease the 4 to 3, averaging it with the average of its neighbors.
This procedure turns out to work almost as well if you only average four neighbors:
111
242
333
1+2+2+3 = 8; 8/4 = 2
Average 2 and 4 --> 3
In the long run, you get the same pattern of digits whether you count 4 neighbors, 6
neighbors in a hexagonal pattern, 8 neighbors, or 8 neighbors in which the diagonal neighbors count less. I have tried all these. You can even count 24 neighbors.
Just keep on averaging the numerical values of nearby locations, and what you get will always be identical to a diffusion gradient.
CONCEPT NUMBER TWO: If nearby cells keep adjusting ANY QUANTITATIVE PROPERTY (or any property that can be ranked, averaged and compared) then
repeated averaging of the property will create a spatial pattern that looks and behaves exactly like a diffusion gradient. This works equally well on one dimension, two dimensions, or three dimensions.
TO REPEAT: Equilibration of quantities produces gradients.
Cells can produce spatial patterns that look just like diffusion gradients, even without
any chemical diffusing from cell to cell. They only need to be able to detect some
property of the cells touching them, and compare that average with their own property.
THEREFORE: When you see a graded property (yolk granule size, pigment density, ephrin and eph concentration gradients, "positional information" of limb buds, and maybe even colinearity of hox gene expression, then don't assume that the property
must be controlled by a diffusion gradient.
Instead, try to figure out a set of nearest-neighbor responses that produce whatever graded pattern you are studying.
Incidentally, this general principle applies to any algorithm for simulating formation of any geometric pattern. The biological cause may be that cells obey the algorithm.
As soon as blood (and lymph) begin to flow through the tissues of an embryo, then
extracellular diffusion gradients will be disrupted like the smell of flowers on a windy day. Even without blood flow, diffusion is vulnerable to random distortion even at less than milimeter dimensions.
Driesch needed to have a conversation with Babbage (but was born too late).
If Turing had lived longer, he would have figured all this out.
Possible Exam Questions:
1) Explain the simple computer algorithm which I used to simulate diffusion of a chemical.
2) How would you need to adjust this simulation program in order to predict the pattern of heat distribution in a piece of metal being heated with a Bunsen burner at one point, and/or cooled by a piece of ice at another point?
3) The retino-tectal projection (and perhaps all neural projections?) is created by the guided locomotion of retinal nerve cell "growth cones" that are stimulated to detach by binding to certain proteins on their own surfaces, to certain other proteins in the tectum. (Remember? Ephs and ephrins? Binding induces detachment?)
But you need to create concentration gradients of both the ephs and the ephrins.
Explain how to create those gradients without needing diffusion by either the ephs,
the ephrins, retinoic acid, or any "positional information" chemicals
4) Figure out how to change the computer program in order to make the diffusion constant lower in the bottom half of the field of numbers, as compared with the upper half.
(Hint: What change in the program would slow down the rate of equilibration between neighboring locations? For example, what change would reduce equilibration to one tenth of the speed which occurs when each cycle of the program changes the number in a a square "mail-box", to make this number closer to the average of the numbers stored in the neighboring squares? Calculate the average of the neighboring numbers; compare this with the number in the square being recalculated; then change this central number by a tenth of its difference from the average of the numbers of the adjacent boxes.)
[The following is just a less specific version of question #3]
5) Suppose that the cells in an epithelial sheet synthesize variable amounts of a certain membrane protein.
Also suppose that this membrane protein cannot diffuse from cell to cell, but that cells can detect the amounts of the protein on the cells that they touch.
Also, suppose that cells adjust the amounts of this protein that they synthesize in such a way that their own concentration approximates the average of the concentration of their immediate neighboring cells.
If all the cells along one side of an epithelium have large concentrations of this substance (regardless of the concentration on their neighbors), what will happen?
What if all the cells along the opposite side have zero concentration of this substance?
Shifting briefly to a purely biological point of view, maybe the cells along one edge have enzymes that destroy or inactivate this substance.
Or what if they can't synthesize it?
Or what if they can't detect the amount this substance on the surfaces of neighboring cells?
Will these possibilities each have a different effect on gradient formation?
Or the same net effect?
Can you invent some other changes in cell properties that would create low extremes of such diffusion gradients?
6) Please invent a better question that tests understanding of the two main concepts.
*7) [A few of you may enjoy this one; but if it doesn't excite your imagination, then please don't spend more than a few minutes on it, getting the general idea.]
Try to simulate Turing's pattern-generating method using two cell surface proteins, (A and B), neither of which can diffuse, but both of which cause neighboring cells to change their concentrations depending on each cell's own current concentration (of A and of B), and also depending on the average concentrations of A and B on the surfaces of their neighboring cells.
HINT:
New_Concentation_Of_A = Previous_Concentration_Of_A +
CONSTANT#1 * (Previous_Concentration_Of_A) +
CONSTANT#2 * (Previous_Concentration_Of_B) +
CONSTANT#3 * (Previous_Average_Concentrations_Of_A_on_Neighboring_Cells) +
CONSTANT#4 * (Previous_Average_Concentrations_Of_B_on_Neighboring_Cells).
New_Concentation_Of_B = Previous_Concentration_Of_B +
CONSTANT#5 * (Previous_Concentration_Of_A) +
CONSTANT#6 * (Previous_Concentration_Of_B) +
CONSTANT#3 * (Previous_Average_Concentrations_Of_A_on_Neighboring_Cells) +
CONSTANT#4 * (Previous_Average_Concentrations_Of_B_on_Neighboring_Cells).
Another HINT:
CONSTANT#1 is positive.
CONSTANT#5 is also positive.
CONSTANT#3 and CONSTANT#4 are both positive, but one of them is bigger.
Please explain whether CONSTANT#2 is positive, zero, or negative.
Please explain whether CONSTANT#6 is positive, zero, or negative.
Please explain whether CONSTANT#3 is positive, zero, or negative.
Please explain whether CONSTANT#3 is larger or smaller than CONSTANT#4.
Why can't CONSTANTs #3 and #4 be negative?
What would happen if one of them were negative?
These questions are much easier than they will probably seem, at first.
Question #8) Imagine that cells can detect the amounts of different hox genes proteins in their neighboring cells (immediate next-door cells), and adjust their own rates of transcription of each hox gene, based on these amounts of each hox protein in their
neighboring cells.
First part of #8) In order to produce what look like diffusion gradients, should detection of large amounts of Hox gene #1 cause an increased or a decreased rate of transcription of Hox gene #1? (and conversely, would detecting small amount of that protein
in neighboring cells cause a cell to increase or decrease the amount of that gene transcribed in a cell?)
Second part of #8) Figure out at least one possible way to produce patterns of gene expression that resemble the phenomenon of co-linearity of times and locations of hox genes in developing embryos.
Hint: Suppose that transcription of hox gene number 2 is either stimulated or inhibited by the amount of hox protein number 1 in neighboring cells.
(Is that such an unreasonable possibility? For certain genes in one cell to adjust transcription depending on concentration of transcripts of neighboring genes in neighboring cells.)
Don't waste a lot of time on #8. But a Nobel Prize may be lurking in these questions.
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