Article pubs.acs.org/IECR
Cybernetic Modeling Revisited: A Method for Inferring the Cybernetic Variables ui from Experimental Data Aravinda R. Mandli† and Jayant M. Modak*
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India ABSTRACT: The cybernetic modeling framework for the growth of microorganisms provides for an elegant methodology to account for the unknown regulatory phenomena through the use of cybernetic variables for enzyme induction and activity. In this paper, we revisit the assumption of limited resources for enzyme induction (∑ui = 1) used in the cybernetic modeling framework by presenting a methodology for inferring the individual cybernetic variables ui from experimental data. We use this methodology to infer ui during the simultaneous consumption of glycerol and lactose by Escherichia coli and then model the fitness trade-offs involved in the recently discovered predictive regulation strategy of microorganisms.
1. INTRODUCTION Modeling the growth of microorganisms in fermentation processes has witnessed renewed attention in recent years, primarily because of the process analytical technology guidelines.1 Several modeling frameworks exist in the literature for modeling the growth of microorganisms.2 Among them, the cybernetic modeling framework of Ramkrishna and co-workers has been receiving increased attention because of its potential applications in the areas of bioprocess optimization and metabolic engineering.3 The cybernetic modeling framework distinguishes itself by taking into account the unknown regulatory processes in the microorganisms. This framework assumes that microorganisms have evolved under tremendous selective pressure to become optimal with respect to certain cellular objectives (for example, maximization of the growth rate, maximization of the substrate uptake rate, etc.) and achieve optimality by actively modulating the induction/repression and activation/inhibition of the key enzymes of substrates available in their external environment.3 Modulation of the key enzymes is achieved through the introduction of cybernetic variables for induction/repression (ui) and activation/inhibition (vi) of the key enzymes in the governing differential equations of cell growth, substrate consumption, and key enzyme production. The cybernetic models also assume that the microorganisms possess limited resources for enzyme induction (∑ui = 1) and that they intelligently allocate these limited resources for the induction of various key enzymes to achieve the cellular objectives. On the basis of the aforementioned assumptions, Ramkrishna and co-workers have proposed the utilization of matching and proportional laws for the induction/repression and activation/inhibition of the key enzymes.4−6 Although possible resources and the reasons for their limitation have been discussed previously,5 they have not been identified precisely3 in the literature. In addition, the extent of limitation of the resources has not been quantified in the literature using experimental data. Dekel and Alon7 have experimentally determined the costs and benefits of enzyme induction during the growth of Escherichia coli on glycerol and lactose. In these experiments (Figure 1A), E. coli cells are grown on glycerol at various © 2015 American Chemical Society
concentrations of IPTG, a nonmetabolizable inducer of the lac operon. Because IPTG results in the induction of enzymes required for lactose consumption, part of the resources invested for the production of key enzymes of glycerol get diverted, and this manifests as a cost (η, defined as the relative reduction in the growth rate because of the inducer) for the growth of microorganisms. Similarly, a benefit (B, defined as the relative increase in the growth rate in the presence of lactose because of IPTG) manifests when lactose at various concentrations is supplemented to the growth medium containing glycerol and a saturating concentration of IPTG. Recent experiments have unraveled the evolutionary advantage of various regulation strategies such as stochastic switching regulation for microorganisms growing in fluctuating environments.8,9 Mitchell et al.10 have explored the advantages of a predictive regulation strategy, termed asymmetric regulation, for microorganisms when the environmental changes are characterized by a predictable temporal order. For E. coli, such a temporal order exists when they pass through the intestines of mammals, wherein they first encounter lactose and subsequently encounter maltose.11 Mitchell et al.10 have shown that when E. coli is exposed to lactose, it anticipates the subsequent arrival of maltose and gains a fitness advantage. Mitchell and Pilpel12 have further explored the advantages of the predictive regulation strategy for the growth of E. coli on glycerol and lactose experimentally. In these experiments (Figure 1B), for predictive regulation, E. coli is first grown in a medium containing glycerol and a saturating concentration of IPTG for a period of Δt minutes. At this juncture, lactose is further added to the medium. The addition of IPTG results in the induction of enzymes for lactose utilization and hence results in a predictive capacity for the cells upon the arrival of lactose. For direct regulation, lactose and IPTG are added simultaneously at Δt minutes to cells growing on glycerol. The Special Issue: Doraiswami Ramkrishna Festschrift Received: Revised: Accepted: Published: 10190
January 22, 2015 March 29, 2015 May 13, 2015 May 13, 2015 DOI: 10.1021/acs.iecr.5b00306 Ind. Eng. Chem. Res. 2015, 54, 10190−10196
Article
Industrial & Engineering Chemistry Research
Figure 1. Schematic of the experimental procedure in (A) Dekel and Alon7 for measuring costs and benefits during the growth of E. coli on glycerol and lactose and (B) Mitchell and Pilpel12 for measuring the relative fitness in the predictive regulation strategy.
