In the continuous mode of cell culture, a constant flow carrying fresh media replaces culture fluid, cells, nutrients and secreted metabolites. Here we present a model for continuous cell culture coupling intra-cellular metabolism to extracellular variables describing the state of the bioreactor, taking into account the growth capacity of the cell and the impact of toxic byproduct accumulation. We provide a method to determine the steady states of this system that is tractable for metabolic networks of arbitrary complexity. We demonstrate our approach in a toy model first, and then in a genome-scale metabolic network of the Chinese hamster ovary cell line, obtaining results that are in qualitative agreement with experimental observations. We derive a number of consequences from the model that are independent of parameter values. The ratio between cell density and dilution rate is an ideal control parameter to fix a steady state with desired metabolic properties. This conclusion is robust even in the presence of multi-stability, which is explained in our model by a negative feedback loop due to toxic byproduct accumulation. A complex landscape of steady states emerges from our simulations, including multiple metabolic switches, which also explain why cell-line and media benchmarks carried out in batch culture cannot be extrapolated to perfusion. On the other hand, we predict invariance laws between continuous cell cultures with different parameters. A practical consequence is that the chemostat is an ideal experimental model for large-scale high-density perfusion cultures, where the complex landscape of metabolic transitions is faithfully reproduced.

While at present most biotechnology industrial facilities adopt batch or fed-batch processes, continuous processing has been vigorously defended in the literature and many predict its adoption in the near future. However, identical cultures may lead to distinct steady states and the lack of comprehension of this multiplicity has been a limiting factor for the widespread application of this kind of processes in the industry. In this work we try to remediate this providing a computationally tractable approach to determine the steady-states of genome-scale metabolic networks in continous cell cultures and show the existence of general invariance laws across different cultures. We represent a continuous cell culture as a metabolic model of a cell coupled to a dynamic environment that includes toxic by-products of metabolism and the cell capacity to grow. We show that the ratio between cell density and dilution rate is the control parameter fixing steady states with desired properties, and that this is invariant accross perfusion systems. The typical multi-stability of the steady-states of this kind of culture is explained by the negative feedback loop on cell growth due to toxic byproduct accumulation. Moreover, we present invariance laws connecting continuous cell cultures with different parameters that imply that the chemostat is the ideal experimental model to faithfully reproduce the complex landscape of metabolic transitions of a perfusion system.

Our goal in this work is to introduce a detailed characterization of the steady states of cell cultures in continuous mode, considering the impact of toxic byproduct accumulation on the culture, and employing a minimum number of essential kinetic parameters. To achieve this and inspired by the success of DFBA in other settings we couple macroscopic variables of the bioreactor (metabolite concentrations, cell density) to intracellular metabolism. However, we explain how to proceed directly to the determination of steady states, bypassing the necessity of solving the dynamical equations of the problem. This spares us from long simulation times and provides an informative overview of the dynamic landscape of the system. The approach, presented here for a toy model and for a genome-scale metabolic network of CHO-K1, but easily extensible to other systems, supports the idea that multi-stability, i.e., the coexistence of multiple steady states under identical external conditions, arises as a consequence of toxic byproduct accumulation in the culture. We find and characterize specific transitions, defined by simultaneous changes in the effective cell growth rate and metabolic states of the cell, and find a wide qualitative agreement with experimental results in the literature. Our analysis implies that batch cultures, typically used as benchmarks of cell-lines and culture media, are unable to characterize the landscape of metabolic transitions exhibited by perfusion systems. On the other hand, our results suggest a general scaling law that translates between the steady states of a chemostat and any perfusion system. Therefore, we predict that the chemostat is an ideal experimental model of high-cell density perfusion cultures, enabling a faithful characterization of the performance of a cell-line and media formulation truly valid in perfusion systems.

