{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Exercise 6.3: Growth curves from a movie\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At the dawn of the molecular revolution in biology, key experiments by Jacques Monod in which he measured growth curves of bacteria under different conditions exposed some of the mechanisms of regulation of gene expression. Those growth curves were measured in a bulk solution. In this exercise, we will measure bacterial growth starting from *two bacteria*. The movie shows *Bacillus subtilis* constitutively expressing mCherry growing under slow growth conditions. These data were kindly donated by Jin Park from the [Elowitz lab](http://www.elowitz.caltech.edu) at Caltech.\n", "\n", "**a)** Load in the series of images contained in the directory `~git/bootcamp/data/bacterial_growth/`. Be sure that however you store them (a list or tuple or other object) has the frames in the proper order." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**b)** Segment the images to separate bacteria from background. You do not need to segment individual bacteria; this would likely require some more advanced techniques involving edge detection that we haven't covered in bootcamp." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**c)** Show a representative image from the stack (with the segmentation overlayed) of images. Be sure to check out the `README` file in the directory containing the images to get the interpixel distance." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**d)** If $m$ is the mass of bacteria, which is proportional to the area in the images, then, for exponential growth,\n", "\n", "\\begin{align}\n", "m(t) = m_0\\,\\mathrm{e}^{rt},\n", "\\end{align}\n", "\n", "where $r$ is the growth rate. Taking the logarithm of both sides gives\n", "\n", "\\begin{align}\n", "\\ln m(t) = \\ln m_0 + rt.\n", "\\end{align}\n", "\n", "So, the slope of the line on a log-log scale is the growth rate.\n", "\n", "With this in mind, plot the bacterial area as a function of time with the area on a log scale. If the plot is linear, the bacteria are performing exponential growth." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**e)** To get the slope of the line, you can use the `np.polyfit()` function and perform the regression on a semilog scale. Alternatively, you can fit the exponential function directly using `scipy.optimize.curve_fit()`. Once you find the growth rate (and intercept), plot the theoretical growth curve along with the data." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Solution\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "(function(root) {\n", " function now() {\n", " return new Date();\n", " }\n", "\n", " var force = true;\n", " var py_version = '3.1.1'.replace('rc', '-rc.');\n", " var is_dev = py_version.indexOf(\"+\") !== -1 || py_version.indexOf(\"-\") !== -1;\n", " var reloading = false;\n", " var Bokeh = root.Bokeh;\n", " var bokeh_loaded = Bokeh != null && (Bokeh.version === py_version || (Bokeh.versions !== undefined && Bokeh.versions.has(py_version)));\n", "\n", " if (typeof (root._bokeh_timeout) === \"undefined\" || force) {\n", " root._bokeh_timeout = Date.now() + 5000;\n", " root._bokeh_failed_load = false;\n", " }\n", "\n", " function run_callbacks() {\n", " try {\n", " root._bokeh_onload_callbacks.forEach(function(callback) {\n", " if (callback != null)\n", " callback();\n", " });\n", " } finally {\n", " delete root._bokeh_onload_callbacks;\n", " }\n", " console.debug(\"Bokeh: all callbacks have finished\");\n", " }\n", "\n", " function load_libs(css_urls, js_urls, js_modules, js_exports, callback) {\n", " if (css_urls == null) css_urls = [];\n", " if (js_urls == null) js_urls = [];\n", " if (js_modules == null) js_modules = [];\n", " if (js_exports == null) js_exports = {};\n", "\n", " root._bokeh_onload_callbacks.push(callback);\n", "\n", " if (root._bokeh_is_loading > 0) {\n", " console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n", " return null;\n", " }\n", " if (js_urls.length === 0 && js_modules.length === 0 && Object.keys(js_exports).length === 0) {\n", " run_callbacks();\n", " return null;\n", " }\n", " if (!reloading) {\n", " console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n", " }\n", "\n", " function on_load() {\n", " root._bokeh_is_loading--;\n", " if (root._bokeh_is_loading === 0) {\n", " console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n", " run_callbacks()\n", " }\n", " }\n", " window._bokeh_on_load = on_load\n", "\n", " function on_error() {\n", " console.error(\"failed to load \" + url);\n", " }\n", "\n", " var skip = [];\n", " if (window.requirejs) {\n", " window.requirejs.config({'packages': {}, 'paths': {'jspanel': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/jspanel', 'jspanel-modal': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/modal/jspanel.modal', 'jspanel-tooltip': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/tooltip/jspanel.tooltip', 'jspanel-hint': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/hint/jspanel.hint', 'jspanel-layout': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/layout/jspanel.layout', 'jspanel-contextmenu': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/contextmenu/jspanel.contextmenu', 'jspanel-dock': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/dock/jspanel.dock', 'gridstack': 'https://cdn.jsdelivr.net/npm/gridstack@7.2.3/dist/gridstack-all', 'notyf': 'https://cdn.jsdelivr.net/npm/notyf@3/notyf.min'}, 'shim': {'jspanel': {'exports': 'jsPanel'}, 'gridstack': {'exports': 'GridStack'}}});\n", " require([\"jspanel\"], function(jsPanel) {\n", "\twindow.jsPanel = jsPanel\n", "\ton_load()\n", " })\n", " require([\"jspanel-modal\"], function() {\n", "\ton_load()\n", " })\n", " require([\"jspanel-tooltip\"], function() {\n", "\ton_load()\n", " })\n", " require([\"jspanel-hint\"], function() {\n", "\ton_load()\n", " })\n", " require([\"jspanel-layout\"], function() {\n", "\ton_load()\n", " })\n", " require([\"jspanel-contextmenu\"], function() {\n", "\ton_load()\n", " })\n", " require([\"jspanel-dock\"], function() {\n", "\ton_load()\n", " })\n", " require([\"gridstack\"], function(GridStack) {\n", "\twindow.GridStack = GridStack\n", "\ton_load()\n", " })\n", " require([\"notyf\"], function() {\n", "\ton_load()\n", " })\n", " root._bokeh_is_loading = css_urls.length + 9;\n", " } else {\n", " root._bokeh_is_loading = css_urls.length + js_urls.length + js_modules.length + Object.keys(js_exports).length;\n", " }\n", "\n", " var existing_stylesheets = []\n", " var links = document.getElementsByTagName('link')\n", " for (var i = 0; i < links.length; i++) {\n", " var link = links[i]\n", " if (link.href != null) {\n", "\texisting_stylesheets.push(link.href)\n", " }\n", " }\n", " for (var i = 0; i < css_urls.length; i++) {\n", " var url = css_urls[i];\n", " if (existing_stylesheets.indexOf(url) !== -1) {\n", "\ton_load()\n", "\tcontinue;\n", " }\n", " const element = document.createElement(\"link\");\n", " element.onload = on_load;\n", " element.onerror = on_error;\n", " element.rel = \"stylesheet\";\n", " element.type = \"text/css\";\n", " element.href = url;\n", " console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n", " document.body.appendChild(element);\n", " } if (((window['jsPanel'] !== undefined) && (!(window['jsPanel'] instanceof HTMLElement))) || window.requirejs) {\n", " var urls = ['https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/jspanel.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/modal/jspanel.modal.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/tooltip/jspanel.tooltip.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/hint/jspanel.hint.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/layout/jspanel.layout.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/contextmenu/jspanel.contextmenu.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/dock/jspanel.dock.js'];\n", " for (var i = 0; i < urls.length; i++) {\n", " skip.push(urls[i])\n", " }\n", " } if (((window['GridStack'] !== undefined) && (!