Bi1: The Great Ideas of Biology Homework 8 Due Date: Thursday, June 1, Transcription rates and sizes of genes

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1 Bi1: The Great Ideas of Biology Homework 8 Due Date: Thursday, June 1, 2017 To simplify is the greatest form of sophistication. - attributed to Leonardo da Vinci 1 Transcription rates and sizes of Ostensibly, this problem concerns mrna transcription rates in Drosophila. By this we mean the speed at which RNA polymerase adds nucleotides to a growing mrna strand as it reads along the DNA template, not the rate at which new transcripts are initiated (although that is surely an interesting question as well!). It will be a return to the street-fighting skills we have neglected the last few weeks. Along the way we will discover some interesting facts about early embryonic development in this workhorse model organism. Question 1a: Let us recall an estimate from week 1 in lecture. If a typical protein is 300 amino acids in length, about how many would you expect to find in E. coli, whose genome is about 5 million base pairs long? How about for a eukaryote such as Drosophila, whose genome is around 100 million base pairs? Compare to the actual numbers: depending on the strain, E. coli has between 4000 and 5500, while Drosophila has around Clearly this simplistic estimate is missing something important about eukaryotes! That something important is the existence of noncoding DNA. While almost all (typically 90%) of bacterial genomes codes for proteins, with most of the remainder involved in regulatory regions like those covered in Homework 7, things are very different in eukaryotes. Noncoding DNA makes up the majority of eukaryotic genomes, and learning more about its many important functions is still an active topic of research. For this problem our interest is focued on an important noncoding part known as introns. Specifically, one of the fascinating discoveries during the period of the emergence of modern molecular biology was the realization that 1

2 eukaryotic feature mrnas that have insertions that do not code for amino acids in the protein. These insertions are known as introns, whereas that part of the mrna that codes for amino acids are known as exons. To proceed we need to know something about typical sizes of introns. Fig. 1 can be used to make an estimate of intron and exon content in a typical gene. Note that what Chambon and coworkers cleverly did was to hybridize the DNA for a gene of interest to the spliced mrna. Hence, those parts of the spliced DNA that are complementary to those on the DNA resulted in base pairing, while those parts of the gene (introns) that had been removed formed unhybridized loops. Question 1b: Use Fig. 1 to estimate what fraction of a typical transcript is composed of exons and, therefore, what fraction is composed of introns. Question 1c: Hunchback (Hb) is larger than most of its fellow gap, at 758 aa. Assuming the exon/intron ratio you estimated in 1b is typical, estimate the size of its prespliced mrna transcript. Then, estimate the minimum possible transcription rate (in nt/s) that would allow the hb gene to be transcribed at all. Note that in early Drosophila development nuclear divisions happen at the incredible rate of every 8 min. 4 of those minutes are spent in M phase, meaning the DNA is condensed into chromosomes, so no transcription can initiate then and any currently elongating transcripts will be aborted. Fig. 2 shows the results of clever experiments to measure the transcription rate in live embryos. The result is that the inferred transcription rate is 25 nt/sec. Depending how you did your estimates, your rate might be comparable or somewhat larger, so it is unclear whether there is time to make the necessary transcripts! The resolution is that hb has significantly shorter introns than our estimated typical intron fraction; if you examine Fig. 2 closely, you will notice that the transcript we are considering here is only about 3.3 kb. Since 2.3 kb of that is coding, only about 30% is intron, much less that we estimated above in 1b. Here is a good question: are the unusually short introns in hb a coincidence? Fig. 3 shows us the network of associated with embryonic development in the fly. In addition, it provides a sense of the sizes of the 2

3 Figure 1: Electron-microscopy image and corresponding schematic for estimating intron and exon content of a typical gene. In this clever experiment, the post-splicing mrna transcript (blue dashed line) is hybridized with the DNA (red solid line). In other words, roughly the entire length of the mrna is bound to the complementary sequence in the DNA, i.e., the exons, while the introns are left unbound to form random loops on their own. The exons are labeled 1 through 7 while the introns are labeled with corresponding Roman numerals. (The fact that there happen to be equal numbers of exons and introns in this example is coincidental and in no way significant.) 3

