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"text/plain": [
"Column\n",
" [0] FloatSlider(name='κ', start=0.01, step=0.01, value=0.5)\n",
" [1] Spacer(height=10)\n",
" [2] FloatSlider(end=2, name='log₁₀ ζ', start=-2, step=0.05)\n",
" [3] Spacer(height=10)\n",
" [4] ParamFunction(function)"
]
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"source": [
"def michaelis_menton_rhs(c, t, kappa, zeta):\n",
" ces, cs, cp = c\n",
" return np.array(\n",
" [\n",
" ((1 - ces) * cs - ces) / kappa,\n",
" zeta * (-(1 - ces) * cs + kappa * ces),\n",
" zeta * (1 - kappa) * ces,\n",
" ]\n",
" )\n",
"\n",
"\n",
"def approx_michaelis_menton_rhs(c, t, kappa, zeta):\n",
" cs, cp = c\n",
" deriv = zeta * (1 - kappa) * cs / (1 + cs)\n",
" return np.array([-deriv, deriv])\n",
"\n",
"\n",
"kappa_slider = pn.widgets.FloatSlider(\n",
" name=\"κ\", start=0.01, end=1, value=0.5, step=0.01\n",
")\n",
"log_zeta_slider = pn.widgets.FloatSlider(\n",
" name=\"log₁₀ ζ\", start=-2, end=2, value=0, step=0.05\n",
")\n",
"\n",
"\n",
"@pn.depends(kappa_slider.param.value, log_zeta_slider.param.value)\n",
"def plot_mm(kappa, log_zeta):\n",
" # Initial condition\n",
" c0 = np.array([0.01, 10, 0.01])\n",
"\n",
" # Time points\n",
" t = np.linspace(0, 100, 400)\n",
"\n",
" # Solve the full system\n",
" zeta = 10 ** log_zeta\n",
" c = scipy.integrate.odeint(michaelis_menton_rhs, c0, t, args=(kappa, zeta))\n",
" ces, cs, cp = c.transpose()\n",
"\n",
" # Solve the approximate system\n",
" c_approx = scipy.integrate.odeint(\n",
" approx_michaelis_menton_rhs, c0[1:], t, args=(kappa, zeta)\n",
" )\n",
" cs_approx, cp_approx = c_approx.transpose()\n",
"\n",
" # Set up plot\n",
" p = bokeh.plotting.figure(\n",
" plot_width=500,\n",
" plot_height=250,\n",
" x_axis_label=\"t\",\n",
" y_axis_label=\"dimensionless concentration\",\n",
" y_axis_type=\"linear\",\n",
" x_range=[0, 100],\n",
" )\n",
"\n",
" colors = bokeh.palettes.d3[\"Category10\"][3]\n",
"\n",
" # Populate glyphs\n",
" p.line(\n",
" t, ces, line_width=2, color=colors[0], legend_label=\"ES\",\n",
" )\n",
" p.line(\n",
" t, cs, line_width=2, color=colors[1], legend_label=\"S\",\n",
" )\n",
" p.line(\n",
" t, cp, line_width=2, color=colors[2], legend_label=\"P\",\n",
" )\n",
" p.line(\n",
" t, cs_approx, line_width=4, color=colors[1], alpha=0.3,\n",
" )\n",
" p.line(\n",
" t, cp_approx, line_width=4, color=colors[2], alpha=0.3,\n",
" )\n",
"\n",
" p.legend.location = \"center_right\"\n",
"\n",
" return p\n",
"\n",
"\n",
"pn.layout.Column(\n",
" kappa_slider,\n",
" pn.Spacer(height=10),\n",
" log_zeta_slider,\n",
" pn.Spacer(height=10),\n",
" plot_mm,\n",
")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In manipulating the sliders, we see that, indeed, if $\\zeta$ is small (that is we do not have too much enzyme), then the concentration of ES quickly goes to a steady level while the concentration of product grows. The approximate solution to the Michaelis-Menten dynamics, at least for production of product, very closely follows that obtained for the full dynamics."