relative fitness is measured as the ratio of the cellular concentration of E. coli cells with predictive regulation to that of E. coli cells with direct regulation 1.5 h after the addition of lactose. In the present study, we will present a methodology for inferring the cybernetic variables ui of the cybernetic model from cost and benefit measurements and utilize it to infer ui during the simultaneous consumption of glycerol and lactose by E. coli.7 Furthermore, we utilize the inferred ui while modeling the fitness trade-offs involved in the predictive regulation strategy of microorganisms.12
levels of key enzyme for substrate Si, αi* and rEi are the constitutive and inductive synthesis rates for the key enzyme ei, and βi is the first-order degradation rate. The expression for the specific growth rate ri is given by the modified Monod kinetics m max ri = rMi(ei/emax i ) = [μi si/(Ki + si)](ei/ei ), where rMi represents Monod’s growth kinetics and μim and Ki represent the maximum growth rate and half-saturation constant for growth on substrate Si, respectively. The inductive enzyme synthesis rates for the key enzymes rEi are given by rEi = αisi/(Ki + si), where αi are the maximum inductive enzyme synthesis rates. From eq 3, the maximum level of the key enzymes can be derived as
2. METHODOLOGY FOR INFERRING ui 2.1. Growth of Microorganisms on N Substitutable Substrates. We consider the growth of microorganisms C on N growth-limiting substitutable substrates Si (i = 1, 2, ..., N). The governing dynamic equations of the cybernetic model for the growth of microorganisms are written as4,13 ⎛N ⎞ dc ⎟ = ⎜⎜∑ rv i i ⎟c = μc dt ⎝ i=1 ⎠
(1)
dsi 1 = − rvc i i , dt Yi
(2)
i = 1, 2, ..., N
dei = αi* + uirEi − μei − βi ei , dt
i = 1, 2, ..., N
eimax =
αi* + rEi βi + rMi
(4)
We assume that a nonmetabolizable inducer Hi exists for the operons of each of the substrates Si. For the growth of microorganisms on multiple substitutable substrates, the cost of microorganisms for growth on substrate Si, ηi, can be measured as the relative growth rate difference between cells growing exclusively on Si and for cells growing on a mixture of Si and the nonmetabolizable inducers of the operons of all of the other substrates ∑j=1−N,j≠iHj. e
rMi − rMi e maxi μSi − μSi +∑j ,j≠i Hj i ηi = = rMi μSi
(3)
where c and si are the concentrations of microorganisms C and substrate Si, respectively, ri and Yi are the specific growth rate and yield on substrate Si, μ is the overall growth rate, ei are the
(5)
Simplifying eq 5, we obtain 10191
DOI: 10.1021/acs.iecr.5b00306 Ind. Eng. Chem. Res. 2015, 54, 10190−10196
Article
Industrial & Engineering Chemistry Research
ηi = 1 −
ei eimax
where h2 is the concentration of the nonmetabolizable inducer H2. u1 can be inferred from eq 11 using the procedure outlined before. u2 can be inferred from the measurements of benefit (B), which is measured as the relative growth rate difference between cells growing on a mixture of S1, H2, and S2 and cells growing exclusively on S1.7 Hence,
(6)
On the basis of the experimental procedure of Dekel and Alon,7 we assume that the concentration of microorganisms is too low and does not cause significant changes in the substrate concentrations until the measurements of cost are made. Hence, dsi/dt ≈ 0. We also assume that all of the cybernetic variables for enzyme activation/inhibition, vi, are equal to 1 because we have obtained in a previous study that the activity of the key enzymes is maintained at the maximum level under the constraints employed in the cybernetic modeling methodology to maximize the final cellular productivity.14 This policy for vi has been termed “indifferent” by Young and Ramkrishna6 and is also used in the cybernetic model formulation of Dhurjati et al.15 Under these assumptions, eqs 1−3 can be simplified to ⎛N ⎞ dc = ⎜⎜∑ ri⎟⎟c dt ⎝ i=1 ⎠
B=
μS1+ H2 + S2 = μS1(B + 1)
μS1+ H2 + S2 = rM1
ei eimax
−βi +
(9)
1 eimax
2rMi
(10)
[μS1(B + 1) + β2] −
eimax [(1 − ηi)2 rMi + (1 − ηi)βi ] − αi* rEi
(17)
(18)
⎛ 1 ⎞⎛ α2* + rE2 ⎞⎛ K 2 + s2 + h2 ⎞ ⎟⎟⎜ ⎟⎜⎜ ⎟ ⎜ ⎝ α2rM2 ⎠⎝ β2 + rM2 ⎠⎝ s2 + h2 ⎠
Solving for the cybernetic variable ui using eqs 6 and 10 and further simplifying, we obtain ui =
e1max
⎧ ⎫ ⎡ α*+ ur ⎤ ⎛ β + r ⎞⎪ ⎪ 1 1 E1 M1 ⎥⎜⎜ 1 ⎟⎟⎬ u 2 = ⎨μS1(B + 1) − rM1⎢ S ⎪ ⎢⎣ μ 1(B + 1) + β1 ⎥⎦⎝ α1* + rE1 ⎠⎪ ⎭ ⎩
( )
=
