
1)
Model construction
Here we model changes in the concentrations of ppGpp and its fluorescent reporter GFP (the biosensor) in E. coli cells. We consider 2 types of nutrient limitation, which might increase intracellular ppGpp levels: growth phasespecific or nonspecific (Fig. 1). The first type is related to a depletion of growthlimiting nutrient (e.g., carbohydrates) in the growth medium at the end of exp. growth phase. The second type is related to a depletion of essential molecules inside a cell due to their excessive diversion into the production of biotechnological products (e.g., depletion of AcylACP during fatty acid (FA) production; [2], Fig. 1). We use the model to describe and explain the existing data on ppGpp and growth in normal and FAproducing cells ([2, 6]; Results) and to explore the applicability of a ppGpp biosensor for the control of growth conditions and optimization of FA production. The parameters of the model were estimated based on available data as described in Additional file 1. The analysis of parameter sensitivity demonstrated that the model is robust to parameter variation (Additional file 1: Fig. S1). The model consists of 9 ordinary differential equations presented below.
The processes included in the model are summarised on Fig. 1 and described below. Briefly, we model the mutual relationship between ppGpp and growth phases. The increase of ppGpp concentration [6, 8] at the end of exp. growth was described by the depletion of exp. phaselimiting nutrient lim (e.g., main carbohydrate, see Results). The decrease in the growth rate during exp. to stat phase transition is described through ppGppmediated decrease of ribosomal synthesis [6, 25]. The termination of growth in stat phase is described by the depletion of a second, growthsupporting nutrient nutr (e.g., secondary carbohydrate, which is released into the medium during exp. phase; Results).
The model also describes ppGpp accumulation due to a depletion of AcylACP product of FAS during FA production (Fig. 1). The accumulation of ppGpp inhibits membrane phospholipids synthesis (PLS), FAS, and growth through several parallel mechanisms in our model (Fig. 1). This includes inhibition of the key PLS enzyme PlsB by ppGpp [26], causing transient accumulation of AcylACP and inhibition of FAS flux by AcylACP [20, 26]; downregulation of FAS and PLS fluxes and growth by low ribosomal activity rib (relative number of active ribosomes; Fig. 1); and inhibition of growth by decreased PLS flux in high FAproduction lines [27].
In addition to ppGpp, FA and growth, the model analyses the kinetics of GFP reporter of ppGpp in FAproducing and nonproducing lines. The ppGpp reporter is designed using the natural property of ppGpp to inhibit transcription from P1 and P2 promoters, responsible for rRNA biosynthesis (Fig. 1; [6, 11, 25]). P1 and P2 promoters are regulated in a similar way in E. coli cells [6, 11], so we considered a tandem of the ribosomal P1 and P2 promoters as a single entity (hereafter called the P1/P2 promoter) in our model. To transmit the increase of ppGpp levels to increase of a fluorescent signal, an artificial inhibitor I (e.g., tetrepressor TetR or lac repressor LacI) is expressed from the P1/P2 promoter. Expression of GFP protein is from an Irepressible promoter (e.g., TetR or LacIinhibited promoter; Fig. 1). The use of rapidly degraded variants of I and GFP proteins ensures that the biosensor reports dynamic changes in the metabolic state of the cells [28, 29], as we discuss in Results. The levels of intracellular compounds are determined in our model by their synthesis and degradation. Their dilution due to cell division was ignored due to its slow rate (less than 0.01 min^{−1}, [8]).
The model equations are presented below.

