Feeding strategies for E. coli fermentations demanding an enriched environment

Abstract

The addition of a carbon nutrient feed to a fed-batch cultivation is often not enough to obtain satisfactory growth and/or production. In some cases, an additional feed with for example supplementary amino acids or complex media is required. This work presents the development of feeding strategies where more than one feed is required and the knowledge of the growth requirements is low. Simulations and cultivations with E. coli are shown using the proposed feed controllers which are based on a probing control concept. The strategies work well and they can be used to shorten the process development phase considerably.

Appendix

Two models are presented. The feeding strategies are designed to operate around μ which is chosen below the critical growth rate where acetate is accumulating. Therefore the overflow metabolism is not modelled. It can be included by introducing a maximum oxygen uptake rate and letting the glucose react to acetic acid when this threshold is passed. In both models the following relations are used:

The relation between K _L a and the stirrer speed N is given by

$$K_{{\rm L}}a(N)=\alpha(N-N_{0}).$$

Henry’s law gives the dissolved oxygen concentration DO in %:

$${\rm DO} = {\rm HC}_{{\rm o}}.$$

The dissolved oxygen sensor dynamics is approximated as:

$$T_{p} \frac{{\rm d}{\rm DO}_{{\rm p}}}{{\rm d}t} + {\rm DO}_{{\rm p}} = {\rm DO}.$$

Mass balances of a fed-batch bio-reactor are given by

$$\frac{{\rm d}V}{{\rm d}t} = F_{{\rm g}} + F_{{\rm sup}} $$

(3)

$$\frac{{\rm d}({\rm VG})}{{\rm d}t} = F_{{\rm g}} G_{{\rm in}} - q_{{\rm g}} (G, {\rm sup} ){\rm VX}$$

(4)

$$\frac{{\rm d}({\rm VX})}{{\rm d}t} = \mu (G, {\rm sup} ){\rm VX}$$

(5)

$$\frac{{\rm d}({\rm VC}_{{\rm o}})}{{\rm d}t} = K_{{\rm L}} a(N)V(C_{{\rm o}}^{*} - C_{{\rm o}}) + q_{{\rm o}} (G, {\rm sup} ){\rm VX},$$

(6)

where sup is the needed supplement, in our case either lysine or complex medium. For notation and values of the parameters, see Table 1.

Model of E. coli DSM1099

The assumptions made in the model are:

• The cells cannot take up glucose without lysine.

• No growth takes place on lysine solely.

The mass balance of lysine (Lys) is given by

$$\frac{{\rm d}({\rm VLys})}{{\rm d}t} = F_{{\rm lys}} {\rm Lys}_{{\rm in}} - q_{{\rm lys}} (G, {\rm Lys}){\rm VX}.$$

The other mass balances are given by Eqs. 3–6, where the supplement (sup) is lysine. The cells cannot take up glucose without lysine, thus q _g(G, Lys) can be described by double Michaelis–Menten:

$$q_{{\rm g}} (G, {\rm Lys}) = \frac{q_{{\rm g}}^{{\rm max}} G{\rm Lys}}{(k_{{\rm sg}} + G)(k_{{\rm slys}} + {\rm Lys})}.$$

The amount of lysine that is consumed together with glucose is described by:

$$q_{{\rm lys}}(G, {\rm Lys})=q_{{\rm g}}(G, {\rm Lys})Y_{{\rm lys}/{\rm g}}.$$

The yield coefficient Y _lys/g is calculated from the experimental Y _x/lys as Y _lys/g  = Y _x/g/Y _x/lys.

A part of the carbon source is used for maintenance:

$$q_{{\rm m}}=\min(q_{{\rm g}}(G, {\rm Lys}),q_{{\rm mc}}).$$

The glucose flow can be divided into two: q ^en_g that is used for energy purposes and q ^an_g that is used in the anabolism

$$\begin{aligned} q_{{\rm g}}^{{\rm an}} & =(q_{{\rm g}} - q_{{\rm m}})Y_{{\rm x/g}} \frac{C_{{\rm x}}}{C_{{\rm g}}}\\ q_{{\rm g}}^{{\rm en}} &= q_{{\rm g}} - q_{{\rm g}} ^{{\rm an}} \end{aligned}$$

Growth μ and oxygen consumption q _o are described by:

$$\begin{aligned} \mu &= (q_{{\rm g}} - q_{{\rm m}})Y_{{\rm x/g}}\\ q_{{\rm o}} &= q_{{\rm g}}^{en} Y_{{\rm o/g}}\\ \end{aligned} $$

Model of E. coli DSM6968

The following assumptions are made in the model:

• The glucose uptake q _g is only used for energy purposes.