To consider the temporal evolution of a culture, FBA may be applied to successive points in time, coupling cell metabolism to the dynamics of extra-cellular concentrations. This is the approach of Dynamic Flux Balance Analysis (DFBA) [ 27 ] and has been applied prominently either to the modeling of batch/fed-batch cultures or to transient responses in continuous cultures, being particularly successful in predicting metabolic transitions in E. Coli and yeast [ 23 , 27 , 28 ]. However, to the best of our knowledge, the steady states of continuous cell cultures have not been investigated before. First, because DBFA for genome-scale metabolic networks may be a computational demanding task, particularly when the interest is to understand long-time behavior. Second, because it assumes knowledge of kinetic parameters describing metabolic exchanges between the cell and culture medium, that are usually unknown in realistic networks. Moreover, although the importance of toxic byproduct accumulation has been appreciated for decades [ 29 , 30 ], its impact on steady states of continuous cultures has been studied mostly in simple metabolic models involving few substrates [ 31 , 32 ], while it has been completely overlooked in DFBA of large metabolic networks. Lactate and ammonia are the most notable examples in this regard and have been widely studied in experiments in batch and continuous cultures [ 30 , 33 – 36 ].

Fortunately, in the last few years it has been possible to exploit an increasingly available amount of information about cellular metabolism at the stoichiometric level to build genome-scale metabolic networks [ 15 , 16 ]. These networks have been modeled by different approaches [ 17 , 18 ] but Flux Balance Analysis (FBA) has been particularly successful predicting cell metabolism in the growth phase [ 19 ]. FBA starts assuming a quasi-steady state of intra-cellular metabolite concentrations, which is easily translated into a linear system of balance equations to be satisfied by reaction fluxes. This system of equations is under-determined and a biologically motivated metabolic objective, such as biomass synthesis, is usually optimized to determine the complete distribution of fluxes through the solution of a Linear Programming problem [ 20 ]. This approach was first used to characterize the metabolism of bacterial growth [ 21 ], but later has been applied also to eukaryotic cells [ 22 , 23 ]. Alternatively, given a set of under-determined linear equations, one can estimate the space of feasible solutions of the system and average values of the reaction fluxes [ 24 – 26 ].

By definition, a continuous cell culture ideally reaches a steady state when the macroscopic properties of the tank (cell density and metabolite concentrations) attain stationary values. Industrial applications place demands on the steady state, usually: high-cell density, minimum waste byproduct accumulation, and efficient nutrient use. However, identical external conditions (dilution rate, media formulation) may lead to distinct steady states with different metabolic properties (a phenomenon known in the literature as multi-stability or multiplicity of steady states) [ 10 – 14 ]. Therefore, for the industry, it becomes fundamental to know in advance, given the cell of interest and the substrates to be used, which are the possible steady states of the system and how to reach them. Moreover, to satisfy production demands, it may be advantageous to extend the duration of a desired steady state indefinitely [ 6 ], implying that their stability properties are also of great interest.

A classical example of continuous cell culture is the chemostat, invented in 1950 independently by Aaron Novick and Leo Szilard [ 7 ] (who also coined the term chemostat) and by Jacques Monod [ 8 ]. In this system, microorganisms reside inside a vessel of constant volume, while sterile media, containing nutrients essential for cell growth, is delivered at a constant rate. Culture medium containing cells, remanent substrates and products secreted by the cells are removed at the same rate, maintaining a constant culture volume. The main dynamical variable in this system is the dilution rate (D), which is the rate at which culture fluid is replaced divided by the culture volume. In a well-stirred tank any entity (molecule or cell) has a probability per unit time D of leaving the vessel. In industrial settings, higher cell densities are achieved by attaching a cell retention device to the chemostat, but allowing a bleeding rate to remove cell debris [ 9 ]. Effectively only a fraction 0 ≤ ϕ ≤ 1 of cells are carried away by the output flow D. This variation of the continuous mode is known as perfusion culture.

