\n", "

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Design principles\n", "* Larger circuits can be understood in terms of recurring functional modules.\n", "* Searching for such motifs can help identify these modules.\n", "\n", "#### Key concepts\n", "* Circuit motifs can be identified by comparing the graphs of actual circuits with randomized ensembles of statistically similar circuits.\n", "\n", "#### Techniques\n", "* Graph theory approaches\n", "\n", "

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Circuit motifs\n", "\n", "A regulatory circuit can be thought of abstractly as a **directed graph**--a set of nodes and arrows connecting them. For the transcriptional circuitry of a bacterium, each node could represent an operon, and each arrow could represent regulation of one operon (tip of the arrow) by a transcription factor in another operon (base of the arrow).\n", "\n", "A **circuit motif** (or ‘network motif’) is defined as an over-represented regulatory pattern--or **sub-graph** within the overall circuit or graph. To be a 'motif', and not just a pattern, it should occur significantly more frequently than one would expect under some reasonable null hypothesis. \n", "\n", "The reason we care about motifs is that their elevated frequency suggests that they have been selected by evolution, perhaps because of their ability to perform important functions within the circuit.\n", "\n", "

\n", "

\n", "

\n", "\n", "_Different views of the same graph, using dots and arrows (left) or as a binary matrix, shown either with 0s and 1s or in black and white (right)_.\n", "\n", "

\n", "\n", "**Questions:**\n", "- What algorithm would you use to generate the randomized graphs (matrices)?\n", "- How would you control for the bias of smaller motifs in the frequency of larger motifs?\n", "\n", "Note: Supplementary information from Milo et al, _Science_, 2002 [here]( http://www.weizmann.ac.il/mcb/UriAlon/sites/mcb.UriAlon/files/uploads/NMpaper/networkmotifssmd.pdf)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Finding 3-node motifs\n", "\n", "Let's focus on 3-node sub-graphs. These are big enough to be quite interesting, but small enough to be easily conceptualized. There are 13 possible 3-node graphs that exclude direct autoregulation, which are shown here in a drawing from the Alon book:\n", "\n", "

\n", "

\n", "\n", "_Image: Alon, U., Introduction to Systems Biology_\n", "\n", "Now look back at the schematic graph above. Can you tell which of these sub-graphs are over-represented? The original graphs was constructed to contain a preponderance of one particular sub-graph:\n", "\n", "

\n", "

\n", "\n", "This sub-graph, termed the **Feed-Forward Loop (FFL)**, has one node that regulates a target node two ways: directly, and indirectly through the third node. \n", "\n", "To visualize this, we can highlight in red all of the FFLs in the graphs shown above: \n", "\n", "

\n", "

\n", "\n", "We see that this motif is over-represented. Of course, this is a schematic graph deliberately constructed to illustrate over-representation of this motif. \n", "\n", "What happens in real circuits?\n", " \n", "To address this question, Alon & colleagues went through the _E. coli_ transcriptional circuit, and systematically counted how many times they observed each of the 13 sub-graphs shown above. For each one, they compared the number of observations in the real circuit to the distribution of the number of times that sub-graph was observed in a large ensemble of randomized circuits. They used a **Z-score** to quantify over- or under-representation in units of the standard deviation of the number of occurrences for the sub-graph in the randomized circuits: $z = \\frac{n_{obs}-

\n", "

\n", "\n", "* Sparsely connected sub-graphs increase in frequency with $N$\n", "* Densely connected sub-graphs decrease in frequency with $N$. This means that as the graph grows larger there are actually *fewer* of these motifs.\n", "* And in between, sub-graphs with the same number of nodes and edges, such as the FFL, actually have a constant expected number regardless of $N$. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Beyond transcriptional circuits\n", "\n", "It's important to remember that these are expectations for (a particular class of) random networks. Real networks will of course differ, because they are designed, or evolved, to perform specific functions. In fact, different types of systems seem to show quite distinct motif profiles:\n", "\n", "

\n", "\n", "

\n", "\n", "Here, the top group are the transcriptional circuits we already saw. The next group are biological signal transduction circuits of different types from different species. The third group are networks like the world-wide web, for which a directed edge indicates a link from one web page to another. Finally, the lowest group represents motifs in language graphs, in which nodes are words and edges are derived from the probability that one word follows another. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## There are many kinds of FFLs\n", "\n", "Our description of regulatory interactions so far has been oversimplified in a number of ways. We have collapsed positive and negative regulation together. We have not considered how multiple regulators combine to control expression of a mutual target operon. And we have ignored the quantitative strength of the regulation. Trying to understand the function of a motif requires thinking about these aspects more carefully. \n", "\n", "We can classify the overall FFL motif into $2^3=8$ different categories depending on which if its 3 arrows are positive or negative:\n", "\n", "

\n", "

\n", "\n", "

\n", "\n", "We can then further classify the FFLS, according to how the regulatory arrows converging on the third node (now labeled \"Z\") combine. In AND regulation, both X and Y need to be simultaneously present at high levels for Z to be expressed. in OR regulation, either input is sufficient to activate Z. \n", "\n", "In the next lecture, we will consider what functions the various FFLs can perform for cells.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Summary\n", "\n", "* Motifs are statistically over-represented patterns in a circuit.\n", "* To find motifs, we construct ensembles of random circuits with the same arrow distributions for comparison.\n", "* The abundances of different sub-graphs can scale in opposite and even counter-intuitive ways as the size of the overall circuit grows\n", "* This approach can be applied to a wide variety of systems that can be represented as graphs.\n", "* Transcriptional regulatory circuits appear to have a single major motif: the feed-forward loop, which can appear in multiple types.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For those who have not previously read it, here is the first paragraph of Moby Dick:\n", " \n", "> Call me Ishmael. Some years ago - never mind how long precisely - having little or no money in my purse, and nothing particular to interest me on shore, I thought I would sail about a little and see the watery part of the world. It is a way I have of driving off the spleen and regulating the circulation. Whenever I find myself growing grim about the mouth; whenever it is a damp, drizzly November in my soul; whenever I find myself involuntarily pausing before coffin warehouses, and bringing up the rear of every funeral I meet; and especially whenever my hypos get such an upper hand of me, that it requires a strong moral principle to prevent me from deliberately stepping into the street, and methodically knocking people's hats off - then, I account it high time to get to sea as soon as I can. This is my substitute for pistol and ball. With a philosophical flourish Cato throws himself upon his sword; I quietly take to the ship. There is nothing surprising in this. If they but knew it, almost all men in their degree, some time or other, cherish very nearly the same feelings towards the ocean with me. " ] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": false }, "varInspector": { "cols": { "lenName": 16, "lenType": 16, "lenVar": 40 }, "kernels_config": { "python": { "delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())" }, "r": { "delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) " } }, "position": { "height": "730.4000244140625px", "left": "486.6000061035156px", "right": "20px", "top": "120px", "width": "517.4000244140625px" }, "types_to_exclude": [ "module", "function", "builtin_function_or_method", "instance", "_Feature" ], "window_display": false } }, "nbformat": 4, "nbformat_minor": 2 }