Ribosome Abundance Control in Prokaryotes

Abstract

Cell growth is an essential phenotype of any unicellular organism and it crucially depends on precise control of protein synthesis. We construct a model of the feedback mechanisms that regulate abundance of ribosomes in E. coli, a prototypical prokaryotic organism. Since ribosomes are needed to produce more ribosomes, the model includes a positive feedback loop central to the control of cell growth. Our analysis of the model shows that there can be only two coexisting equilibrium states across all 23 parameters. This precludes the existence of hysteresis, suggesting that the ribosome abundance changes continuously with parameters. These states are related by a transcritical bifurcation, and we provide an analytic formula for parameters that admit either state.

A Parameter Values Justification

In order to parameterize the model we use numbers for E. coli. There are two challenges to successful parameterization. The first is that the numbers depend on growth condition of E. coli so that some of the parameters (i.e. number of ribosomes in a cell) may change by two orders of magnitude. The second challenge is that our model, as every model, is a simplification and thus many parameters have to be interpreted appropriately in terms of measured data. Below is the detailed justification parameter values in Table 1.

• $\beta = 0.14\; \min ^{-1} = 0.00233 \; \textrm{s}^{-1}$ Degradation rate of mRNA and rRNA. In E. coli mRNA lifespan is about 5 min (Milo and Phillips 2016) (BNID 106869).

• $\xi =0.014 \; \min ^{-1} = 0.000233 \; \textrm{s}^{-1}$ Degradation rate of proteins. Rapidly degraded proteins in E. coli have lifespan about an hour or 10 times that of mRNA will do 50 minutes (Milo and Phillips 2016) (BNID 108404). Note that this constant can be also interpreted as a dilution rate of proteins due to volume growth of the cell.

• $\eta = 0.2-0.35 \; \min ^{-1} = 0.0033-0.0058 \; \textrm{s}^{-1}$ ppGpp degradation rate (Gallant et al. 1972).

• $\gamma =1100$ is the number of nucleotides that are assembled into mRNA or rRNA per initiation event. Typical mRNA in E. coli has length 370 nm and a single nucleotide has length 0.33 nm for about 1100 nucleotides per mRNA.

• $U =1.3 \; \upmu$M amino acid average concentration in E. coli. The article (Yuan et al. 2006) lists concentrations for some amino acids and they range from 0.2 $\upmu$mol/g for phenylalanine to 6.81 $\upmu$mol/g for alanine. With the conversion factor 0.36g/L we get a range $0.072-2.45$ $\upmu$mol/L which is just $\upmu$M. We selected number 1.3 to be the average of this range.

• $K_m =0.5$ $\upmu$M. We chose comparable to value of U.

• $I_0 =6212 \; \upmu$M. Concentration of NTP required for balanced growth has ben measured by Buckstein et al. (2008), who cite numbers ATP = 3560 $\upmu$M, CTP = 325 $\upmu$M, GTP = 1660 $\upmu$M, UTP = 667 $\upmu$M.

• $I =$ free parameter

• $\kappa _2,\kappa _4 \in [30,113] \; \upmu$M are half-saturation constants for ppGpp repression. During the transition from exponential to stationary phase the ppGpp concentaration is between $30-113 \; \upmu$M (Buckstein et al. 2008). We select comparable values of $\kappa _2,\kappa _4$.

• $\kappa _1, \kappa _3 =[10^3,10^4] \; \upmu$M are half-concentrations of nucleotides when these become limiting. Since we derive $I_0 =6212 \; \upmu$M:concentration of NTP required for balanced growth below, we select $\kappa _1, \kappa _3$ to be on the same order of magnitude.

• $B_{2}$ is maximal rate of mRNA initiation in $s^{-1}$. Typical mRNA initiation rate is $20 \, \min ^{-1} = 0.33 \textrm{s}^{-1}$ (Moran et al. 2010) (ID 111997). We will assume that the maximal rate is larger, but does not match the rate of the rrn gene and set $B_{2} \in [1,3] \; \textrm{s}^{-1}$. Since $B_{max}=B_2\kappa _2$, the range of $B_{max}$ is $B_{max} \in [30, 339]$.