Biotechnological products are obtained by treating cells as little factories that transform substrates into products of interest. There are three major modes of cell culture: batch, fed-batch and continuous. In batch, cells are grown with a fixed initial pool of nutrients until they starve, while in fed-batch the pool of nutrients is re-supplied at discrete time intervals. Cell cultures in the continuous mode are carried out with a constant flow carrying fresh medium replacing culture fluid, cells, unused nutrients and secreted metabolites, usually maintaining a constant culture volume. While at present most biotechnology industrial facilities adopt batch or fed-batch processes, the advantages of continuous processing have been vigorously defended in the literature [ 1 – 5 ], and currently some predict its widespread adoption in the near future [ 6 ].

Numerical simulations were carried out in Julia [ 77 ]. Linear programs were solved with Gurobi [ 78 ]. The CHO-K1 metabolic network [ 70 ] was read and setup with all relevant parameters using a script written in Python with the COBRApy package [ 79 – 81 ]. All scripts (which also include parameter values) are freely available in a public Github repository [ 82 ].

The two toxic byproducts most commonly studied in mammalian cell cultures are ammonia and lactate. Their toxicity is primarily attributed to their effects in osmolarity and pH [ 33 – 36 ]. It has been suggested that the accumulated toxicity may result in increased maintenance demands [ 55 , 74 ] and in reduced biomass yields [ 55 ]. Parameters describing these effects quantitatively vary over an order of magnitude [ 57 , 75 , 76 ] depending on culture conditions and cell-line. In our model we incorporate these effects through the factor K and for the sake of specificity in this example we use: (11) with K nh4 = 1.05mM, K lac = 8mM [ 67 ], and set μ = K × z.

In order to enforce Eq 4 , we complemented this network with a set of reaction costs. Following T. Shlomi et. al [ 22 ], we assigned costs as follows: , where and are the molecular weight and catalytic rate of the enzyme catalyzing reaction k in the given direction. The parameters , were gathered by T. Shlomi et. al from public repositories of enzymatic data. Missing values are set to the median of available values. An estimate of the enzyme mass fraction C = 0.078mg/mgDW was obtained for mammalian cells by the same authors. A constant maintenance energetic demand (cf. term e i in Eq 3 ) was added in the form of an ATP hydrolysis drain at a flux rate 2.24868mmol/gDW/h [ 39 ] (the reported value is for mouse LS cells, which we converted by accounting for the dry weight of CHO cells [ 70 ]). The maximum uptake rate of glucose was set at V glc = 0.5 mmol/gDW/h, from previous models of cultured CHO cells fitted to experimental data [ 71 , 72 ] (which also closely matches the values obtained from kinetic measurements on other mammalian cell lines [ 66 ]). However, kinetic parameters needed to estimate V i for most metabolites are not known at present. Based on data in the literature [ 33 , 68 , 73 ], we estimated that the uptake rates of amino acids is typically one order of magnitude slower than the uptake rate of glucose, accordingly we set V i = V glc /10 for amino acids. Other metabolites have an unbounded uptake (V i = ∞). In the simulations we used Iscove’s modified Dulbecco’s medium (IMDM), and set infinite concentrations for water, protons and oxygen. We converted between grams of dry weight and cell number using a cellular dry weight of 350pg [ 70 ].

Exploiting the increasingly available information about cellular metabolism at the stoichiometric level [ 15 , 16 ], a metabolic network can be reconstructed containing the biochemical reactions occurring inside a cell of interest. These reconstructions typically contain data about stoichiometric coefficients (N ik , cf. Eq 3 ), thermodynamic bounds (lb k , ub k , L i , cf. Eqs 5 and 6 ), and a biomass synthesis pseudo-reaction (y i , cf. Eq 3 ) [ 18 , 41 , 69 ]. Motivated by the fact that most therapeutic proteins requiring complex post-transnational modifications in the biotechnological industry are produced in Chinese hamster ovary (CHO) cell lines [ 48 ], we analyzed the steady states of a genome scale model of the CHO-K1 line [ 70 ]. Based on the latest consensus reconstruction of CHO metabolism available at the time of writing, containing 1766 genes and 6663 reactions, a cell-line-specific model for CHO-K1 was built by Hefzi et al. [ 70 ], comprising 4723 reactions (including exchanges) and 2773 metabolites (with cellular compartmentalization). It accounts for biomass synthesis through a virtual reaction that contains the moles of each metabolite required to synthesize one gram of biomass. The network recapitulates experimental growth rates and cell-line-specific amino acid auxotrophies.