(window['GridStack'] instanceof HTMLElement))) || window.requirejs) {\n", " var urls = ['https://cdn.holoviz.org/panel/1.1.0/dist/bundled/gridstack/gridstack@7.2.3/dist/gridstack-all.js'];\n", " for (var i = 0; i < urls.length; i++) {\n", " skip.push(urls[i])\n", " }\n", " } if (((window['Notyf'] !== undefined) && (!(window['Notyf'] instanceof HTMLElement))) || window.requirejs) {\n", " var urls = ['https://cdn.holoviz.org/panel/1.1.0/dist/bundled/notificationarea/notyf@3/notyf.min.js'];\n", " for (var i = 0; i < urls.length; i++) {\n", " skip.push(urls[i])\n", " }\n", " } var existing_scripts = []\n", " var scripts = document.getElementsByTagName('script')\n", " for (var i = 0; i < scripts.length; i++) {\n", " var script = scripts[i]\n", " if (script.src != null) {\n", "\texisting_scripts.push(script.src)\n", " }\n", " }\n", " for (var i = 0; i < js_urls.length; i++) {\n", " var url = js_urls[i];\n", " if (skip.indexOf(url) !== -1 || existing_scripts.indexOf(url) !== -1) {\n", "\tif (!window.requirejs) {\n", "\t on_load();\n", "\t}\n", "\tcontinue;\n", " }\n", " var element = document.createElement('script');\n", " element.onload = on_load;\n", " element.onerror = on_error;\n", " element.async = false;\n", " element.src = url;\n", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " document.head.appendChild(element);\n", " }\n", " for (var i = 0; i < js_modules.length; i++) {\n", " var url = js_modules[i];\n", " if (skip.indexOf(url) !== -1 || existing_scripts.indexOf(url) !== -1) {\n", "\tif (!window.requirejs) {\n", "\t on_load();\n", "\t}\n", "\tcontinue;\n", " }\n", " var element = document.createElement('script');\n", " element.onload = on_load;\n", " element.onerror = on_error;\n", " element.async = false;\n", " element.src = url;\n", " element.type = \"module\";\n", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " document.head.appendChild(element);\n", " }\n", " for (const name in js_exports) {\n", " var url = js_exports[name];\n", " if (skip.indexOf(url) >= 0 || root[name] != null) {\n", "\tif (!window.requirejs) {\n", "\t on_load();\n", "\t}\n", "\tcontinue;\n", " }\n", " var element = document.createElement('script');\n", " element.onerror = on_error;\n", " element.async = false;\n", " element.type = \"module\";\n", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " element.textContent = `\n", " import ${name} from \"${url}\"\n", " window.${name} = ${name}\n", " window._bokeh_on_load()\n", " `\n", " document.head.appendChild(element);\n", " }\n", " if (!js_urls.length && !js_modules.length) {\n", " on_load()\n", " }\n", " };\n", "\n", " function inject_raw_css(css) {\n", " const element = document.createElement(\"style\");\n", " element.appendChild(document.createTextNode(css));\n", " document.body.appendChild(element);\n", " }\n", "\n", " var js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.1.1.min.js\", \"https://cdn.holoviz.org/panel/1.1.0/dist/panel.min.js\"];\n", " var js_modules = [];\n", " var js_exports = {};\n", " var css_urls = [];\n", " var inline_js = [ function(Bokeh) {\n", " Bokeh.set_log_level(\"info\");\n", " },\n", "function(Bokeh) {} // ensure no trailing comma for IE\n", " ];\n", "\n", " function run_inline_js() {\n", " if ((root.Bokeh !== undefined) || (force === true)) {\n", " for (var i = 0; i < inline_js.length; i++) {\n", " inline_js[i].call(root, root.Bokeh);\n", " }\n", " // Cache old bokeh versions\n", " if (Bokeh != undefined && !reloading) {\n", "\tvar NewBokeh = root.Bokeh;\n", "\tif (Bokeh.versions === undefined) {\n", "\t Bokeh.versions = new Map();\n", "\t}\n", "\tif (NewBokeh.version !== Bokeh.version) {\n", "\t Bokeh.versions.set(NewBokeh.version, NewBokeh)\n", "\t}\n", "\troot.Bokeh = Bokeh;\n", " }} else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(run_inline_js, 100);\n", " } else if (!root._bokeh_failed_load) {\n", " console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n", " root._bokeh_failed_load = true;\n", " }\n", " root._bokeh_is_initializing = false\n", " }\n", "\n", " function load_or_wait() {\n", " // Implement a backoff loop that tries to ensure we do not load multiple\n", " // versions of Bokeh and its dependencies at the same time.