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t,i,o,a;if(r)for(i=0,o=b(this),a=yt(e);t=a[i++];)o.hasClass(t)?o.removeClass(t):o.addClass(t);else void 0!==e&&\"boolean\"!==n||((t=vt(this))&&G.set(this,\"__className__\",t),this.setAttribute&&this.setAttribute(\"class\",t||!1===e?\"\":G.get(this,\"__className__\")||\"\"))}))},hasClass:function(e){var t,n,r=0;for(t=\" \"+e+\" \";n=this[r++];)if(1===n.nodeType&&(\" \"+gt(vt(n))+\" \").indexOf(t)>-1)return!0;return!1}});var mt=/\\r/g;b.fn.extend({val:function(e){var t,n,r,i=this[0];return arguments.length?(r=h(e),this.each((function(n){var i;1===this.nodeType&&(null==(i=r?e.call(this,n,b(this).val()):e)?i=\"\":\"number\"==typeof i?i+=\"\":Array.isArray(i)&&(i=b.map(i,(function(e){return null==e?\"\":e+\"\"}))),(t=b.valHooks[this.type]||b.valHooks[this.nodeName.toLowerCase()])&&\"set\"in t&&void 0!==t.set(this,i,\"value\")||(this.value=i))}))):i?(t=b.valHooks[i.type]||b.valHooks[i.nodeName.toLowerCase()])&&\"get\"in t&&void 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o,a,s,u,l,f,p,d,y=[r||v],m=c.call(t,\"type\")?t.type:t,x=c.call(t,\"namespace\")?t.namespace.split(\".\"):[];if(a=d=s=r=r||v,3!==r.nodeType&&8!==r.nodeType&&!xt.test(m+b.event.triggered)&&(m.indexOf(\".\")>-1&&(x=m.split(\".\"),m=x.shift(),x.sort()),l=m.indexOf(\":\")<0&&\"on\"+m,(t=t[b.expando]?t:new b.Event(m,\"object\"==typeof t&&t)).isTrigger=i?2:3,t.namespace=x.join(\".\"),t.rnamespace=t.namespace?new RegExp(\"(^|\\\\.)\"+x.join(\"\\\\.(?:.*\\\\.|)\")+\"(\\\\.|$)\"):null,t.result=void 0,t.target||(t.target=r),n=null==n?[t]:b.makeArray(n,[t]),p=b.event.special[m]||{},i||!p.trigger||!1!==p.trigger.apply(r,n))){if(!i&&!p.noBubble&&!g(r)){for(u=p.delegateType||m,xt.test(u+m)||(a=a.parentNode);a;a=a.parentNode)y.push(a),s=a;s===(r.ownerDocument||v)&&y.push(s.defaultView||s.parentWindow||e)}for(o=0;(a=y[o++])&&!t.isPropagationStopped();)d=a,t.type=o>1?u:p.bindType||m,(f=(G.get(a,\"events\")||Object.create(null))[t.type]&&G.get(a,\"handle\"))&&f.apply(a,n),(f=l&&a[l])&&f.apply&&X(a)&&(t.result=f.apply(a,n),!1===t.result&&t.preventDefault());return t.type=m,i||t.isDefaultPrevented()||p._default&&!1!==p._default.apply(y.pop(),n)||!X(r)||l&&h(r[m])&&!g(r)&&((s=r[l])&&(r[l]=null),b.event.triggered=m,t.isPropagationStopped()&&d.addEventListener(m,bt),r[m](),t.isPropagationStopped()&&d.removeEventListener(m,bt),b.event.triggered=void 0,s&&(r[l]=s)),t.result}},simulate:function(e,t,n){var r=b.extend(new b.Event,n,{type:e,isSimulated:!0});b.event.trigger(r,null,t)}}),b.fn.extend({trigger:function(e,t){return this.each((function(){b.event.trigger(e,t,this)}))},triggerHandler:function(e,t){var n=this[0];if(n)return b.event.trigger(e,t,n,!0)}}),d.focusin||b.each({focus:\"focusin\",blur:\"focusout\"},(function(e,t){var n=function(e){b.event.simulate(t,e.target,b.event.fix(e))};b.event.special[t]={setup:function(){var r=this.ownerDocument||this.document||this,i=G.access(r,t);i||r.addEventListener(e,n,!0),G.access(r,t,(i||0)+1)},teardown:function(){var r=this.ownerDocument||this.document||this,i=G.access(r,t)-1;i?G.access(r,t,i):(r.removeEventListener(e,n,!0),G.remove(r,t))}}}));var wt=e.location,Tt={guid:Date.now()},Ct=/\\?/;b.parseXML=function(t){var n;if(!t||\"string\"!=typeof t)return null;try{n=(new e.DOMParser).parseFromString(t,\"text/xml\")}catch(e){n=void 0}return n&&!n.getElementsByTagName(\"parsererror\").length||b.error(\"Invalid XML: \"+t),n};var 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],