The parameters were set as follows. The stoichiometric coefficients N F = 2, N R = 38 are the characteristic ATP yields of glycolysis and respiration, respectively [ 61 ]. Maintenance demand is modeled as a constant drain of ATP at a rate e = 1.0625 mmol/gDW/h, typical of mammalian cells [ 39 ]. The maximum respiratory capacity is computed as r max = F thr × Vol × DW = 0.45 mmol/g/h, where F thr = 0.9 mM/min is a glucose uptake threshold (per cytoplasmic volume) beyond which mammalian cells secrete lactate [ 58 ], Vol = 3 pL and DW = 0.9 ng are the volume [ 62 ] and dry weight, respectively, of mammalian (HeLa) cells, the later estimated from the dry mass fraction (≈ 30%, [ 63 , BNID100387]) and total weight (= 3 ng [ 64 ]) of one HeLa cell. The concentration of substrate in the medium was set c = 15 mM, which is a typical glucose concentration in mammalian cell culture media (for example, RPMI-1640 [ 65 ]). Next, V = 0.5 mmol/gDW/h, also measured for HeLa cells [ 66 ] (the measured flux is per protein weight, so we multiplied by 0.5 to obtain a flux per cell dry weight, since roughly half of a generic cell dry weight is protein [ 63 , BNID101955]). The parameter y = 348 mmol/gDW was adjusted so that the maximum growth rate was ≈1 day −1 , which is within the range of duplication rates in mammalian cells [ 67 , 68 ]. Finally, the toxicity of waste was set as τ = 0.0022 h −1 mM −1 , obtained from linearizing the death rate dependence on lactate in a mammalian cell culture reported by S. Dhir et al. [ 68 ]. To convert between grams of dry weight and number of cells, we used the cellular dry weight estimated above for HeLa cells (0.9ng).

A primary carbon source S is consumed by the cell at a rate u ≥ 0. It is processed into an intermediate P, generating N F energy units per unit S consumed. The intermediate P can be excreted in the form of a waste product W (rate −v ≤ 0), or it can be completely oxidized (rate r ≥ 0) generating an additional N R energy units (N R ≫ N F ). The respiration rate is capped, r ≤ r max .

A diagram of the network is shown in Fig 2 . There are four metabolites: a primary nutrient S, an energetic currency E, an intermediate P, and a waste product W. Only S and W can be exchanged with the extracellular medium, and their concentrations in the tank will be denoted by s and w, respectively. The cell can consume S from the medium at a rate u ≥ 0. The nutrient is first processed into P, generating N F units of E per unit S processed. The intermediate can have two destinies: it can be excreted in the form of W (rate −v ≤ 0), or it can be further oxidized (rate r ≥ 0), generating N R units of E per unit P. These two pathways are reminiscent of fermentation and respiration. We assume that N F ≪ N R , which is consistent with the universally lower energy yield of fermentation versus respiration. Therefore, a maximization of energy output implies that the respiration mode is preferred. However, the enzymatic costs required to enable respiration are very high compared to fermentation. Therefore in Eq 4 only the costs of respiration are significant [ 58 ], which implies that this flux is bounded: (7) Metabolic overflow occurs when the nutrient uptake is higher than the respiratory bound r max . The remaining S must then be exported as waste, W. A balance constraint (cf. Eq 3 ) at the intermediate metabolite P requires that: (8) where stoichiometric coefficients are set to 1 for simplicity. Another balance constraint at the internal energetic currency metabolite, E, leads to: (9) where e denotes an energetic maintenance demand. The currency E is a direct precursor of biomass, at a yield y. Finally, the waste byproduct W is considered toxic, inducing a death rate proportional to its concentration: (10)

To gain insight into the kind of solutions expected, we examined first a simple metabolic network that admits an analytic solution. It is based on the simplified network studied by A. Vazquez et al. to explain the Warburg effect [ 58 ], and serves as a minimal model of metabolic transitions in the cell [ 58 – 60 ].