• $A_2\in [0.46,6.76] \; \textrm{s}^{-1}$ is the maximal rate of rrn transcription initiation. There are $4-58$ initiations/min/gene for rrn gene (Bremer and Dennis 1996), with 7 copies of rrn gene 28-406 initiations per min. Since $A_{max}=A_2\kappa _2$, the range of $A_{max}$ is $A_{max} \in [13.8, 763.88]$.

• $B_{0} =0$ minimal rate of incorporation in the absence of ppGpp.

• $A_{0} =0$ minimal rate of incorporation in the absence of ppGpp.

• $\ell = 5.14$ The average size of ribosomal proteins in E. coli is 132 amino acids (Reuveni et al. 2017) and since each of them is encoded by 3 nt, the average length of ribosomal protein mRNA is 396 nt. On the other hand, the average spacing of ribosomes is 77 nt in E. coli (Siwiak and Zielenkiewicz 2013), and therefore we assume that there are $\ell = 396/77=5.14$ ribosomes on mRNA.

For the remainder of the parameters, we will use frequently the following numbers. All conversions to micromolar are done using Remark 5.

• 1. Number of ribosomes per cell is $6800{-}72000$ lower value is for slow division rate (100 min) and higher value is for fast division rate (24 min) (Moran et al. 2010) (BNID 101441). This is $11.29{-}119.56 \upmu$M

• 2. 80% of ribosomes are actively translating (Moran et al. 2010) (BNID 102344)

• 3. There are 7 rrn genes (Condon et al. 1995). This is $0.011624\,\upmu$M, see Remark 5.

• 4. The volume of E. coli is about 1 fL (Moran et al. 2010) (BNID 101788)

• 5. Avogadro number is $6.02214076*10^{23}\,mol^{-1}$.

• 6. There are $2400{-}7800$ copies of mRNA per E. coli cell, depending on the growth medium (Moran et al. 2010) (BNID 112795). This is $3.98{-}12.95 \upmu$M.

• 7. The mean initiation rate of translation in E. coli is about 5 initiations/min/mRNA (Moran et al. 2010) (BNID 112001), which is 0.08 initiations/sec/mRNA.

Remark 5

We describe how to convert number per cell to units of micromolar $\upmu$M. As an example we take the number of copies of rrn gene in E. coli, which is 7 (Condon et al. 1995). Since the volume of E. coli is about 1 fL (Moran et al. 2010) (BNID 101788), which is $10^{-15}L$ there are $7 10^{15}$ rrn genes per liter. Dividing by the Avogadro number $6.02214076*10^{23}\textrm{mol}^{-1}$ we get concentration $1.1624 * 10^{-8}\textrm{mol}/L$. Converting the units to micromolar, which is equivalent by multiplying by $10^6$ we get $0.011624\,\upmu$M.

• $\kappa$ is Michealis-Menten constant for protein repression of its own mRNA. We computed above typical concentration of free ribosomal protein as $11.29{-}119.56 \; \upmu$M. We assume $\kappa$ belongs to the same range.

• $\alpha$ in units $(\upmu \textrm{M s})^{-1}$ is the rate of ribosome assembly from mRNA and proteins. This process is rapid requiring 2 min for production of a single ribosome (Chen et al. 2012; Lindahl 1975). There are between $11.29-119.56 \upmu$M ribosomes per cell (Moran et al. 2010). Based on Chen et al. (2015) on average $1{-}2\%$ of ribosomal proteins are in the free pool, the rest is in ribosomes. So the typical size of the protein pool (these are "protein complexes" of 55 proteins) is $0.11{-}1.19 \upmu$M; the same would be true for free ribosomal RNA. We assume that

  $$\begin{aligned} \alpha \in \left[ \frac{1}{120} s^{-1} * \frac{1}{0.011}(\upmu \textrm{M})^{-1},\frac{1}{120} s^{-1} * \frac{1}{1.19}(\upmu \textrm{M})^{-1}\right] = [0.007, 0.754]. \end{aligned}$$

• ${\hat{a}}$ is the $1/K_D$ where $K_D$ is the dissociation constant for binding ribosomes to ribsomal mRNA in units $(\upmu \textrm{mol})^{-1}$. In other words,

  $$\begin{aligned} {\hat{a}} = \frac{\text{ active } \text{ ribsomes } \text{ on } \text{ ribsomal } \text{ mRNA }}{ \text{ free } \text{ ribosomes } \text{* } \text{ free } \text{ ribsomal } \text{ mRNA }} \end{aligned}$$