"text/plain": [
":NdLayout [Strain]\n",
" :HoloMap [Antibody]\n",
" :NdOverlay [Variant]\n",
" :Scatter [Ab_Conc_ng_mL,OD450] (Antibody,Variant,Strain)"
]
},
"execution_count": 11,
"metadata": {
"application/vnd.holoviews_exec.v0+json": {
"id": "1001"
}
},
"output_type": "execute_result"
}
],
"source": [
"hv.extension(\"bokeh\")\n",
"\n",
"scatter_plots = hv.Scatter(\n",
" data=df, \n",
" kdims=[\"Ab_Conc_ng_mL\", \"OD450\"], vdims=[\"Antibody\", \"Variant\", \"Strain\"],\n",
").groupby(\n",
" [\"Antibody\", \"Variant\", \"Strain\"]\n",
").opts(\n",
" logx=True,\n",
" frame_height=200,\n",
" frame_width=300,\n",
" xlabel=\"Antibody Concentration (ng/mL)\",\n",
" size=7,\n",
").overlay(\n",
" \"Variant\"\n",
").layout(\n",
" \"Strain\"\n",
").cols(\n",
" 1\n",
")\n",
"\n",
"scatter_plots"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Estimation of the EC50\n",
"\n",
"From ELISA data, we also typically want the half maximal effective concentration, or **EC50**. It tells you how much antibody concentration you need to reach half of the maximum OD450. This is typically accomplished by obtaining a maximum likelihood estimate (MLE) of parameters of a Hill function,\n",
"\n",
"\\begin{align}\n",
"y = d + \\frac{a-d}{1 + \\left(\\frac{x}{c} \\right)^b}.\n",
"\\end{align}\n",
"\n",
"The function has an inverse that can be used to compute the EC50 from the MLE parameters.\n",
"\n",
"\\begin{align}\n",
"x = c \\left(\\frac{y - a}{d - y}\\right)^{1/b}\n",
"\\end{align}\n",
"\n",
"This is a phenomenological approach widely used in the field (but not something Justin recommends). In the functions below, we compute the MLE. We will visit MLE in depth later in the term; for now, take this calculation as a given and as a way to demonstrate application groupby operations.\n",
"\n",
"Importantly, the function `compute_ec50()` takes as arguments x-values and y-values, performs a MLE, and then returns the resulting EC50."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"def four_param_hill(x, a, b, c, d):\n",
" \"\"\"Code up the above 4-parameter Hill function\"\"\"\n",
" return d + (a - d) / (1 + (x / c) ** b)\n",
"\n",
"\n",
"def inverse_four_param_hill(y, a, b, c, d):\n",
" \"\"\"Inverse of the four parameter fit in order to compute concentration from absorbance\"\"\"\n",
" return c * ((y - a) / (d - y)) ** (1 / b)\n",
"\n",
"\n",
"def compute_ec50(x, y):\n",
" # Perform the fit using scipy and save the optimal parameters (popt)\n",
" popt, _ = scipy.optimize.curve_fit(four_param_hill, x, y)\n",
"\n",
" # Compute the expected values from the logistic regression\n",
" fit = np.array(four_param_hill(x, *popt))\n",
"\n",
" # Compute the half maximal value in order to find the EC50\n",
" halfway = (np.min(fit) + np.max(fit)) / 2\n",
" ec50 = inverse_four_param_hill(halfway, *popt)\n",
"\n",
" return ec50"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We wish to compute the EC50 for each strain/variant/antibody combination. To do so, we group by through three variables and apply the `compute_ec50()` function. Any function we wish to apply needs to take a group (a sub-data frame) as an argument. So, we can write a lambda function to call `compute_fit()`. "
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Strain Variant Antibody \n",
"1a157 Q412R HEPC153 80.001005\n",
" HEPC153rua 1191.675379\n",
" HEPC74 37.760901\n",
" HEPC74rua 63.641514\n",