e2
Substituting eqs 4, 16, and 18 in eq 17, we obtain
we obtain
βi 2 + 4(αi* + uirEi)rMi
+ rM2
α (s + h ) αi* + ui K i +i s + ih dei = 0 ⇒ ei = S + H + Si i i dt μ 1 2 2 + βi
During exponential growth on Si in the presence of inducers of all of the other substrates, eq 8 for key enzyme ei becomes
Simplifying eq 9 for
e1 max e1
During exponential growth on a mixture of S1, H2, and S2, the enzyme levels can be calculated using
(8)
ei/emax i ,
(16)
The specific growth rate in the presence of S1, H2, and S2, μS1+H2+S2, is written as
i = 1, 2, ..., N
dei ei 2 = 0 ⇒ αi* + uirEi − rMi max − βi ei = 0 dt ei
(15)
Simplifying eq 15 for μS1+H2+S2, we obtain
(7)
⎛N ⎞ dei = αi* + uirEi − ⎜⎜∑ ri⎟⎟ei − βi ei , dt ⎝ i=1 ⎠
μS1+ H2 + S2 − μS1 μS1
Equation 11 can be used to determine the cybernetic variables ui from the measurement of costs ηi when nonmetabolizable inducers Hi exist for the operons of each of the substrates Si. 2.2. Growth of Microorganisms on Two Substitutable Substrates. For the growth of microorganisms on two substitutable substrates, ui for both the substrates can be inferred even if a nonmetabolizable inducer Hi exists for only one of the substrates. Without loss of generality, we assume that the substrate without a corresponding nonmetabolizable inducer is denoted as S1 and the other substrate as S2. The governing dynamic equations (7) and (8) for the growth of microorganisms on the two substrates can be written as (12)
de1 = α1* + u1rE1 − (r1 + r2)e1 − β1e1 dt
(13)
(19)
The values of u2 can be inferred from eq 19 using u1 inferred previously and measurements of benefit (B).
(11)
dc = (r1 + r2)c dt
α2* ⎛ K 2 + s2 + h2 ⎞ ⎟ ⎜ α2 ⎝ s 2 + h 2 ⎠
3. RESULTS We infer the cybernetic variables for enzyme induction ui during the growth of E. coli on glycerol (denoted as S1) and lactose (denoted as S2) from the cost and benefit data measured by Dekel and Alon.7 IPTG (denoted as H2) is used as the nonmetabolizable inducer of the lactose operon. On the basis of the experimental conditions in Dekel and Alon,7 we assume that the concentrations of the substrate S1 and inducer H2, when present in the growth medium, are high compared to the half-saturation constants so that s1/(K1 + s1), (s2 + h2)/(K2 + s2 + h2) ≈ 1. We further assume that the rate of degradation of the key enzymes for both substrates is zero because we recently showed that the optimal degradation policy for the key enzymes is regulated and plays a role only when the growthlimiting substrate in the medium is depleted to very low levels.14 Under these assumptions, eqs 11 and 19 can be simplified to
⎛ s2 + h2 ⎞ de 2 = α2* + u 2α2⎜ ⎟ − (r1 + r2)e 2 − β2e 2 dt ⎝ K 2 + s2 + h2 ⎠
u1 =
(14) 10192
(1 − η)2 (α1* + α1) α* − 1 α1 α1
(20)
DOI: 10.1021/acs.iecr.5b00306 Ind. Eng. Chem. Res. 2015, 54, 10190−10196
Article
Industrial & Engineering Chemistry Research u2 =
(μ1m )2 ⎡ α1* + u1α1 ⎤⎛ K + s2 2 ⎢ ⎥⎜α2* 2 + − ( B 1) m 2 * s2 α1 + α1 ⎦⎝ (μ2 ) ⎣ ⎞ K + s2 α* + α2⎟ 2 − 2 α2 ⎠ α2s2
1.2, higher than the value of 1 assumed by the cybernetic framework. This result is in contrast to assumptions of the cybernetic modeling framework, wherein the total resources for enzyme induction are always assumed to be limited. The value of ∑ui thus inferred from the experimental data quantifies the extent of limitation of the key resources. We now model the predictive regulation strategy of microorganisms explored by Mitchell and Pilpel.12 For cells with direct regulation, u1 = 1 and u2 = 0 in the interval 0−Δt minutes, whereas in the interval Δt−(Δt + 90) min, u1 and u2 are determined using eqs 20 and 21 from the cost and benefit measurements of Dekel and Alon7 (Table 2). For cells with
(21)
u1 is determined using eq 20 from the cost measured at a saturating concentration of IPTG. We assume that this cost does not vary when lactose is added to the medium. u2 is determined using eq 21 at various levels of lactose. u2 thus determined represents the partial utilization of resources diverted by the saturating concentration of IPTG. Hence, at saturating lactose levels, u2 provides a measure of the total resources diverted by the saturating concentration of IPTG. We assume that the half-saturation constant for growth of E. coli on lactose, K2, is 0.0591 mM16 and further neglect the data of benefit at very low substrate concentrations (