]
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Approximate dynamics of the Che signaling system\n",
"\n",
"Because of the more verbose names of the species in the Che system (e.g., \"$X_m^*$\" instead of \"S\"), we will denote concentrations of respective species with lower case letter, e.g. $x_m^*$.\n",
"\n",
"We will assume that the quasi-steady state approximation holds for all enzyme-catalyzed reactions. We assume further that CheR operates at saturation. This means that there is so much substrate (in this case unmethlyated CheA receptor complex) such that the rate of methylation by CheR is directly proportional to the CheR concentration, or\n",
"\n",
"\\begin{align}\n",
"\\frac{r/K_R}{1+r/K_R} \\approx 1.\n",
"\\end{align}\n",
"\n",
"When this is the case, it is said that CheR **operates at saturation**.\n",
"\n",
"We can write an ODE for the rate of change of _total_ methylated CheA. At steady state, this derivative vanishes.\n",
"\n",
"\\begin{align}\n",
"\\frac{\\mathrm{d}(x^*_m + x_m)}{\\mathrm{d}t} = \\frac{\\mathrm{d}x_m^\\mathrm{tot}}{\\mathrm{d}t} = 2v_R r - v_B b\\left(\\frac{x_m^*/K_m}{1 + x_M^*/K_m} + \\frac{x_m/\\kappa_m}{1 + x_m/\\kappa_m}\\right) = 0,\n",
"\\end{align}\n",
"\n",
"where $K_m$ and $\\kappa_m$ are Michaelis constants for demethylation by CheB. This equation is readily solved for $x_m^*$.\n",
"\n",
"\\begin{align}\n",
"x_m^* = K_m\\,\\frac{2 v_R r - v_B b\\,\\frac{x_m/\\kappa_m}{1 + x_m/\\kappa_m}}{v_B b - 2v_R r + v_B b\\,\\frac{x_m/\\kappa_m}{1 + x_m/\\kappa_m}}\n",
"\\end{align}\n",
"\n",
"Note that as $x_m$ grows, the numerator decreases, while the denominator increases. So, for increasing $x_m$, $x_m^*$ decreases. Because the amount of $x_m$ is dependent on the amount of attractant present (more attractant means the equilibrium shifts toward the inactive conformations, $x$ and $x_m$), the concentration of active CheA receptor complex is dependent on the attractant concentration in this model."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Fine-tuned exact adaptation\n",
"\n",
"\n",
"We an imagine a scenario where we could get exact adaptation in tumbling frequency from this circuit. If the attractant affects the phosphotransfer and the CheY response, we can write\n",
"\n",
"\\begin{align}\n",
"\\text{tumbling frequency} \\equiv A = a(L)\\,x_m^*,\n",
"\\end{align}\n",
"\n",
"where $L$ is the concentration of attractant ligand and $a(L)$ is some function. Before the attract was present, we have a tumbling frequency of $A_0 = a_0 x_{m,0}^*$, and after we have $A_1 = a_1 x_{m,1}^*$, where $a_0 = a(L_0)$ and $a_1 = a(L_1)$. Note that addition of attractant also shifts the balance between methylated and unmethylated CheA receptor complex, so $x_m$ is affected. We define \n",
"\n",
"\\begin{align}\n",
"\\theta = 2v_r r - v_b b\\,\\frac{x_m/\\kappa_m}{1 + x_m/\\kappa_m}.\n",
"\\end{align}\n",
"\n",
"Then, the concentration of active methylated CheA complex is\n",
"\n",
"\\begin{align}\n",
"x_m^* = K_m\\,\\frac{\\theta}{v_B b - \\theta}.\n",
"\\end{align}\n",
"\n",
"With this expression, we can write down the two tumbling frequencies.\n",
"\n",
"\\begin{align}\n",
"A_0 = a_0 x_{m,0}* = a_0 K_m \\,\\frac{\\theta_0}{v_B v \\theta_0}, \\\\[1em]\n",
"A_1 = a_1 x_{m,1}* = a_1 K_m \\,\\frac{\\theta_1}{v_B v \\theta_1}.\n",
"\\end{align}\n",
"\n",
"For exact adaptation, for this particular concentration of attractant, $A_0 = A_1$, or\n",
"\n",
"\\begin{align}\n",
"\\frac{a_1}{a_0} = \\frac{\\theta_1(v_B b - \\theta_0)}{\\theta_0(v_B b - \\theta_1)}.\n",
"\\end{align}\n",
"\n",
"The problem is that this ratio must hold for _any_ $a_0$ and $a_1$ within an operating regime if exact adaptation is to be robust. This results in a delicate balancing act, since $\\theta$ changes with varying attractant concentration. This is an example of **fine-tuning**. Exact adaptation is only possible by precisely setting parameters for a given pair of attractant concentrations, and the exact adaptation is lost if those attractant concentrations change at all."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Robust exact adaptation\n",