\n", " // In recent versions we use the root._bokeh_is_initializing flag\n", " // to determine whether there is an ongoing attempt to initialize\n", " // bokeh, however for backward compatibility we also try to ensure\n", " // that we do not start loading a newer (Panel>=1.0 and Bokeh>3) version\n", " // before older versions are fully initialized.\n", " if (root._bokeh_is_initializing && Date.now() > root._bokeh_timeout) {\n", " root._bokeh_is_initializing = false;\n", " root._bokeh_onload_callbacks = undefined;\n", " console.log(\"Bokeh: BokehJS was loaded multiple times but one version failed to initialize.\");\n", " load_or_wait();\n", " } else if (root._bokeh_is_initializing || (typeof root._bokeh_is_initializing === \"undefined\" && root._bokeh_onload_callbacks !== undefined)) {\n", " setTimeout(load_or_wait, 100);\n", " } else {\n", " Bokeh = root.Bokeh;\n", " bokeh_loaded = Bokeh != null && (Bokeh.version === py_version || (Bokeh.versions !== undefined && Bokeh.versions.has(py_version)));\n", " root._bokeh_is_initializing = true\n", " root._bokeh_onload_callbacks = []\n", " if (!reloading && (!bokeh_loaded || is_dev)) {\n", "\troot.Bokeh = undefined;\n", " }\n", " load_libs(css_urls, js_urls, js_modules, js_exports, function() {\n", "\tconsole.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n", "\trun_inline_js();\n", " });\n", " }\n", " }\n", " // Give older versions of the autoload script a head-start to ensure\n", " // they initialize before we start loading newer version.\n", " setTimeout(load_or_wait, 100)\n", "}(window));" ], "application/vnd.holoviews_load.v0+json": "(function(root) {\n function now() {\n return new Date();\n }\n\n var force = true;\n var py_version = '3.1.1'.replace('rc', '-rc.');\n var is_dev = py_version.indexOf(\"+\") !== -1 || py_version.indexOf(\"-\") !== -1;\n var reloading = false;\n var Bokeh = root.Bokeh;\n var bokeh_loaded = Bokeh != null && (Bokeh.version === py_version || (Bokeh.versions !== undefined && Bokeh.versions.has(py_version)));\n\n if (typeof (root._bokeh_timeout) === \"undefined\" || force) {\n root._bokeh_timeout = Date.now() + 5000;\n root._bokeh_failed_load = false;\n }\n\n function run_callbacks() {\n try {\n root._bokeh_onload_callbacks.forEach(function(callback) {\n if (callback != null)\n callback();\n });\n } finally {\n delete root._bokeh_onload_callbacks;\n }\n console.debug(\"Bokeh: all callbacks have finished\");\n }\n\n function load_libs(css_urls, js_urls, js_modules, js_exports, callback) {\n if (css_urls == null) css_urls = [];\n if (js_urls == null) js_urls = [];\n if (js_modules == null) js_modules = [];\n if (js_exports == null) js_exports = {};\n\n root._bokeh_onload_callbacks.push(callback);\n\n if (root._bokeh_is_loading > 0) {\n console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n return null;\n }\n if (js_urls.length === 0 && js_modules.length === 0 && Object.keys(js_exports).length === 0) {\n run_callbacks();\n return null;\n }\n if (!reloading) {\n console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n }\n\n function on_load() {\n root._bokeh_is_loading--;\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n run_callbacks()\n }\n }\n window._bokeh_on_load = on_load\n\n function on_error() {\n console.error(\"failed to load \" + url);\n }\n\n var skip = [];\n if (window.requirejs) {\n window.requirejs.config({'packages': {}, 'paths': {'jspanel': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/jspanel', 'jspanel-modal': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/modal/jspanel.modal', 'jspanel-tooltip': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/tooltip/jspanel.tooltip', 'jspanel-hint': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/hint/jspanel.hint', 'jspanel-layout': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/layout/jspanel.layout', 'jspanel-contextmenu': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/contextmenu/jspanel.contextmenu', 'jspanel-dock': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/dock/jspanel.dock', 'gridstack': 