4 various as a function of time during embryonic development, showing that the longer have to await those stages of embryonic development that take longer time. So hb and its cousins (the other gap and also pair-rule ) have low intron content out of necessity. By contrast, the which are only transcribed later, after the rapid division cycles are complete, can be much larger; note the positively enormous Hox weighing in at 40 kb on average! It is also interesting to note that hb, like many eukaryotic, has multiple splicing isoforms. This means that the exons are not always spliced together in the same number or order, so in essence different proteins can be produced from a single gene! Sometimes the different isoforms only differ by one or a few small exons and are therefore very similar, but sometimes the differences can be extreme. For hb, the isoform produced by the embryo itself (the 3 kb one analyzed here) has very little intron. But the maternal isoform (deposited by the mother into the early embryo) is double the length at about 6.5 kb. (If you re interested, checkout flybase.org/reports/fbgn html and look under Gene Model and Products, where you can see a map of the different transcripts.) Here is another good question: how do cells decide which isoforms to produce? Obviously this is a deep and complicated subject. For hb, there are two entirely different promoter sites, i.e., RNA polymerase can start transcription at two different sites separated by 2-3 kb. The start site for the short isoform is located in one of the introns of the long isoform. It is suspected that other nearby regions and some distant regions of the genome are involved in regulating the transcription from one promoter or the other, probably by binding transcription factors, but the full story is still an active subject of research. The take-home message is that, for most eukaryotic, there is a complex dance that regulates the excision of introns and the determination of which exons to include in what order. 2 Reaction-diffusion: morphogen gradients and spreading the butter In class, we have now discussed the paramount importance of diffusion in biological processes on multiple occasions. One of the more interesting examples of diffusion is in the context of setting up animal body plans by establishing 4

5 Figure 2: Dynamics of transcription in the fly embryo. (A) Schematic of the experiment showing how a loop in the nascent RNA molecule serves as a binding site for a viral protein that has been fused to GFP. (B) Depending upon whether the RNA loops are placed on the 5 or 3 end of the mrna molecule, the time it takes to begin seeing GFP puncta will be different. The delay time is equal to the length of the transcribed region divided by the speed of the polymerase. (C) Microscopy images showing the appearance of puncta associated with the transcription process for both constructs shown in (B). (D) Distribution of times of first appearance for the two constructs yielding a delay time of 2.2 minutes, from which a transcription rate of 25 nt/s is inferred. Measurements performed at room temperature of 22 C. (adapted from H. G. Garcia, et al., Current Biology, 23, , 2013.) 5

6 GRADIENTS maternal caudal bicoid nanos 0 average gene length (kb) DOMAINS gap giant hunchback Kruppel knirps gap 7 STRIPES pair-rule fushi tarazu even-skipped runt hairy pair-rule 14 STRIPES segment polarity engrailed hedgehog wingless Segment polarity SEGMENTS Hox labial proboscipedia Deformed Sex combs reduced Antennapedia Ultrabithorax abdominal A Abdominal B Hox Figure 3: Gene regulatory network dictating fruit fly development. The network is organized in a cascade of expressed in increasingly sharp spatiotemporal patterns. (Adapted from Carroll et al., From DNA to diversity: molecular genetics and the evolution of animal design, Blackwell Science, 2001 and Jaeger, Cell Mol Life Sci 68:243, 2011.) Average gene length in the fruit fly developmental program. As development progresses, active in the regulatory network have increasing lengths. (Adapted from FlyBase Release ) 6