"\n",
"Let us now assume a different architecture of the Che circuit. We will assume that CheB only demethylates _active_ receptors, rendering them inactive. The circuit diagram in this case is\n",
"\n",
"\n",
" \n",
"\n",
"\n",
"
\n",
" \n",
"Taking the same approach as before, we can again write down the dynamics for the total amount of methylated CheA complex. We will again assume that CheR is operating at saturation.\n",
"\n",
"\\begin{align}\n",
"\\frac{\\mathrm{d}(x_m +x_m^*)}{\\mathrm{d}t} = v_R r - v_B b\\,\\frac{x_m^*/K_m}{1+x_m^*/K_m}.\n",
"\\end{align}\n",
"\n",
"At steady state, the time derivative vanishes, and we have\n",
"\n",
"\\begin{align}\n",
"x_m^* = K_m\\,\\frac{v_R r}{v_B b - v_R r}.\n",
"\\end{align}\n",
"\n",
"This is independent of $x_m$, and is dependent only on the parameters and concentrations pertaining to the CheB and CheR enzymes. If we have the simplest model where $a(L) = \\text{constant}$, then exact adaptation is achieved over all attractant concentrations. Furthermore, exact adaptation is robust to variations in parameter values (like CheR concentration, $v_B$, etc.), provided the Michaelis-Menten kinetics are valid and CheR operates at saturation.\n",
"\n",
"Note, however, that the steady state tumbling frequency is _not_ robust to changes in parameter values. The exact adaptation is robust to changes in parameter values, but the level of the steady state has explicit dependence on them and is therefore not robust to fluctuations in parameter values."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## A more complex model for chemotaxis\n",
"\n",
"In the model for chemotaxis that we have considered, there are only two methylation states of the receptor, either unmethylated or methylated. In a more general model, we could consider multiple methylation states of the receptor. That is, it could be unmethylated, once methylated, twice methylated, ..., m-times methylated. Such a system was analyzed computationally by [Barkai and Leibler (1997)](https://doi.org/10.1038/43199).\n",
"\n",
"Because the model for generic number of methylation states is not tractable analytically as the one we have considered, Barkai and Leibler instead did a computational exploration of the parameter space. They began with a set of $n$ parameters $k_1^0$, $k_2^0$, ..., $k_n^0$. They randomly varied these parameters to get a new set of parameters $k_1$, $k_2$, ..., $k_n$. They defined the total parameter variation $K$ according to\n",
"\n",
"\\begin{align}\n",
"\\ln K = \\sum_{i=1}^n \\left|\\ln\\,\\frac{k_i}{k_i^0}\\right|.\n",
"\\end{align}\n",
"\n",
"For this new set of parameters, they solved the system of ODEs numerically from a steady state in response to an abrupt step up in ligand. If the probability of being in an active state for the original set of parameters is $A_0$ and for the perturbed parameters is A, then the precision of adaptation, $P$, is defined as $P = A_0 / A$. \n",
"\n",
"The results from thousands of such perturbations and calculations of the precision of adaptation is shown below. This image was taken from the Barkai and Liebler paper (which unfortunately has the Nature Publishing Group watermark on it). \n",
"\n",
"\n",
" \n",
"\n",
"\n",
"
\n",
"\n",
"\n",
"In the top plot, the diamond represented the original set of parameters. In the inset, the response for the original set of parameters is shown (solid line) along with an example response, corresponding to the white box.\n",
"\n",
"To make clear the overlapping points in the top plot, the bottom plot shows the fraction of parameter sets for which the precision of adaptation $P < 0.95$ for various values of the total parameter variation $K$. These plots demonstrate that even over large degrees of parameter variation, up to a total of four or five orders of magnitude, the system can still achieve rather exact adaptation.\n",