'https://cdn.jsdelivr.net/npm/gridstack@7.2.3/dist/gridstack-all', 'notyf': 'https://cdn.jsdelivr.net/npm/notyf@3/notyf.min'}, 'shim': {'jspanel': {'exports': 'jsPanel'}, 'gridstack': {'exports': 'GridStack'}}});\n require([\"jspanel\"], function(jsPanel) {\n\twindow.jsPanel = jsPanel\n\ton_load()\n })\n require([\"jspanel-modal\"], function() {\n\ton_load()\n })\n require([\"jspanel-tooltip\"], function() {\n\ton_load()\n })\n require([\"jspanel-hint\"], function() {\n\ton_load()\n })\n require([\"jspanel-layout\"], function() {\n\ton_load()\n })\n require([\"jspanel-contextmenu\"], function() {\n\ton_load()\n })\n require([\"jspanel-dock\"], function() {\n\ton_load()\n })\n require([\"gridstack\"], function(GridStack) {\n\twindow.GridStack = GridStack\n\ton_load()\n })\n require([\"notyf\"], function() {\n\ton_load()\n })\n root._bokeh_is_loading = css_urls.length + 9;\n } else {\n root._bokeh_is_loading = css_urls.length + js_urls.length + js_modules.length + Object.keys(js_exports).length;\n }\n\n var existing_stylesheets = []\n var links = document.getElementsByTagName('link')\n for (var i = 0; i < links.length; i++) {\n var link = links[i]\n if (link.href != null) {\n\texisting_stylesheets.push(link.href)\n }\n }\n for (var i = 0; i < css_urls.length; i++) {\n var url = css_urls[i];\n if (existing_stylesheets.indexOf(url) !== -1) {\n\ton_load()\n\tcontinue;\n }\n const element = document.createElement(\"link\");\n element.onload = on_load;\n element.onerror = on_error;\n element.rel = \"stylesheet\";\n element.type = \"text/css\";\n element.href = url;\n console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n document.body.appendChild(element);\n } if (((window['jsPanel'] !== undefined) && (!(window['jsPanel'] instanceof HTMLElement))) || window.requirejs) {\n var urls = ['https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/jspanel.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/modal/jspanel.modal.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/tooltip/jspanel.tooltip.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/hint/jspanel.hint.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/layout/jspanel.layout.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/contextmenu/jspanel.contextmenu.js', 'https://cdn.holoviz.org/panel/1.1.0/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/dock/jspanel.dock.js'];\n for (var i = 0; i < urls.length; i++) {\n skip.push(urls[i])\n }\n } if (((window['GridStack'] !== undefined) && (!(window['GridStack'] instanceof HTMLElement))) || window.requirejs) {\n var urls = ['https://cdn.holoviz.org/panel/1.1.0/dist/bundled/gridstack/gridstack@7.2.3/dist/gridstack-all.js'];\n for (var i = 0; i < urls.length; i++) {\n skip.push(urls[i])\n }\n } if (((window['Notyf'] 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\\n\"+\n \"

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\\n\"+\n \"\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
\"}};\n\n function display_loaded() {\n const el = document.getElementById(\"d874144b-6fd1-41dd-98ff-b7f30011fb10\");\n if (el != null) {\n el.textContent = \"BokehJS is loading...\";\n }\n if (root.Bokeh !== undefined) {\n if (el != null) {\n el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n }\n } else if (Date.now() < root._bokeh_timeout) {\n setTimeout(display_loaded, 100)\n }\n }\n\n function run_callbacks() {\n try {\n root._bokeh_onload_callbacks.forEach(function(callback) {\n if (callback != null)\n callback();\n });\n } finally {\n delete root._bokeh_onload_callbacks\n }\n console.debug(\"Bokeh: all callbacks have finished\");\n }\n\n function load_libs(css_urls, js_urls, callback) {\n if (css_urls == null) css_urls = [];\n if (js_urls == null) js_urls = [];\n\n root._bokeh_onload_callbacks.push(callback);\n if (root._bokeh_is_loading > 0) {\n console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n return null;\n }\n if (js_urls == null || js_urls.length === 0) {\n run_callbacks();\n return null;\n }\n console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n function on_load() {\n