Finally, the net growth rate of cells μ (see Eq 1 ) is essentially determined by the cellular capacity to synthesize biomass (rate z), but it may also be affected by environmental toxicity. In the examples presented below we considered that: or , corresponding to two different mechanisms explored in the literature [ 36 , 55 ]. In the first case is easily interpreted as the death rate of the cell, while represents a fraction of biomass that must be expended on non-growth related activities, for example, due to increased maintenance demands on account of environmental toxicity (but see also Refs. [ 56 , 57 ] and in particular B. Ben Yahia et al. [ 37 ] for a recent review of the subject). Both and depend on the concentrations of toxic metabolites in the culture, such as lactate and ammonia.

Next, we reason that, although cellular clones in biotechnology are artificially chosen according to various productivity-related criteria [ 48 ], the growth rate is typically under an implicit selective pressure. We will consider then that the flux distribution of metabolic reactions inside the cell maximizes the rate of biomass synthesis, z, subject to all the constrains enumerated above. Note that to carry out this optimization it is enough to solve a linear programming problem, for which efficient algorithms are available [ 49 ]. This formulation is closely related to Flux Balance Analysis (FBA) [ 20 , 21 , 50 , 51 ], but some of the constrains imposed here might be unfamiliar. In particular, Eq 4 has been used before to explain switches between high-yield and high-rate metabolic modes under the name FBA with molecular crowding (FBAwMC) [ 43 , 52 , 53 ], while the right-hand side of Eq 6 is a novel constraint introduced in this work to model continuous cell cultures. If multiple metabolic flux distributions are consistent with a maximal biomass synthesis rate [ 54 ], the one with minimum cost α (cf. Eq 4 ) is selected [ 17 ]. Summarizing, from the complete solution of the linear program we obtain the optimal z, and the metabolic fluxes feeding the synthesis of biomass.

Still, we require a functional connection between variables describing the macroscopic state of the tank (X, ) and the average behavior of cells ( , μ). We start assuming that metabolites inside the cell attain quasi-steady state concentrations [ 38 ], so that fluxes of intra-cellular metabolic reactions balance at each metabolite. If N ik denotes the stoichiometric coefficient of metabolite i in reaction k (N ik > 0 if metabolite i is produced in the reaction, N ik < 0 if it is consumed), and r k is the flux of reaction k, then the metabolic network produces a net output flux of metabolite i at a rate ∑ k N ik r k , where N ik = 0 if metabolite i does not participate in reaction k. This output flux must balance the cellular demands for metabolite i. In particular we consider a constant maintenance demand at rate e i which is independent of growth [ 39 , 40 ], as well as the requirements of each metabolite for the synthesis of biomass components. If y i units of metabolite i are needed per unit of biomass produced [ 41 , 42 ], and biomass is synthesized at a rate z, we obtain the following overall balance equation for each metabolite i: (3) It is also well known that a cell has a limited enzymatic budget [ 17 ]. The synthesis of new enzymes, needed to catalyze many intracellular reactions, consumes limited resources, including amino acids, energy, cytosolic [ 22 , 43 , 44 ] or membrane space [ 45 ] (for enzymes located on membranes), ribosomes [ 46 ], all of which can be modeled as generic enzyme costs [ 17 ]. We split reversible reaction fluxes into negative and positive parts, , with , and quantify the total cost of a flux distribution in the simplest (approximate) linear form [ 17 ]: (4) where are constant flux costs. The limited budget of the cell to support enzymatic reactions is modeled as a constraint α ≤ C, where C is a constant maximum cost. Thermodynamics places additional reversibility constrains on the flux directions of some intra-cellular reactions [ 47 ], which can be written as: (5) Additionally, some uptakes u i are limited by the availability of nutrients in the culture. We distinguish two regimes. If the cell density is low, nutrients will be in excess and uptakes are only bounded by the intrinsic kinetics of cellular transporters. In this case u i ≤ V i , where V i is a constant maximum uptake rate determined by molecular details of the transport process. These will be the only kinetic parameters introduced in the model. When the cell density increases and the concentrations of limiting substrates reach very low levels, a new regime appears where cells compete for resources. In this regime the natural condition s i ≥ 0 together with the mass balance equation ( Eq 2 in steady state) imply that u i ≤ c i D/X. In summary, (6) where L i = 0 for metabolites that cannot be secreted, and L i = ∞ otherwise. Thus, an important approximation in our model is that low concentrations of limiting nutrients are replaced by an exact zero. The ratio D/X in Eq 6 establishes the desired coupling between internal metabolism and external variables of the bioreactor. The appendix contains an alternative derivation of Eq 6 .