"\n",
"Conversely, Barkai and Leibler found that the time to steady state is *not* robust to parameter variations. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exact adaptation in the Che system is robust to CheR levels, as shown experimentally\n",
"\n",
"In a follow-up experimental paper ([Alon, et al., 2000](https://doi.org/10.1038/16483)), the authors varied CheR expression levels by putting CheR under control of the *lac* promoter and varying the concentrations of IPTG. They used microscopy to observe the tumbling frequency of cells. They spiked in a saturating amount of attractant and observed the tumbling frequency over time. The results from their experiment are shown below, reproduced from the paper.\n",
"\n",
"\n",
" \n",
"\n",
"\n",
"
\n",
"\n",
"Commensurate with the model predictions, the precision of adaptation is robust to varying levels of CheR over about two orders of magnitude. They also found that the time it took to reach a steady state tumbling frequency and the steady state frequency itself were *not* robust to varying levels of CheR."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Integral feedback\n",
"\n",
"The chemotaxis system we have studied in this lesson achieves exact adaptation robustly via a mechanism called **integral feedback control**, as first noticed by [Yi and coworkers](https://doi.org/10.1073/pnas.97.9.4649). The key features of integral feedback control are:\n",
"\n",
"- The steady state is independent of external perturbation.\n",
"- The system tends toward steady state as time tends toward infinity; i.e., it is asymptotically stable.\n",
"- The time integral of error is fed back into the system.\n",
"\n",
"A block diagram for integral control is shown below. The desired output is $y_0$ and the actual output is $y$, so the error is $e = y_0 - y$. This error is integrated over time, multiplied by a gain $k$, and then added to the a perturbation $u$. In the context of the chemotaxis system, the perturbation is the change in attractant concentration.\n",
"\n",
"\n",
" \n",
"\n",
"\n",
"
\n",
"\n",
"Image from Del Vecchio and Murray, *Biomolecular Feedback Systems*, Princeton University Press, 2015.\n",
"\n",
"
\n",
"\n",
"If we write ODEs for this diagram, we have that\n",
"\n",
"\\begin{align}\n",
"\\frac{\\mathrm{d}z}{\\mathrm{d}t} = e(t) = y_0 - y,\n",
"\\end{align}\n",
"\n",
"with $y = kz + u$, giving\n",
"\n",
"\\begin{align}\n",
"\\frac{\\mathrm{d}z}{\\mathrm{d}t} = e(t) = y_0 - kz - u.\n",
"\\end{align}\n",
"\n",
"The steady state is then $y = y_0$, which is asymptotically stable if the gain $k$ is positive and the perturbation is constant (such as a change in attractant concentration). In the present case, the term $kz$ represents the linearized balance between methylated and unmethylated CheA receptor complex. Importantly, the integral feedback ensures that the steady state is independent of the perturbation $u$.\n",
"\n",
"You can read more about integral feedback in [section 3.2 of Del Vecchio and Murray's book](http://www.cds.caltech.edu/~murray/books/AM08/pdf/bfs-dynamics_14Sep14.pdf) and in [this very nice primer on control theory in synthetic biology](https://doi.org/10.1109/MCS.2018.2810459) by Swaminathan, Hsiao, and Murray."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Computing environment"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"tags": [
"hide_input"
]
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPython 3.7.6\n",
"IPython 7.12.0\n",
"\n",
"numpy 1.18.1\n",
"scipy 1.4.1\n",
"bokeh 1.4.0\n",
"panel 0.8.3\n",
"biocircuits 0.0.8\n",
"jupyterlab 1.2.6\n"
]
}
],
"source": [
"%load_ext watermark\n",
"%watermark -v -p numpy,scipy,bokeh,panel,biocircuits,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": "3.7.6"
}
},
"nbformat": 4,
"nbformat_minor": 4
}