• Growth can take place on complex medium.

• The complex uptake q _c is independent of the glucose uptake.

• The complex medium is used as building blocks firstly and as energy source secondly.

The mass balance of the complex medium (C) is:

$$\frac{{\rm d}({\rm VC})}{{\rm d}t} = F_{{\rm c}} {\rm Comp}_{{\rm in}} - q_{{\rm c}} (C){\rm VX}.$$

The other mass balances are given by the Eq. 3–6, where the supplement (sup) is a complex medium. The glucose uptake rate when there is enough complex medium present is given by

$$q_{{\rm g}}^{{\rm norm}} (G) = \frac{q_{{\rm g}}^{{\rm max}} G}{k_{{\rm sg}} + G}.$$

The complex uptake rate is described by:

$$q_{{\rm c}} (C) = \frac{q_{{\rm c}}^{{\rm max}} C}{k_{{\rm scomp}} + C}.$$

The complex medium flow into the cells is divided into two: q ^an_c that describes the amount of complex medium that is used as building blocks in the cell growth, and q ^en_c that describes the amount of complex medium used for energy purposes. The complex medium goes to q ^en_c if there is not enough glucose to handle the energy requirements. This leads to a fraction, denoted φ, of q _c that goes to cell growth. It is also assumed that twice (η = 2) as much complex medium as glucose is demanded for an efficient cell growth, i.e. glucose is used as energy and the complex medium is used as building blocks. After introduction of ${q_{{\rm g}}^{{\rm limit}} = \frac{q_{{\rm c}}}{\eta} + q_{{\rm mc}}},$the model is given by

If q ^norm_g >  q ^limit_g

$$\begin{aligned} q_{{\rm c}}^{{\rm an}^{*}}&= q_{{\rm c}} \\ q_{{\rm c}}^{{\rm en}}&=0\\ q_{{\rm g}} &= q_{{\rm g}}^{{\rm limit}}\\ \end{aligned}$$

else if q _mc <  q ^norm_g <  q ^limit_g

$$\begin{aligned} q_{{\rm g}}& = q_{{\rm g}}^{{\rm norm}}\\ q_{{\rm c}}^{{\rm an},1^{*}} &= \eta (q_{{\rm g}}^{{\rm norm}} - q_{{\rm mc}})\\ q_{{\rm c}}^{{\rm an}, 2^{*}}& = (q_{{\rm c}} - \eta (q_{{\rm g}}^{{\rm norm}} - q_{{\rm mc}}))\phi \\ q_{{\rm c}}^{{\rm an}^{*}}& = q_{{\rm c}}^{{\rm an}, 1^{*}} + q_{{\rm c}}^{{\rm an}, 2^{*}}\\ q_{{\rm c}}^{{\rm an}, 2}& = q_{{\rm c}}^{{\rm an}, 2^{*}} Y_{{\rm x/c}} \frac{C_{{\rm x}}}{C_{{\rm c}}}\\ q_{{\rm c}}^{{\rm en}}&= q_{{\rm c}} - \eta (q_{{\rm g}}^{{\rm norm}} - q_{{\rm mc}}) - q_{{\rm c}}^{{\rm an}, 2}\\ \end{aligned} $$

else q ^norm_g <  q _mc

$$\begin{aligned} q_{{\rm g}} &= q_{{\rm g}}^{{\rm norm}}\\ q_{{\rm c}}^{{\rm an}^{*}}& = \max (0,(q_{{\rm c}} + \eta (q_{{\rm g}}^{{\rm norm}} - q_{{\rm mc}}))\phi)\\ q_{{\rm c}}^{{\rm an}} &= q_{{\rm c}}^{{\rm an}^{*}} Y_{{\rm x/c}} \frac{C_{{\rm x}}}{C_{{\rm c}}}\\ q_{{\rm c}}^{{\rm en}} &= q_{{\rm c}} - q_{{\rm c}}^{{\rm an}}\\ \end{aligned}. $$