Eq 1 describes the dynamics of the cell density as a balance between cell growth and dilution, while Eq 2 describes the dynamics of metabolite concentrations in the culture as a balance between cell consumption (or excretion if u i < 0) and dilution. One must notice that at variance with the standard formulation of DFBA, the terms involving the dilution rate in the right-hand side of both equations enable the existence of non-trivial steady states (with non-zero cell density) which are impossible in batch. These are the steady states that are relevant for industrial applications adopting the perfusion model and that we study in what follows.

A cell culture is grown in a vessel that is continuously fed with a constant flux (blue arrows) of fresh media coming from a reservoir. An equivalent flux carries used media and cells away from the culture vessel, maintaining a constant volume in the culture. The effluent contains cells, secreted metabolites and unused substrates. The figure displays the simple case of no cell retention, where the cell density in the effluent is the same as in the culture. Notation: substrate concentrations in media reservoir (c i ), cell density and metabolite concentrations in the culture (X, s i ), dilution rate (D = F/ν, where F is the input/output flux and ν the culture volume).

We study an homogeneous population of cells growing inside a well-mixed bioreactor [ 37 ], where fresh medium continuously replaces culture fluid ( Fig 1 ). The fundamental dynamical equations describing this system are: (1) (2) where X denotes the density of cells in the bioreactor (units: gDW/L), μ the effective cell growth rate (units: 1/hr), u i the specific uptake of metabolite i (units: mmol/gDW/hr), and s i the concentration of metabolite i in the culture (units: mM). The external parameters controlling the culture are the medium concentration of metabolite i, c i , the dilution rate, D (units: 1 / time), and the bleeding coefficient, ϕ (unitless), which in perfusion systems characterizes the fraction of cells that escape from the culture through a cell-retention device [ 9 ] or a bleeding rate. For convenience of notation, in what follows an underlined symbol like will denote the vector with components {s i }.