1)
AcylACP and FA production
The intracellular kinetics of the key intermediate of fatty acid metabolism AcylACP is described in our model as:
$$ \frac{dAcyl\hbox{} ACP}{dt}={V}_{FA S}{V}_{PLS}{k}_{fa}\cdot {V}_{FA} $$
(1)
Where V
_{
FAS
} and V
_{
PLS
} are the intracellular rates of FAS and PLS (in μM/min); V
_{
FA
} is the rate of FA production in a cell culture, normalized to cell number N (in mg/l/min/OD); and k
_{
fa
} is a volume coefficient for recalculation of the V
_{
FA
} rate into intracellular units of μM/min (Additional file 1).
The rate of FAS is described by the simplified lump equation, which includes the MichaelisMenten dependence of FAS rate on concentration of substrates  AcCoA and ACP protein and feedback inhibition of FAS rate by AcylACP product [26]:
$$ {V}_{FAS}={V}_{m FAS}\cdot rib\cdot \frac{ACP}{ACP+{K}_{m ACP}}\cdot \frac{AcCoA}{AcCoA+{K}_{m AcCoA}}\cdot \frac{1}{1+ Acyl\hbox{} ACP/{K}_{i Acyl ACP}} $$
(1')
Where AcCoA and ACP are concentrations of acetylCoA and free active ACP protein; AcylACP is the total concentration of longchain acyl product of FAS.
We assumed that FAS and PLS fluxes and growth rate are proportional to the ribosomal activity rib due to the dependence of the protein synthesis on rib.
PLS rate is assumed to be determined by the rate of the first committed enzyme, PlsB, with MichaelisMenten dependence on AcylACP substrate and inhibition by ppGpp [19, 26]:
$$ {V}_{PLS}={V}_{m PLS}\cdot rib\cdot \frac{Acyl\hbox{} ACP}{Acyl\hbox{} ACP+{K}_{m Acyl ACP}}\cdot \frac{1}{1+{\left( ppGpp/{K}_{i ppGpp}\right)}^n} $$
(1'')
The use of the Hill function for the inhibition of PLS by ppGpp is motivated by existing data on multiple levels of negative regulation of PlsB and related enzymes by ppGpp [22].
The rate of FA production is described by MichaelisMenten kinetics of thioesterase Tes, which hydrolyses thioester bond in the molecule of AcylACP and releases free FA:
\( {V}_{FA}={V}_{tes}\cdot \frac{Acyl\hbox{} ACP}{Acyl\hbox{} ACP+{K}_{m tes}} \) (1″‘).
The activity of Tes (V
_{
Tes
}) depends on Tes concentration. In simulated Tesox and normal lines V
_{
Tes
} values were estimated from the measured FA yields ([2]; see Results).
The rate of the total FA production by a cell culture is described as:
$$ \frac{dFA}{dt}=N\cdot {V}_{FA} $$
(2)
where N is a cell number.

2)
Cell growth
Our model describes the kinetics of cell growth (cell number, N) [2, 6], which affects nutrient levels and FA yields. The accumulation of ppGpp at the end of exp. phase (Fig. 1) was described by the depletion of a limiting nutrient (variable lim, eqs. 5, 6, see below). Increase of ppGpp in turn leads to decrease of the ribosomal activity (eqs. 7, 7′) and inhibition of growth (eq. 3, Fig. 1) during exp. to stationary (stat) phase transition. The depletion of a second, growthsupporting nutrient (variable nutr; eq. 4) determines the cessation of growth and entrance to stat phase. Additionally, we assumed that a certain minimal rate of phospholipid synthesis (V
_{
0
}) is required to sustain growth [27]. In addition to limited production of total membrane PL, the membrane composition might be unbalanced in high Tesox lines due to higher proportion of unsaturated FA, which was suggested to be a key factor of FA toxicity and growth limitation in high Tesox lines [30]. In our model, the growth limitations in high Tesox lines [2] were collectively accounted by restricting growth rate at low rates of PLS (V
_{
PLS
}) (eq. 3‘below). The cell growth is described as:
$$ \frac{dN}{dt}={v}_g\cdot N $$
(3)
$$ {v}_g={K}_{gr}\cdot nutr\cdot rib\cdot {V}_{PLS}/\left({V}_{PLS}+{V}_0\right) $$
(3')
Where v
_{
g
} is the growth rate (in min^{−1}) and N is a cell number, expressed in units of OD measured at a wavelength of 600 nm.
The kinetics of nutr and lim depletion was assumed to be proportional to cell number N:
$$ \frac{dnutr}{dt}={k}_{nutr}\cdot N\cdot \frac{nutr}{nutr+0.001} $$
(4)
$$ \frac{d\mathit{\lim}}{dt}={k}_{lim}\cdot N\cdot \frac{\mathit{\lim}}{\mathit{\lim}+0.001} $$
(5)
where nutr and lim are relative amounts of the growthsupporting and exp. phaselimiting nutrients respectively, initially both set to 1. To avoid negative values of nutr and lim, their depletion is restricted when their concentrations reach levels of 0.001.