root._bokeh_is_loading--;\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n run_callbacks()\n }\n }\n\n function on_error(url) {\n console.error(\"failed to load \" + url);\n }\n\n for (let i = 0; i < css_urls.length; i++) {\n const url = css_urls[i];\n const element = document.createElement(\"link\");\n element.onload = on_load;\n element.onerror = on_error.bind(null, url);\n element.rel = \"stylesheet\";\n element.type = \"text/css\";\n element.href = url;\n console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n document.body.appendChild(element);\n }\n\n for (let i = 0; i < js_urls.length; i++) {\n const url = js_urls[i];\n const element = document.createElement('script');\n element.onload = on_load;\n element.onerror = on_error.bind(null, url);\n element.async = false;\n element.src = url;\n console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n document.head.appendChild(element);\n }\n };\n\n function inject_raw_css(css) {\n const element = document.createElement(\"style\");\n element.appendChild(document.createTextNode(css));\n document.body.appendChild(element);\n }\n\n const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.1.1.min.js\", \"https://unpkg.com/@holoviz/panel@1.1.0/dist/panel.min.js\"];\n const css_urls = [];\n\n const inline_js = [ function(Bokeh) {\n Bokeh.set_log_level(\"info\");\n },\nfunction(Bokeh) {\n }\n ];\n\n function run_inline_js() {\n if (root.Bokeh !== undefined || force === true) {\n for (let i = 0; i < inline_js.length; i++) {\n inline_js[i].call(root, root.Bokeh);\n }\nif (force === true) {\n display_loaded();\n }} else if (Date.now() < root._bokeh_timeout) {\n setTimeout(run_inline_js, 100);\n } else if (!root._bokeh_failed_load) {\n console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n root._bokeh_failed_load = true;\n } else if (force !== true) {\n const cell = $(document.getElementById(\"d874144b-6fd1-41dd-98ff-b7f30011fb10\")).parents('.cell').data().cell;\n cell.output_area.append_execute_result(NB_LOAD_WARNING)\n }\n }\n\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n run_inline_js();\n } else {\n load_libs(css_urls, js_urls, function() {\n console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n run_inline_js();\n });\n }\n}(window));" }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import copy\n", "import os\n", "import glob\n", "\n", "import numpy as np\n", "import pandas as pd\n", "\n", "import skimage.io\n", "import skimage.morphology\n", "import skimage.segmentation\n", "import skimage.measure\n", "\n", "import holoviews as hv\n", "hv.extension('bokeh')\n", "\n", "notebook_url = 'localhost:8888'\n", "\n", "import bokeh.io\n", "bokeh.io.output_notebook()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**a)** We will load the images into a tuple, where the index of the tuple corresponds to the frame number. Under normal circumstances, we would do data validation to make sure there are no skipped frames, but in the interest of brevity, we will proceed assuming all data are properly labeled and organized." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "There are 55 images.\n" ] } ], "source": [ "# Use a list comprehension to read in the images\n", "ims = [\n", " skimage.io.imread(fname)\n", " for fname in sorted(glob.glob(\"data/bacterial_growth/bacillus_*.tif\"))\n", "]\n", "\n", "# Store it as a tuple so we don't mess with it\n", "ims = tuple(ims)\n", "\n", "# How many?\n", "print(\"There are\", len(ims), \"images.\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To visualize the images before we get going, and also to visualize our success in segmentation, we can build a dashboard that shows the images and optionally an overlay of the segmentation mask. We first need a function to create the overlay." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "def make_overlay(im, im_bw):\n", " \"\"\"Create RGB image with cyan channel saturated via segmentation mask.