" WT HEPC153 40.358598\n",
" HEPC153rua 3353.090522\n",
" HEPC74 26.682279\n",
" HEPC74rua 33.763761\n",
" Y527H HEPC153 34.673164\n",
" HEPC153rua 391.143463\n",
" HEPC74 24.829800\n",
" HEPC74rua 27.293204\n",
"1a53 Q412R HEPC153 36.874215\n",
" HEPC153rua 509.791761\n",
" HEPC74 30.436993\n",
" HEPC74rua 33.391581\n",
" WT HEPC153 36.111559\n",
" HEPC153rua 599.556461\n",
" HEPC74 27.502283\n",
" HEPC74rua 31.517035\n",
"dtype: float64"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ec50_results = df.groupby([\"Strain\", \"Variant\", \"Antibody\"]).apply(\n",
" lambda group: compute_ec50(group[\"Ab_Conc_ng_mL\"].values, group[\"OD450\"].values)\n",
")\n",
"\n",
"# Take a look\n",
"ec50_results"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The result is a Pandas series with a multiindex. We can convert this to a data frame by resetting the index, and then sort by strain and antibody for convenience."
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
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\n",
" \n",
"
\n",
"
"
],
"text/plain": [
" Strain Variant Antibody EC50\n",
"0 1a157 Q412R HEPC153 80.001005\n",
"1 1a157 WT HEPC153 40.358598\n",
"2 1a157 Y527H HEPC153 34.673164\n",
"3 1a157 Q412R HEPC153rua 1191.675379\n",
"4 1a157 WT HEPC153rua 3353.090522\n",
"5 1a157 Y527H HEPC153rua 391.143463\n",
"6 1a157 Q412R HEPC74 37.760901\n",
"7 1a157 WT HEPC74 26.682279\n",
"8 1a157 Y527H HEPC74 24.829800\n",
"9 1a157 Q412R HEPC74rua 63.641514\n",
"10 1a157 WT HEPC74rua 33.763761\n",
"11 1a157 Y527H HEPC74rua 27.293204\n",
"12 1a53 Q412R HEPC153 36.874215\n",
"13 1a53 WT HEPC153 36.111559\n",
"14 1a53 Q412R HEPC153rua 509.791761\n",
"15 1a53 WT HEPC153rua 599.556461\n",
"16 1a53 Q412R HEPC74 30.436993\n",
"17 1a53 WT HEPC74 27.502283\n",
"18 1a53 Q412R HEPC74rua 33.391581\n",
"19 1a53 WT HEPC74rua 31.517035"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Set the name of the series\n",
"ec50_results.name = \"EC50\"\n",
"\n",
"# Convert to data frame and sort\n",
"ec50_results.reset_index().sort_values(by=[\"Strain\", \"Antibody\"]).reset_index(drop=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We've almost completely recreated the ELISA plotting in Prism, a common subscription software used by biologists to plot all sorts of data. We also didn't have to copy and paste columns and rows from Excel over and over again.\n",
"\n",
"In EC50 values, we are interested in the smaller numbers because they indicate that LESS antibody was required to achieve the same signal. It appears that our mutants have a mix of larger and smaller EC50 values relative to the wild-type strains. \n",
"\n",
"We could also plot the regression lines on the same plots with the points to do a visual test of the EC50 estimates. We do not do this here, opting instead to wait for the discussion on MLE and visualization of the results later in the course."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Computing Environment"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPython 3.8.5\n",
"IPython 7.18.1\n",
"\n",
"numpy 1.19.1\n",
"scipy 1.5.0\n",
"pandas 1.1.3\n",
"bokeh 2.2.3\n",
"holoviews 1.13.4\n",
"bebi103 0.1.1\n",
"jupyterlab 2.2.6\n"
]
}
],
"source": [
"%load_ext watermark\n",
"%watermark -v -p numpy,scipy,pandas,bokeh,holoviews,bebi103,jupyterlab"
]
}
],
"metadata": {
"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": "3.8.5"
}
},
"nbformat": 4,
"nbformat_minor": 4
}