3)
ppGpp kinetics
The kinetics of ppGpp is determined by the balance between its synthesis and hydrolysis [5]:
$$ \frac{dppGpp}{dt}={k}_{+ ppGpp}\cdot rib ppGpp\cdot \left({k}_{ ppGpp}\cdot \mathit{\lim}+k{0}_{ ppGpp}\right)\cdot \frac{Acyl\hbox{} ACP}{Acyl\hbox{} ACP+{K}_{m Ac\_ pp}} $$
(6)
Here ppGpp, rib and lim are the levels of ppGpp, ribosome activity and limiting nutrient lim, respectively. Several molecular mechanisms are integrated during ppGpp synthesis and degradation. ppGpp is synthesises on ribosomes [5, 6], therefore in our model we assumed that the synthesis rate of ppGpp is proportional to the number of active ribosomes rib [5, 6], with the rate constants k
_{
+ppGpp
}. We next assumed that depletion of lim nutrient at the end of the exp. phase slows down ppGpp hydrolysis (rate constant k
_{
ppGpp
}), presumably via the inhibition of ppGpp hydrolase SpoT [31, 32]. This results in the increase of ppGpp concentration at the end of exp. phase in our model ([6, 8], Fig. 1). The background hydrolysis of ppGpp in the absence of lim is accounted for by the rate constant k0
_{
ppGpp
}. Finally, ppGpp levels are upregulated in our model by the depletion of AcylACP, due to the inactivation of SpoT hydrolase activity [5, 10, 21, 33].

4)
Ribosomal and P1/P2 promoter activities
The equation for the ribosomal activity (relative number of active ribosomes) rib is:
$$ \frac{drib}{dt}={k}_{+ rib}\cdot P1P2{k}_{ rib}\cdot rib $$
(7)
where P1P2 and rib are the relative activities of the P1/P2 promoter and the ribosomes. The rib synthesis is determined by the rate of rRNA transcription from the P1/P2 promoter [6, 11, 34]; therefore, we assume a linear dependence of rib synthesis rate on P1/P2 activity.
Based on the existing data we assumed that during cell growth transcription from P1/P2 promoter is regulated by ppGpp inhibition [6, 11, 25]. P1 and P2 show similar responses to changes in nutrient levels, but the tandem of P1 and P2 promoters (P1/P2) shows a stronger response compared to single P1 and P2 promoters [11]. This was accounted for in the model by using a Hill coefficient m = 2 in the equation for P1/P2. The data [6, 11] suggest that P1/P2 activity quickly (in minutes) responds to changes in ppGpp concentration. Therefore transcriptional activity of P1/P2 was expressed via ppGpp by the algebraic equation:
$$ P1P2=\frac{1}{1+{\left( ppGpp/{k}_{iP1P2 ppGpp}\right)}^m} $$
(7')

5)
ppGpp sensor
ppGpp sensing was implemented through the inhibition of GFP expression by the inhibitor I, which is expressed from the P1/P2 promoter (Fig. 1). Since the abundance of most proteins changes much more slowly than the abundance of their mRNAs [28], we assumed that the amount of I mRNA is simply proportional to the transcriptional activity of P1/P2 promoter, so that the kinetics of protein I is described by the following equation:
$$ \frac{dI}{dt}={k}_I\cdot \left(P1P2I\right) $$
(8)
where I is the relative amount of the inhibitor I (changing from 0 to 1) and k
_{
I
} is a rate constant of protein I degradation. The rate constant of protein I synthesis was assumed to be equal to k
_{
I
} to achieve the maximal level of I = 1.
The amount of GFP mRNA is determined by the amount of the inhibitor I. The equation for the relative amount of fluorescent GFP protein is:
$$ \frac{dGFP}{dt}={k}_{GFP}\cdot \left(\frac{1}{1+{\left(I/{Ki}_I\right)}^l} GFP\right) $$
(9)
where GFP is the relative amount of GFP fluorescence (changing from 0 to 1) and k
_{
GFP
} is the rate constant of GFP protein degradation. The rate constant of GFP synthesis was assumed to be equal to k
_{
GFP
} to achieve the maximal level of GFP = 1. A Hill coefficient l = 4 accounts for the formation of tetrameric inhibitor complex (e.g., lacR) on a doublestranded DNA [35].
The constant Ki
_{
I
} for inhibition of GFP expression by the relative amount of the inhibitor I integrates two unknown parameters of the system: the absolute expression level of I and its inhibition strength. Ki
_{
I
} was varied as discussed in Results, with the optimal value of Ki
_{
I
} = 0.1. The system of ODEs was solved using MATLAB, integrated with the stiff solver ode15s (The MathWorks UK, Cambridge). The MATLAB code of the model is provided in Additional file 1.