Persister Cells – a Plausible Outcome of Neutral Coevolutionary Drift

The phenomenon of bacterial persistence – a non-inherited antibiotic tolerance in a minute fraction of the bacterial population, was observed more than 70 years ago. Nowadays, it is suggested that “persister cells” undergo an alternative scenario of the cell cycle; however, pathways involved in its emergence are still not identified. We present a mathematically grounded scenario of such possibility. We have determined that population drift in the space of multiple neutrally coupled mutations, which we called “neutrally coupled co-evolution” (NCCE), leads to increased dynamic complexity of bacterial populations via appearance of cells capable of carrying out a single cell cycle in two or more alternative ways and that universal properties of the coupled transcription-translation system underlie this phenotypic multiplicity. According to our hypothesis, modern persister cells have derived from such cells and regulatory mechanisms that govern the consolidation of this phenomenon represented the trigger. We assume that the described type of neutrally coupled co-evolution could play an important role in the origin of extremophiles, both in bacteria and archaea.

and the dilution rate of protein p concentration due to cell volume growth is described with equality 1 () dp As a result, the rate of change in protein concentration, taking into account its synthesis (Eq. S1) and dilution (Eq. S3), can be written in the form We shall note that protein degradation is not considered in Eq. S4, which is an important element involved in the cell cycle regulation of modern cells, considering the external and internal environmental factors. But, for simplicity, we excluded protein degradation from consideration, although degradation reduces protein concentration and under certain conditions can lead to complex chaotic changes in intracellular protein concentration (Likhoshvai et al., 2016).
We assume that if we can identify the mechanism for the formation of phenotypic multiplicity of the cell cycle, which does not include degradation processes, this would mean that protein degradation is not the primary phenomenon-forming factor, although it can play a similar role under certain conditions.
From H we choose a protein, which is consumed for the cell growth and denote it by r. Assume there is at least one such protein. This assumption is quite realistic. For example, in modern E. coli cells it is Lpp membrane protein (Inouye et al., 1972); such proteins exist in S. typhimurium, B. subtilis, Mycobacterium tuberculosis, etc. In our opinion, this statement is valid for the majority of cells (if not all) and therefore does not limit the generality of reasoning in any way. The consumption rate for such protein during cell volume growth is described by the equation Where, r is the stoichiometric coefficient equal to the number of protein r molecules consumed for the cell volume growth during one cycle. Then the general equation describing the rate of change in protein r concentration is written in the form Let us assume that cell growth is sufficiently effective in a sense that newly synthesized protein r molecules are rapidly consumed during cell growth, that is, the rate of cell growth is approximately equal to the rate of protein r synthesis Equality (S7) allows to exclude the law of cell volume growth from consideration and rewrite Eqs. S2 and S4 in the form Thus, we obtain a system of Eqs. S8,S9, which describes the rate of change in protein concentration during the cell cycle.
Let us consider the cell cycle as a dynamic process. A cell born at the time t0 develops for some time T, after which it divides and two daughter cells appear instead of the original (mother) cell at the time t1=t0+T. Each daughter cell at the time of division receives all the necessary components for subsequent growth. We assume that division occurs in a simplest symmetrical way: daughter cells receive exactly half of all molecules at the time of division and the daughter cell volume is exactly half the volume of the mother cell just before division. Repeatedly, each cell undergoes growth and subsequent division. During the cell cycle, the rate of change in protein concentration varies according to Eqs. S8 and S9. Now let us consider a cell cycle, in which daughter cells are identical to the mother cell at the time of its birth. That is, cell cycle of the daughter cell repeats the cell cycle of the mother cell. In other words, the cell cycle represents a stationary system. It can be expressed in the form of equalities Where, t is any time point of the cell cycle that is convenient to calculate according to an internal clock, taking the birth moment as 0.
It follows from Eqs. S10 that curves x(t) are cyclic. Hence, for each x there is 0txT, for which Let us now consider a protein that does not belong to the group B. Let us denote its concentration by m. Then, from Eq. S11, for this protein we obtain the following equality at the point tm    . It follows from Eq. S12 that if functions Sr and Sm do not depend on m, then m is uniquely expressed in terms of concentration of proteins from the group B. That is, if protein m does not belong to the group B, its synthesis and dilution can act as factors that engender phenotypic multiplicity of the cell cycle only if there are specific feedback regulatory loops in the process of its synthesis (Shearwin, 2009;Klumpp et al., 2009). Otherwise, synthesis and dilution of proteins not belonging to the group B are not capable of generating phenotypic multiplicity.
Let us now consider a protein from the group B. We denote its concentration by c. Then, at the point tc we have the equality In Eq. S13, B\c denotes a group of proteins B excluding protein c. It can be seen that functions Sr and Sс automatically depend on c and the number of solutions to equation S13 depends significantly on . It follows that synthesis and dilution of proteins from group B (RNA polymerases, ribosomal proteins) can potentially act as factors that engender phenotypic multiplicity of the cell cycle.
SI2. Analysis of the behavior of the adaptability functional W in the model (13) (see the main text of the article). We assume that evolutionary adaptation is directed towards increasing the specific growth rate of the «cell», the metabolism of which is in equilibrium. Therefore, for the model (13), the adaptability functional has the following form where, c is a positive root of Eq. 14, which corresponds to a stable steady-state of Eqs. 13 (see the main text of the article). If Eq. 14 has more than one steady-state, then the root value, for which the value of W is higher, is taken as c. Let us study the behavior of W (Eq. S14) for the synthesis rate functions described by Eq. 20. In this case Eq. S14 is written as    arbitrarily large due to the physical limitations of the rate of molecular processes); max K  (interaction between the synthesis factor and its target sites can be arbitrarily weak); 0 < min K (interaction between the synthesis factor and its target sites can not be arbitrarily effective due to the physical limitations of the rate of molecular processes). It is also obvious that there are objective physical limitations to the unlimited growth of the parameter c value. The simplest justification of this statement lies in a physical fact that in any finite volume there can be a finite (may be large but finite) number of molecules having nonzero volume. In fact, for any type of molecule, the real physiological boundary is much smaller, since there are thousands of molecules in a cell and all of them must function properly. Excessive density of molecules in a limited volume physically limits their mobility and negatively affects the reaction rate. As a result, we can assume that in the course of evolution the concentration value of the resource factor can not exceed a certain limiting value сcfis.
Let us consider the case in which the maximum root of the system Eq. 21 (the main text of article) for any fixed values r k and Because c k <kmax, then for any Therefore, maximum of the adaptability functional W is achieved when  . Under certain conditions, the maximum value of the adaptability functional W is realized not at a single point, but in some nontrivial parametric region, which should be called the region of neutrality. In the areas of neutrality, the parameter values can be chosen arbitrarily, considering the fulfillment of certain relations.
Let us interpret the obtained results given the paradigm that the evolutionary selection of the most adopted cells leads to an increase in the value of the adaptability functional W. Let us consider two cases: ,, x K x c r  . That is, cells carrying such combinations of mutations are not subjected to negative selection and all cells carrying these mutations have absolutely identical chances to survive or be eliminated from the population. Therefore, such coupled mutations are neutral. Accordingly, in a developing cell population, cells with any set of parameter values from the physiological range of parameters of the non-changing (maximal) adaptability functional W will appear over time.