7 morphogen gradients in the embryo. In class, we wrote a simple dynamical description of the diffusion-withdegradation process for a morphogen such as Bicoid in the form p(x, t + t) = p(x, t) + k tp(x a, t) + k tp(x + a, t) 2k tp(x, t) t τ p(x, t), (1) where k is the diffusive jump rate and τ is the degradation time for the Bicoid protein. For a protein such as Bicoid, the diffusion constant has a value of roughly D = ka 2 5 μm 2 /s and the degradation time is τ 3000 s. In this problem, your objective will be to consider a one-dimensional fly and to solve the chemical master equation written above over time (i.e. write a code in Python to work out the time evolution of the concentration field) to explore in more detail the hypothesis that a reaction-diffusion mechanism could be responsible for setting up the morphogen gradient. In the tutorial you learned how to simulate the plain diffusion equation, i.e., exactly Eq. (1) but without the degradation term involving τ. Now your task is to put the reaction into reaction-diffusion by adding production and degradation. Specifically, you will need to discretize the 500 μm fly embryo into a one-dimensional lattice with a lattice parameter of 10 μm. (This is a natural length scale to discretize space: it roughly corresponds to the spacing between nuclei along the anterior-posterior axis, so there is no physical point in adopting a finer grid.) Next, at the anterior end of the embryo, mimic the presence of the mrna deposited by the mother as a source term, i.e, in the anterior-most bin of your simulation, Bicoid is produced at a rate of s 1 from the roughly bicoid mrnas deposited by the mother at the anterior end of the embryo. In other words, s 1 is effectively the total translation rate of Bicoid protein from all the maternal mrna. There is one other subtlety compared with the tutorial: for the pure diffusion problem in the tutorial, the total number of particles in the system, N tot, was constant in time since none were created or destroyed. So it made sense to simply talk about the probability distribution p(x, t) of particle locations. But here, particles are created and destroyed, so the natural variable is concentration c(x, t) rather than probability p(x, t). The relation between them is simple, since p(x, t) = c(x, t)/n tot (t), (2) 7

8 where N tot is the total number of particles. But now N tot is itself a function of time. There are two ways of approaching the problem. The easy way is to simply work with concentration rather than probability, and all the code from the tutorial will translate verbatim in this language. The hard way is to work with probability and renormalize the distribution after each time step. So we might as well do the former, but it is important that you understand the distinction. Question 2a: Take the diffusion code from the tutorial and incorporate production and degradation using the parameters above. Note the degradation term in Eq. (1) occurs in every lattice bin and is proportional to how much Bicoid is already present. By contrast, production will occur only in the anterior-most bin and is independent of how much Bicoid is already present. Plot the Bicoid profile as a function of position along the embryo at 1, 10, 30, 100, 300, and 1000 minutes. Plot all the concentration profiles as curves on the same plot for ease of comparison. Comment on the result, especially noting if the profile has reached steady-state and the rough time scale required to reach steady-state. How does this compare to the timescale of early embryo development, i.e., does a real embryo have time to get (somewhat) close to steady-state? Now that you have simulated and plotted the Bicoid profile to hopefully gain some intuition, we would like to derive the same result analytically. More specifically, it would be useful to have a closed-form solution for the stead-state concentration profile of Bicoid as a function of position along the embryo. This is essentially the derivation we did in lecture. The diffusion equation with degradation has the form c t = D 2 c x c 2 τ, (3) where concentration c is a function of position x and time t. In your simulation, you were able to observe the time-dependence, as the concentration profile progressed towards steady state. That is a difficult problem analytically, so we will limit ourselves to the simpler question of what is the steady state profile, and never mind how long it takes to reach it (also we already have some sense of this timescale from the simulation). By definition, at 8