The cell growth μ is described by:

$$\mu = q_{{\rm c}}^{{\rm an}^{*}} Y_{{\rm x/c}}.$$

The specific oxygen consumption is described by q _o, where q ^en_g = q _g,

$$q_{{\rm o}} = q_{{\rm g}}^{{\rm en}} Y_{{\rm o/g}} + q_{{\rm c}}^{{\rm en}} Y_{{\rm o/c}}.$$

Linearised model, pulse response and stability analysis of E. coli DSM6968

Linearised model

Linearised equations when q _mc <  q ^norm_g <  q ^limit_g are

$$\begin{aligned} T_{{\rm o}} \frac{{\rm d}\Delta {\rm DO}}{{\rm d}t} + \Delta {\rm DO} &= K_{{\rm og}} \Delta q_{{\rm g}} + K_{N} \Delta N + K_{{\rm oc}} \Delta q_{{\rm c}} \\ T_{{\rm g}} \frac{{\rm d}\Delta q_{{\rm g}}}{{\rm d}t} + \Delta q_{{\rm g}} &= K_{{\rm gf}} \Delta F_{{\rm g}} \\ T_{{\rm c}} \frac{{\rm d}\Delta q_{{\rm c}}}{{\rm d}t} + \Delta q_{{\rm c}} &= K_{{\rm cf}} \Delta F_{{\rm c}} \\ Tp\frac{{\rm d}\Delta {\rm DO}_{{\rm p}}}{{\rm d}t} + \Delta {\rm DO}_{{\rm p}} &= \Delta {\rm DO}\\ \end{aligned}$$

$$\begin{aligned} K_{{\rm og}} &= - \frac{{\rm HX}}{K_{{\rm L}} a}(Y_{{\rm o/g}} - \eta Y_{{\rm o/c}}^{*})\quad K_{{\rm oc}} = - \frac{{\rm HX}}{K_{{\rm L}} a}Y_{{\rm o/c}}^{*} \\ K_{{\rm N}}& = \frac{{\rm DO}^{*} - {\rm DO}}{K_{{\rm L}} a}\frac{\delta K_{{\rm L}} a}{\delta N}\quad K_{{\rm gf}} = \frac{G_{{\rm in}}}{{\rm VX}}\\ K_{{\rm cf}} &= \frac{{\rm Comp}_{{\rm in}}}{{\rm VX}}\quad T_{{\rm g}} = \left(\frac{\delta q_{{\rm g}}}{\delta G}X\right)^{- 1}\\ T_{{\rm o}} &= K_{{\rm L}} a^{- 1} \quad T_{{\rm c}} = \left(\frac{\delta q_{{\rm c}}}{\delta C}X\right)^{- 1}\\ \end{aligned}$$

where ${Y_{{\rm o/c}}^{*} = \left(1 - Y_{{\rm x/c}} \phi \frac{C_{{\rm x}}}{C_{{\rm c}}}\right)Y_{{\rm o/c}}}$ is introduced.

The pulse response and stability analysis

The pulse response is given by three equations when q _g > q _mc:

$$\Delta {\rm DO} = \left\{\begin{array}{*{20}l} |K_{{\rm og}} |q^{{\rm pulse}}_{{\rm g}}&\hbox{if}\;q_{{\rm g}} < q_{{\rm g}}^{{\rm limit}} - q_{{\rm g}}^{{\rm pulse}}\\ |K_{{\rm og}} |(q_{{\rm g}}^{{\rm limit}} - q_{{\rm g}})&\hbox{otherwise}\\ 0& q_{{\rm g}} > q_{{\rm g}}^{{\rm limit}} \\ \end{array}\right.$$

where q ^pulse_g =  K _gf ΔF ^pulse_g . The glucose dynamics (T _g), the complex medium dynamics (T _c), the oxygen dynamics (T _o) and the dissolved oxygen sensor dynamics (T _p) are assumed to be fast compared to the pulse length and control phase. This gives the following response when q _mc <  q _g <  q ^limit_g − q ^pulse _g:

$$\begin{aligned} \Delta {\rm DO} = |K_{{\rm og}} |K_{{\rm gf}} \Delta F_{{\rm g}} &= \frac{{\rm HG}_{{\rm in}}}{{\rm VK}_{L} a}(Y_{{\rm o/g}} - \eta Y_{{\rm o/c}}^{*})\Delta F_{{\rm g}}\\ & = \frac{{\rm DO}^{*} - {\rm DO}_{{\rm sp}}}{{1 + \frac{{q_{{\rm c}} + \eta q_{{\rm mc}}}}{{q_{{\rm g}} }}\frac{{Y_{{\rm o/c}}^{*}}}{{Y_{{{\rm o/g}}} - \eta Y_{{\rm o/c}}^{*}}}}}\Delta F_{{\rm g}} \approx 8 \end{aligned}, $$

when DO^* =  80, DO_sp =  30 and q _g ≈ q ^limit_g . This is the maximum pulse response that can be obtained with these values of the parameters.

For q ^limit_g − q ^pulse_g <  q _g <  q ^limit_g , the response is given by

$$\Delta {\rm DO}(k + 1) = \Delta {\rm DO}(k) + \frac{|K_{{\rm og}} |}{\eta}(q_{{\rm c}} (k + 1) - q_{{\rm c}} (k)),$$

when q _g is assumed to be constant (F _g is an exponential feed corresponding to the exponential cell growth) and q _c(k + 1) = q _c(k) + K _cfΔF _c −ɛ. ε describes the influence of the changing cell mass VX. With the controller, Eq. 2, the closed loop response is:

$$\Delta {\rm DO}(k + 1) = \left(1 - \frac{|K_{{\rm og}}| K_{{\rm cf}} \kappa F_{{\rm c}}}{{\eta ({\rm DO}^{*} - {\rm DO}_{{\rm sp}})}}\right)\Delta {\rm DO}(k) + \frac{|K_{{\rm og}} |K_{{\rm cf}} \kappa F_{{\rm c}} }{{\eta ({\rm DO}^{*} - {\rm DO}_{{\rm sp}})}}y_{{\rm r}} - |K_{{\rm og}} |\varepsilon.$$

As

$$|K_{{\rm og}} |K_{{\rm cf}} = \frac{{\rm DO}^{*} - {\rm DO}_{{\rm sp}}}{{F_{{\rm c}} \frac{q_{{\rm g}}}{q_{{\rm c}} } + \frac{q_{{\rm c}} + \eta q_{{\rm mc}}}{q_{{\rm c}}}\frac{Y_{{\rm o/c}}^{*}}{{Y_{{\rm o/g}} - \eta Y_{{\rm o/c}}^{*}}}}} \approx \frac{{\rm DO}^{*} - {\rm DO}_{{\rm sp}}}{{F_{{\rm c}} \left(\frac{1}{\eta} + \beta \right)}}$$

with ${\frac{q_{{\rm g}}}{q_{{\rm c}}} \approx \frac{1}{\eta}, \frac{\eta q_{{\rm mc}} + q_{{\rm c}}}{q_{{\rm c}}} \approx 1}$ and ${\frac{Y_{{\rm o/c}}^{*}}{Y_{{\rm o/g}} - \eta Y_{{\rm o/c}}^{*}} \approx \beta }.$ This gives:

$$\Delta {\rm DO}(k + 1) = \left(1 - \frac{\kappa}{1 + \eta \beta}\right)\Delta {\rm DO}(k) + \frac{\kappa}{1 + \eta \beta}y_{{\rm r}} - |K_{{\rm og}} |\varepsilon.$$

The stability boundaries of κ are then given by: 0 < κ < 2(1 + ηβ) and the convergence point y ^* is given by: ${y^{*} = y_{{\rm r}} - \frac{(1 + \eta \beta )|K_{{{\rm og}}} |}{\kappa }\varepsilon }.$

The influence from ɛ helps to reduce the risk of overfeeding the complex medium.