DOC PREVIEW
The Emergence of Geometric Order

This preview shows page 1-2-3-4 out of 13 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Supplementary Material The Emergence of Geometric Order in Proliferating Metazoan Epithelia Matthew C. Gibson, Ankit B. Patel, Radhika Nagpal & Norbert Perrimon This supplementary document presents two extensions of the mathematical model, as well as a more detailed description of the Drosophila experimental data. In addition, we present some clarifications on the graph model and a detailed methods section. Table of Contents: 1. Graph Model: Derivation in the Presence of Boundary Conditions 2. Extended Markov Model: Error in Interface Formation 3. Extended Markov Model: Alternative Cleavage Plane Models 4. Imaginal Disc Polygonal Cell Counts 5. Detailed Methods 1. Graph Model: Derivation in the Presence of Boundary Conditions In the paper we approximate st, the average number of sides per cell at generation t as: st  2et / ft = 2(et-1 + 3ft-1) / 2ft-1 = (st-1 / 2) + 3 (1) We solve this recurrence to get: st  6 + 2-t(s0 – 6) (2) However, this equation for st is only an upper bound approximation because it double counts edges on the boundary of the epithelium. If we account for boundary cells, then the exact equation for st is: st = (2et - etbd) / ft = 6 + 2-t(s0 – 6) - etbd / ft (3) where etbd is the number of boundary edges and ft is the number of cells at generation t. This basically is the same as equation (1) but corrects for the over counted edges. In a 2D epithelium comprised of similarly-sized cells, the number of boundary edges approximately measures the perimeter and the number of cells approximately measures area. Hence etbd ~ O(r) and ft ~ O(r2), where r is the radius of the epithelium (in cells). The third term in equation (3) is therefore O(1/r) and this approaches zero as t gets large and the epithelium gets large. This implies that the average number of neighbours percell will approach six from below. In fact, the convergence rate is exponential since ft ~ O(2t) and etbd ~ O(ft). [Note that this would not be true for a strip of cells that proliferate in a single direction since then etbd ~ O(ft).] Thus, the original approximate formula for st in (2) is still asymptotically correct. For a roughly circular region of 200 cells, the difference between equations (2) and (3) is less than 4% and for 30,000 cells it is less than 0.4%. Therefore equation (2) is a good approximation for the sizes of epithelial regions we sample, and a very good approximation for the imaginal disc itself, which eventually has well over 30,000 cells. An important aspect of this analysis is that it is independent of cleavage plane orientation (condition 6), and is instead a deeper implication of the formation of mostly tricellular junctions (vertices of degree 3). Notice however that this result by itself is not sufficient to explain the predominance of hexagons or even their existence– for example, a planar graph with equal numbers of octagons and quadrilaterals has only tricellular junctions and an average of six neighbours, but no hexagons. 2. Extended Markov Model: Error in Interface Formation One of the discrepancies between the mathematically predicted distribution and the experimentally observed distribution is the absence of 4-sided cells. The Markov model allows the creation of 4-sided cells through division but assumes that at the end of a round of division each cell has gained exactly one edge. As a result all 4-sided cells become 5-sided in each round, after gaining edges from neighbors, and thus the model predicts no 4-sided cells. In our experimental data, a small fraction of 4-sided cells are observed (2-3%). There are several potential factors that can explain the difference between the prediction and observation: 1. The experimental data does not represent the end of a round of division, and therefore not all cells will have gained a neighbor in the current round. Thus one should expect to see some 4-sided cells that have been created through splitting but not gained an edge from a neighbor yet. 2. In the Markov model, we assume that each cell gains exactly one edge. However in reality a cell may gain more or less. More precisely, the number of sides added via neighbor division is a random variable obeying a particular distribution. We prove that this distribution has a mean of exactly 1. By using the shift matrix S we are making the mean-field approximation, i.e. that this mean is “good enough” to calculate the equilibrium distribution. Our experiments with aged and mitotic cells suggest that the mean-field approximation is reasonable. This approximation likely adds a small amount of error to the prediction for all cell types, but is most obvious for 4-sided cells. 3. Our model assumes that all divisions result in the formation of a new interface between the daughters (condition 3). However, it is possible that this fails with some small rate as shown in our data on the occurrence of non-interface forming clones (Text Fig. 1). While we cannot easily measure the exact rate of failure, wecan model the effect of different error rates in interface formation, as shown below Experimental data on interface formation In the text we define three types of cell division, Type I, II and III (Text Fig. 1). Type I is a division that results in the formation of a normal interface between daughters. Type II division results in two daughters but no new interface is formed so they are connected only at a vertex. Type III similarly has two daughters that fail to form a new interface but also get separated. In our experimental data on two-cell clones, 94% of the divisions are unambiguously Type I. However this is a conservative estimate, and the remaining cases are not unambiguously Type II/III. This is primarily because it is difficult to distinguish Type II from cytokinesis that has not yet resolved into an interface. More importantly, it is also difficult to distinguish a Type III division from two independent clone events that occur within a 1-cell radius (which was our scoring criteria for Type III). For these experiments, clones were tested within a 10-hour period, during which they are likely to divide once. However, some cells may not divide and some cell clones may occur close to each other. Therefore there are a small number of single cells and larger clones. If we consider only two-cell clones, then the fraction of non-Type I events is < 3%, whereas using all clones results in 6% non-Type I events. Including the larger clones


The Emergence of Geometric Order

Download The Emergence of Geometric Order
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view The Emergence of Geometric Order and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view The Emergence of Geometric Order 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?