9 steady state the time dependence disappears and we have the simpler problem 0 = D d2 c dx 2 c τ, (4) where now concentration is a function only of position. Question 2b: Solve this differential equation for c as a function of x. You can simply use a trial solution of the form c(x) = Ae σx and obtain a condition on the value σ must take for the equation to be satisfied. A is the constant to be fit by boundary conditions. Make sure to note the units of Dτ. You should have two independent solutions, but one is ruled out by boundary conditions. Relate A to the concentration of Bicoid at the anterior end (this could be related to the translation rate, but this is unnecessary for our purposes). Question 2c: Armed with the solution from the previous part, derive the position of the cephalic furrow x CF under the French-flag model. Recall that this model assumes that the furrow forms at a given critical concentration c. You should end up with x CF as a function of the gene dosage G relative to wild-type, the diffusion constant D, and the degradation timescale τ. 3 Morphogen gradients and positioning of features in development Problem 1 addressed how morphogens are transcribed, while Problem 2 focused on the theory of how morphogens diffuse through developing organisms. Of course, before we can decide the extent to which this morphogen model describes embryois, we have to see how well it accords with experiment. In this problem we consider how the development of morphological features are influenced by morphogens. For this we will analyze images of developing fly embryos with varying amounts of Bicoid. Perhaps the best summary of one way that Bicoid affects this development is simply given in the title of the Driever et al paper The Bicoid 9

10 Protein Determines Position in the Drosophila Embryo in a Concentration- Dependent Manner. In a nutshell, the authors found that altering the amount of Bicoid present in a fruit fly embryo caused a shift in the location of the cephalic furrow, a body feature that typically develops in a very spatially precise manner. As a reminder, the cephalic furrow effectively defines the barrier between which cells will develop into the head of the fly and which cells will develop into the rest of the body. In order to measure the effects of the concentration of Bicoid on the location of the cephalic furrow, we will analyze data very similar to what was used in the Driever et al. paper. By introducing mutations into the chromosome that contains the Bicoid gene, it is possible to knock out the production of Bicoid. Alternately, one can add extra copies of the gene to new positions in the chromosome, or to different chromosomes. Then through the breeding of flies with different copy numbers of these mutated chromosome(s), it is possible to obtain fly embryos with varying concentrations of Bicoid. In the data set posted on the course website, we have included images of embryos with wild-type Bicoid concentrations (two functional copies of the gene), half (one functional, one knocked out), and double (four functional). In this problem, your task is to analyze these images to measure the location of the cephalic furrow and determine if it fits the model you derived in Problem 2. Had you been luckier and won the lottery for Bi 1x, you would have had the delightful experience of breeding flies with different copy numbers of the bicoid gene yourself. The fun would have continued a few weeks later when you got to collect the fly embryos and take pictures of them with microscopes that cost more than what you pay in tuition in a given year. Finally, you would have been able to sift through all of your images to determine the average location of the cephalic furrow across many different embryos so that you would have good data to compare to the theoretical model. Since we are sadly not able to give everyone this hands-on experience, we have decided to ease up on the data analysis as our way of apologizing. Question 3a: On the Bi 1 website, you ll find two images of Drosophila embryos taken in Bi 1x, one with wild-type and one with mutant (half wild-type) bicoid levels. Following the tutorial, compute the locations of the cephalic furrow as a fraction of embryo length for each image. 10

11 The purpose of the previous problem is to give you a taste of the raw data without excessive suffering. Very hard working students in the lab of Thomas Gregor have created approximately 30 lines of Drosophila with a wide range of Bicoid dosage. They achieved this by varying both the copy number and position of the gene within the fly genome; varying where in the chromosome the gene is located turns out to have a measurable and repeatable effect on how much the gene is transcribed. They measured the position of the cephalic furrow in much the same way for thousands of embryos. The data file we provide reports the average furrow position of each genetically distinct line, each of which is the average of measurement of many (dozens to perhaps a hundred) individual embryos. (Obviously we spared you that amount of clicking!) Question 3b: Use pandas to extract the data from the Gregor lab, posted on the course website. Plot the furrow position as a function of bicoid dosage. (It follows from your result in Problem 2 that plotting the log of bicoid dosage will make things clearer.) Also add the three data points from the three images you analyzed. Finally add the prediction curve from the French Flag model. Finally, we have experimental data to test our model with. Question 3c: How do the experimental cephalic furrow locations compare to the predictions made in Problem 2? If there is a discrepancy, explain in a few sentences how you think that may have arisen. 11