  There are between $11.29{-}119.56 \upmu$M ribosomes per cell (Moran et al. 2010), 80% of which are actively translating (Moran et al. 2010) (BNID 102344). There are between $2400{-}7800$ copies of mRNA per E. coli cell (Moran et al. 2010) (BNID 112795). Since the ribosomal protein is $9{-}22\%$ of total protein, we assume that the same proportion holds for ratio between ribosomal mRNA and total mRNA. Therefore there are between 216 and 1716 ribosomal mRNA per cell which is $0.216{-}1.716 \upmu$M. If we take the lower number of ribosomes and higher nunber of ribosomal mRNA we get ${\hat{a}} = \frac{0.8 11.29 *c }{0.2 11.29 1.716 *c}$ where c is the proportion of ribsomal mRNA/total mRNA. This constant cancels and we get ${\hat{a}} = \frac{0.8}{0.2 1.716} = 2.33 (\upmu \textrm{mol})^{-1}$. If we take the higher number of ribosomes and lower number of ribosomal mRNA we get ${\hat{a}}= \frac{0.8}{0.2 0.216} = 18.52 (\upmu \textrm{mol})^{-1}$. Therefore

  $$\begin{aligned} {\hat{a}} \in [2.33,18.52].\end{aligned}$$

• $k_2 \in [ 1.05, 27.18] s^{-1}$ This is $K_{cat}$ for enzymatic reaction producing a representative ribosomal protein. The start with the number of ribosomes per cell which is 6800 to 72000 (Moran et al. 2010) (BNID 101441). We then assume that 80% or ribosomes are actively translating (Moran et al. 2010) (BNID 102344), and the elongation rate is 16 amino acids per ribosome per second. Then there are between 87, 040 and 921, 600 amino acids needed per second for protein synthesis. If we take this range and assume that 9–22% of protein in the cell are ribosomal proteins than ribosomal protein pool is created at the rate in the range [7834, 202752] amino acids/sec. Since ribosomes contain 7459 amino acids (Moran et al. 2010) (BNID 101175), the rate of production of ribosomes is $1.05{-}27.18$ ribosomes per second. Since in our model p represents a single protein that enters ribosome with stochiometry 1, this range also represents rate of production of this representative protein, which is $k_2$.

• $i_r= 0.011624\,\upmu$M concentration of rrn genes, since there are 7 rrn genes (Condon et al. 1995). See Remark 5 for the conversion.

• $i_m$ concentration of the ribosomal protein genes. We assume there are 55 ribosomal protein genes, one per each protein. Using conversion one gets $0.09133\,\upmu$M.

• $\omega \in [0.33,2.64]\,\textrm{s}^{-1}$ is the rate of formation of complex mR to produce representative ribosomal protein. We equate this number with the number of translation initiations on mRNA of that protein. The mean initiation rate of translation in E. coli is about 5 initiations/min/mRNA (Moran et al. 2010) (BNID 112001), which is 0.08 initiations/sec/mRNA. There are between $2400{-}7800$ copies of mRNA per E. coli cell, depending on the growth medium (Moran et al. 2010) (BNID 112795). Since the ribosomal protein is $9{-}22\%$ of total protein, we assume that the same proportion holds for ratio between ribosomal mRNA and total mRNA. Therefore there are between 216 and 1716 ribosomal mRNA per cell. Using these numbers as typical concentrations we estimate overall initiation to be in the range $[0.08*216, 0.08* 1716] = [17.28,137.28]$ initiations/sec across all ribosomal mRNA in the cell. Since we model a representative ribosomal protein p, this range has to be divided by 52 which is the number of ribosomal proteins in a ribosome. This gives the range [0.33, 2.64].

• $K =\frac{k_2}{\omega }E(I)$ where $E(I) =\frac{qI}{K_m+qI}$, see (9). K is not constant since it depends on the nutrient level I. Note that the function $E(I) \in [0,1]$. It is also known that the lower estimate of rate $k_2$ and the lower estimate of $\omega$ occur at slow growth. Since both upper estimates of these rates occur at high rates, we set $K \in [0, 10.295]$ where 10.295 is the ratio of the higher estimate of $k_2$ and the higher estimate of $\omega$.