![](\"figs/caltech_rosen.png\")
\n",
"\n",
"![](\"figs/stochastic_states_and_environments.png\")
![](\"figs/two_state_system_AvO.png\")
\n", "\n", "If the cells simply switch randomly between states, then the key physiological parameters are the switching rates, i.e. the probability per unit time that an individual cell switches from one state to the other. If the switching rate of the cell is much faster than the switching rate of the environment, then one would tend to have a mixed population of cells, as shown in the next figure on the left. On the other hand, if the cells switch infrequently compared to the rate of environmental change then one would have the situation on the right:\n", "\n", "
![](\"figs/fast_and_slow.png\")
\n", "\n", "The most basic question one can ask about such a system is: Given the environmental switching rate, and the growth rate (fitness) of each state in each environment, what are the optimal cellular switching rates?\n", "\n", "To address this question, one can construct a population dynamic model, in which we write down differential equations for the number of cells, or the number of cells in each state, rather than the number of proteins in the cell. Just as proteins can be synthesized and diluted/degraded, so too cells can proliferate and dilute/die. However, note that the rate of proliferation (at least in the exponential growth regime, far below the 'carrying capacity' of the environment), is proportional to the amount of cells one already has, rather than a constant production term as we have for protein expression.\n", "\n", "We write down a minimal model for two cellular states. For simplicity, we assume a symmetry: state 0 is equally as fit in environment 0 as state 1 is in environment 1. There are then only two (rather than four) growth rates we have to think about: the growth rate of either type of cell in its more ($\\gamma_1$) or less ($\\gamma_0$) fit environment. The other cellular parameters are the rates of switching from the more fit to the less fit state ($k_0$), or vice versa ($k_1$).\n", "\n", "\\begin{align}\n", "\\frac{dn_0}{dt} &= \\gamma_0(E) n_0 -k_1 n_0 + k_0 n_1 \\\\[1em]\n", "\\frac{dn_1}{dt} &= \\gamma_1(E) n_1 +k_1 n_0 - k_0 n_1\n", "\\end{align}\n", "\n", "The overall growth rate, which we take to be the fitness, is calculated as the growth rate of the total population size, $n=n_0+n_1$:\n", "\n", "\\begin{align}\n", "\\gamma &= \\frac{1}{n} \\frac{dn}{dt} \\\\[1em]\n", "&=\\frac {\\frac{dn_0}{dt}+\\frac{dn_1}{dt}}{n_0+n_1}\n", "\\end{align}\n", "\n", "Analyzing this model shows that the optimal switching rate depends, as you might expect, on the switching rates of the environment, as shown here:\n", "\n", "
![](\"figs/acar_fluctuating.png\")
![](\"figs/kussell_leibler.png\")
\n", "\n", "The dynamics are then described by a set of equations for the size of each cell state population, collected together in a cellular state vector, $\\vec{x}$. The dynamics of $\\vec{x}$ depend in turn on the environments, the switching rates, and the growth rates of each cell state in each environment, all collected together into the matrix $\\mathbf{A_k}$ as shown in the diagram above.\n", "\n", "\\begin{align}\n", "\\frac{\\mathrm{d}\\vec{x}}{\\mathrm{d}t} = \\mathbf{A_k} \\vec{x}\n", "\\end{align}\n", "\n", "Each strategy incurs a different type of cost:\n", "\n", "* Sensing requires a constant cost in the production of the sensing machinery, which is not \"paid\" by the stochastic switchers. \n", "* The stochastic switching strategy incurs a \"diversity cost\", due to populating sub-optimal states in each environment. \n", "\n", "They derive expressions for the growth rates in the two strategies:\n", "\n", "* Responsive switching: Long-term growth = Fastest growth - **sensing cost** - delay time cost\n", "\n", "* Stochastic switching: Long-term growth = Fastest growth - **diversity cost** - delay time cost\n", "\n", "Note the different costs that are incurred in the two strategies. Generally speaking, sensing is favored when the diversity costs would be high, sensing costs are low, and the environment is highly variable. By contrast, stochastic switching can be favored in regimes in which sensing costs are higher, and environments are stable over longer times. (These statements are made mathematically precise in the paper). \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Antibiotic persistence: a stochastic switching strategy\n", "\n", "When one adds antibiotics to a bacterial culture, one often finds that they can kill almost, but not quite, all of the cells. A small fraction of cells can survive the antibiotic. Furthermore, these cells turn out to _not_ be resistant mutants. When one grows them up into a new population in the absence of the antibiotic, and then again exposes them to antibiotic, only the same tiny fraction of cells again survive. \n", "\n", "Here is a definition of persistence from Allison et al. ([*Current Opinion in Microbiology*, 2011](https://doi.org/10.1016/j.mib.2011.09.002)):\n", "\n", "> Bacterial persistence is a phenomenon in which a sub-population of cells survives antibiotic treatment.... In contrast to resistant bacteria, persisters do not grow in the presence of antibiotics and their tolerance arises from physiological processes rather than genetic mutations... Persistence was first described by [Joseph Bigger in 1944](https://doi.org/10.1007/BF02948462) while attempting to sterilize cultures of pathogenic *Staphylococcus aureus* with penicillin. He found that a small number of cells \"persisted\" and could later form colonies even after treatment with high antibiotic concentrations.\n", "\n", "Persistence is a major problem in many infectious diseases: \n", "* Persistence leads to recurrent infections\n", "* It extends duration of treatments\n", "* It plays an important role in the therapy choice for immunocompromised patients.\n", "* It is important for understanding infections poorly controlled by the immune system, such as tuburculosis.\n", "\n", "Persistence is not specific to bacteria. A similar phenomenon allows cancer cells to survive chemotherapeutics. For a recent review, see: Maisonneuve and Gerdes ([Cell, 2014](https://doi.org/10.1016/j.cell.2014.02.050)).\n", "\n", "The general phenomenon is diagrammed here:\n", "\n", "
![](\"figs/persistence_diagram.png\")
\n", "\n", "There are different potential explanations for persistence:\n", "\n", "* Spatial heterogeneity: aggregates, etc.\n", "* Temporal heterogeneity: adaptive response to the antibiotic\n", "* Pre-existing heterogeneity?\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Persistence in movies\n", "\n", "The most direct way to visualize persistence is to look at it directly using time-lapse movies. Notice here the large variation in the time at which different cells start growing, even though they are all in essentially the same environment.\n", "\n", "
![](\"figs/balaban_device.jpeg\")
\n", "\n", "Here is their diagram of what they saw in the movie:\n", "\n", "
![](\"figs/balaban_movie_diagram.png\")
![](\"figs/type_I_or_II.jpg\")