\"\"\"\n", " # Make float and normalized image\n", " im = np.copy(im).astype(float)\n", "\n", " # Build RGB image by stacking grayscale images\n", " im_rgb = np.dstack(3 * [im / im.max()])\n", "\n", " # Saturate cyan channel wherever there are white pixels in thresh image\n", " im_rgb[im_bw, 0] = 1.0\n", "\n", " return im_rgb" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And now we can write a function to make the dashboard. It would be easier to make the dashboard and display it here. It will get automatically updated whenever we change the thresholding. So, after updating our thresholding method (with thresholded images stored in `ims_bw`), we should return to the dashboard to check it out." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "tags": [] }, "outputs": [ { "data": { "application/vnd.bokehjs_exec.v0+json": "", "text/html": [ "" ] }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "server_id": "a2edb0bf8f3f4094ad3d9012a409c60b" } }, "output_type": "display_data" } ], "source": [ "# Make thresholded images\n", "ims_bw = [im > skimage.filters.threshold_otsu(im) for im in ims]\n", "\n", "# PArameters for the display of images\n", "ip_distance = 0.0645\n", "dt = 15\n", "frame_width = 200\n", "frame_height = frame_width * ims[0].shape[0] // ims[1].shape[1]\n", "bounds = [\n", " 0,\n", " 0,\n", " ims[0].shape[1] * ip_distance,\n", " ims[0].shape[0] * ip_distance,\n", "]\n", "opts = dict(\n", " xlabel=\"µm\",\n", " ylabel=\"µm\",\n", " frame_width=frame_width,\n", " frame_height=frame_height,\n", ")\n", "\n", "# Slider for frame number\n", "frame_slider = bokeh.models.Slider(title=\"frame\", start=1, end=55, value=1, step=1)\n", "\n", "# Build initial plot\n", "frame = frame_slider.value\n", "title = f\"t = {dt*(frame-1)} min\"\n", "im = ims[frame - 1]\n", "\n", "im_rgb = make_overlay(im, ims_bw[frame - 1])\n", "p_overlay = hv.render(hv.RGB(im_rgb, bounds=bounds).opts(title=\"overlay\", **opts))\n", "p_phase = hv.render(\n", " hv.Image(im, bounds=bounds).opts(title=title, cmap=\"viridis\", **opts)\n", ")\n", "p_phase.x_range = p_overlay.x_range\n", "p_phase.y_range = p_overlay.y_range\n", "\n", "\n", "def callback(attr, old, new):\n", " frame = frame_slider.value\n", "\n", " title = f\"t = {dt*(frame-1)} min\"\n", " im = ims[frame - 1]\n", "\n", " im_rgb = make_overlay(im, ims_bw[frame - 1])\n", " dummy_p_overlay = hv.render(hv.RGB(im_rgb, bounds=bounds).opts(title=title, **opts))\n", " dummy_p = hv.render(\n", " hv.Image(im, bounds=bounds).opts(title=\"overlay\", cmap=\"viridis\", **opts)\n", " )\n", "\n", " p_overlay.renderers[0].data_source.data = dict(\n", " dummy_p_overlay.renderers[0].data_source.data\n", " )\n", " p_phase.renderers[0].data_source.data = dict(dummy_p.renderers[0].data_source.data)\n", "\n", "\n", "# Link callback\n", "frame_slider.on_change(\"value\", callback)\n", "\n", "# Build layout\n", "layout = bokeh.layouts.column(\n", " bokeh.layouts.column(\n", " frame_slider,\n", " bokeh.layouts.row(p_phase, p_overlay),\n", " )\n", ")\n", "\n", "\n", "def app(doc):\n", " doc.add_root(layout)\n", "\n", "\n", "# Away we go!\n", "bokeh.io.show(app)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**b)** For segmentation, we will take a similar approach as in [Lesson 44](l44_intro_to_image_processing.ipynb) to segment the image. We will simply threshold the image using Otsu's method. The dashboard above demonstrates that this works well." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**c)** See the dashboard above.\n", "\n", "The segmentation overshoots the bacteria. We might want to try to refine this a bit. We could do **adaptive thresholding**. We perform automatic thresholding on subimages throughout the image. We will take subimages to be 51 $\\times$ 51, since that would span the width of one bacterium, and then some, thereby avoiding the issue of thresholding within a single bacterium (and adaptive thresholding needs an odd image block size). Adaptive thresholding is achieved using `skimage.filters.threshold_local()`. Let's give it a shot." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "# List of thresholded images\n", "ims_bw = [im > skimage.filters.threshold_local(im, 51) for im in ims]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Checking out the dashboard again.... Whoa! It did a great job with the bacteria, but the background got really messed up. We could include pixels that are unity in both the adaptive and Otsu thresholding to take care of this. We then use `np.logical_and()` to do an element-by-element `AND` operation on the arrays." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "def thresh(im, block_size=51):\n", " \"\"\"Combined adaptive and Otsu thresholding.\"\"\"\n", " thresh_otsu = skimage.filters.threshold_otsu(im)\n", " im_bw = im > thresh_otsu\n", " return np.logical_and(im > skimage.filters.threshold_local(im, block_size), im_bw)\n", "\n", "# Threshold images\n", "ims_bw = [thresh(im) for im in ims]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now looking at the dashboard, it is very nice!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**d)** To plot the growth curve, we can plot the total bacterial area on the $y$-axis (on a log scale), versus time on the $x$-axis. To get the total area, we need to compute the total number of \"bacterial\" pixels in a given image, multiplied by the pixel area, which is $(64.5\\text{ nm})^2 = 4160.25\\text{ nm}^2$. Also, according to the metadata, we have one frame every 15 minutes." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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Plot the result\n", "p = bokeh.plotting.figure(\n", " height=250,\n", " width=400,\n", " y_axis_type='log',\n", " x_axis_label='time (hours)',\n", " y_axis_label='bacterial area (µm²)'\n", ")\n", "\n", "p.circle(\n", " x=t,\n", " y=bac_area\n", ")\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This looks pretty linear on the semilog plot, suggesting that the bacteria are, indeed, performing exponential growth." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**e)** To perform the regression on the growth curve, we use `np.polyfit()` with the `x` values being time, and the `y` values being the natural log of the bacterial area. 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root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n", " }\n", " if (root.Bokeh !== undefined) {\n", " embed_document(root);\n", " } else {\n", " let attempts = 0;\n", " const timer = setInterval(function(root) {\n", " if (root.Bokeh !== undefined) {\n", " clearInterval(timer);\n", " embed_document(root);\n", " } else {\n", " attempts++;\n", " if (attempts > 100) {\n", " clearInterval(timer);\n", " console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n", " }\n", " }\n", " }, 10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "p1150" } }, "output_type": "display_data" } ], "source": [ "# Use polyfit to get slope and intercept\n", "slope, intercept = np.polyfit(t, np.log(bac_area), 1)\n", "\n", "# Pull out and print parameters\n", "print(\"\"\"\n", "m0 = {0:.2f} sq. µm\n", " r = {1:.2f} 1/hours\n", "\"\"\".format(np.exp(intercept), slope))\n", "\n", "# Generate smooth curve\n", "t_smooth = np.linspace(0, t.max(), 200)\n", "y_smooth = np.exp(intercept + slope * t_smooth)\n", "\n", "# Add to the plot\n", "p.line(\n", " x=t_smooth,\n", " y=y_smooth,\n", " color='orange',\n", " line_width=2,\n", ")\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "There is some systematic error, it appears, with the data being consistently above the curve at short times and then below for intermediate times. This could be due to small changes in conditions over time, most likely delay in getting into exponential growth phase at short times." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computing environment" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Python implementation: CPython\n", "Python version : 3.11.3\n", "IPython version : 8.12.0\n", "\n", "numpy : 1.24.3\n", "pandas : 1.5.3\n", "skimage : 0.20.0\n", "bokeh : 3.1.1\n", "holoviews : 1.16.2\n", "jupyterlab: 3.6.3\n", "\n" ] } ], "source": [ "%load_ext watermark\n", "%watermark -v -p numpy,pandas,skimage,bokeh,holoviews,jupyterlab" ] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.11.3" } }, "nbformat": 4, "nbformat_minor": 4 }