Results and discussion

General properties of steady states In this section we present the general procedure to determine the steady states of Eqs 1 and 2 and discuss some general results of our model that are independent of the specificities of the metabolic networks of interest. The first step is to set the time-dependence in Eqs 1 and 2 to zero, (12) (13) Note that Eq 13 depends on X and D only through the ratio 1/ξ = D/X (known in the literature as cell-specific perfusion rate, or CSPR [83]), such that ξ is the number of cells sustained in the culture per unit of medium supplied per unit time (the units of ξ are cells × time / volume). If we recall that in our cellular model, u i is constrained by a term that also depends on X and D only through ξ (cf. Eq 6), it immediately follows that the values of the uptakes and metabolite concentrations in steady state must be functions of ξ, which we denote by and respectively. To compute , solve the linear program of maximizing the biomass synthesis rate (z) subject to Eqs 3–5, but replacing Eq 6 with: (14) The resulting optimal value of z will be denoted by z*(ξ). Moreover, once is known, the stationary metabolite concentrations in the culture follow from Eq 13: (15) Then, given z*(ξ), the effective growth rate in steady state can also be given as a function of ξ, μ*(ξ), by evaluating K or σ using the concentrations from Eq 15. Next, Eq 12 implies that the dilution rate at which a steady state occurs must also be a function of ξ, that we denote by D*(ξ). Combining this with the relation ξ = X/D, we obtain the steady state cell density, X*(ξ), as well: (16) Note that while Eqs 12 and 13 are satisfied by any D ≥ 0 when X = 0, the value D max = D*(0) given by Eq 16 is actually the washout dilution rate, i.e., the minimum dilution rate that washes the culture of cells. Clearly all steady states with non-zero cell density are required to satisfy D*(ξ) < D max . From Eq 16 it is evident that the function μ*(ξ) encodes all the information needed to get the values of X in steady state at different dilution rates and for any value of the bleeding coefficient ϕ. On the other hand, if multiple values of ξ are consistent with the same dilution rate (i.e. if the function D*(ξ) is not one-to-one), the system is multi-stable (i.e., multiple steady states coexist under identical external conditions). A necessary condition multi-stability is that μ*(ξ) is not monotonously decreasing. Since the biomass production rate z*(ξ) is a non-increasing function of ξ (proved in the Appendix), a change in the monotonicity of μ*(ξ) must be a consequence of toxic byproduct accumulation, modeled through the terms K and σ. A noteworthy consequence of Eq 16 is that a plot displaying the parametric curve (ϕD*(ξ), ϕX*(ξ)) as a function of ξ is invariant to changes in ϕ. This means that for a given cell line and medium formulation, this curve can be obtained from measurements in a chemostat (which corresponds to ϕ = 1), and the result will also apply to any perfusion system with an arbitrary value of ϕ. Moreover, since and are independent of ϕ, cellular metabolism in steady states is equivalent in the chemostat and any perfusion system (with an arbitrary value of ϕ), provided that the values of ξ = X/D in both systems match. Finally, we mention that generally a threshold value ξ m exists, such that a steady state with ξ = X/D is feasible only if ξ ≤ ξ m . When ξ > ξ m , some of the constrains in Eqs 3, 5 and 6 cannot be met. In degenerate scenarios we could have ξ m = ∞ (e.g., this could be the case if the maintenance demand in Eq 3 is neglected) or ξ m ≤ 0 (e.g., if the medium is so poor that the maintenance demand cannot be met even with a vanishingly small cell density). The parameter ξ m arising in this way in our model, coincides with the definition of medium depth given in the literature [84], and it quantifies for a given medium composition the maximum cell density attainable per unit of medium supplied per unit time. Stability. To determine the stability of steady states we compute the Jacobian eigenvalues of the system of Eqs 1 and 2. If the real parts are all negative the state is stable, but if at least one eigenvalue has a positive real part, the state is unstable [85]. The critical case where all eigenvalues have non-negative real parts but at least one of them has a zero real part is dealt with using the Center Manifold Theorem [86, § 8.1]. The Appendix contains a detailed mathematical treatment. Briefly, a steady state is unstable if μ*(ξ) is increasing in a neighborhood, and stable otherwise. We stated above that steady states of a given cell line in continuous culture, using a fixed medium formulation, can be given as functions of ξ. The condition for stability stated here is also uniquely determined by ξ and, in particular, it is independent of ϕ. Therefore, a steady state in a chemostat (with ϕ = 1) is stable if and only if the same steady state in perfusion (with a matching value of ξ, but arbitrary ϕ) is also stable. Since our results are qualitatively invariant to changes in ϕ, we set ϕ = 1 in what follows.