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Optimal fed-batch operation of recombinant cells subject to plasmid instability and death


Optimal feed rate strategy is studied for fed-batch culture of recombinant cells with plasmid instability and with different death rates for the plasmid-free cells (PFC) and plasmid-bearing cells (PBC). Most of the fed-batch fermentation is known to have first-order singularity and therefore a single singular arc. However, this study shows that a singular arc with second-order singularity and therefore two distinct singular arcs are possible for a recombinant cell process if PFC and PBC are subjected to death, and their specific growth rates are proportional to each other. Two types of singular arcs are elucidated and analyzed. The optimal policies over the singular arcs are theoretically explored as these findings reveal qualitative information on the singular arc, which is critically important in providing the optimal initial conditions in numerical computation of optimal feed rate profile.

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F :

Feed rate (cm3/s)

F max :

Maximum feed rate (cm3/s)

F min :

Minimum feed rate (cm3/s)

F singular :

Singular feed rate (cm3/s)

H :

Hamiltonian function


Plasmid-bearing cells (g)


Plasmid-free cells (g)

k d :

Constant death rate (s)

S :

Substrate concentration (g/cm3)

S F :

Feed substrate concentration (g/cm3)

t :

Time (s)

t f :

Final time (s)

V :

Bioreactor volume (cm3)

V max :

Maximum bioreactor volume (cm3)

x 1 :

Total plasmid-containing cells (g)

x 2 :

Total plasmid-free cells (g)

x 3 :

Total substrate (g)

x 4 :

Bioreactor volume (cm3)

x 5 :

Time (t)

X :

Cell mass concentration (g/cm3)

Y x/s :

Cell mass yield (g/g)


Switching function


Specific growth rate (s)


Fraction of plasmid-containing cells that revert


The ratio, μ(S) = βμ+(S)


λ4(t f ) = η is a unknown adjoint variable corresponding to x 4(t f ) = V max


Plasmid-bearing cells, PBC


Plasmid-free cells, PFC


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One (HSS) of the authors was supported in part by a fellowship from Korea Science and Engineering Foundation (KOSEF) and a grant from the University of California, Irvine.

Author information

Correspondence to Hwa Sung Shin.


Appendix 1

Re-writing the first ODE of Eq. (6),

$${\dot{\lambda}}_1 =-\lambda_1 (1-\alpha)\mu^+-(\lambda_2 \alpha -{\lambda_3} \mathord{\left/ {\vphantom {{\lambda_3} {Y_{x/s}^+}}} \right. } {Y_{x/s}^+})\mu^+=m\lambda_1 +n$$

where m = −(1−α)μ+ and \(n=-(\lambda_2 \alpha -{\lambda_3} \mathord{\left/ {\vphantom {{\lambda_3} {Y_{x/s}^-}}} \right.}{Y_{x/s}^-})\mu^+.\) On the singular arc m is always negative since α < 1 and μ+ > 0. If n = 0, λ1(t) approaches zero asymptotically. However, n is always negative over the same region because λ2 > 0 and λ3 < 0. This means λ1(t) is less than the negative value of λ1(t) when n = 0 for all times over the singular arc. Therefore, we can say that λ1(t) is always negative over a singular arc. Correspondingly, B is always positive over the singular arc.

Appendix 2

Equation (6) gives rise to the final adjoint variable vector, [−1, 0, 0, η, 0] and the Hamiltonian at the final time is −μ+ x 1(t f). Therefore, the Hamiltonian is negative constant over all time because it must be constant. Over the singular arc where ϕ = 0 and since λ5(t) = 0, we conclude that

$$A\mu^+x_1 +B\mu^-x_2 < \lambda_2 k_{\rm d} x_2$$

Since λ2 is positive over the singular arc, λ2 k d x 2 > 0 and the sufficient condition for Eq. (A2) is

$$A\mu^+x_1 +B\mu^-x_2 < \,0$$

Substituting Eq. (9) into Eq. (A3) and recalling that B is positive over the singular arc, we obtain

$$({Bx_2} \mathord{\left/ {\vphantom {{Bx_2} {\mu_s^+)({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^-)_s}}} \right. } {\mu^-)_s}}}} \right. } {\mu_s^+)({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^-)_s}}} \right. } {\mu^-)_s }}(\mu^-)^2 < \,0\,\,\,\,\Rightarrow \,\,\,\,sign(\mu_s^+)=-sign({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^-)_s}}} \right. } {\mu^-)_s}$$

Before going through a further analysis of the singular arc, let us restrict the form of specific growth rates μ+ and μ. As stated in the Introduction, the process uses the difference in death rates of the PBC and PFC. At very low substrate concentrations the PFC suffer death whereas the PBC do not. At low substrate concentrations, no inhibition takes place, and both specific growth rates μ+and μ increase with the substrate concentration so that both μ s + and μ s are positive over the affected ranges (very low substrate concentration). Then, \(sign({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^-)_s}}} \right. } {\mu^-)_s}\) is negative from Eq. (A4) because sign (μ s + ) is positive. This means that if substrate concentration increases with time \((\dot{S} > 0),\mu^+\) increases and \({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^-}}} \right. } {\mu^-}\) decreases correspondingly, but if \(\dot {S} < \,0, \mu^+\) decreases and \({\mu^+} \mathord{\left/ {\vphantom {{\mu ^+} {\mu^-}}} \right. } {\mu^-}\) increases. Although not simple to determine the definite sign of \(\dot{S},\) we can say that \(\dot {S} < \,0\) on a singular arc so that the substrate concentration moves in the direction of increasing the ratio of specific rates \(({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^-)}}} \right. } {\mu^-)}\) gradually at the expense of μ+. It is in line with the fact that increasing \({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^-}}} \right. } {\mu^-}\) is favorable to PBC formation, even though low \(\mu^+\) and decreasing substrate concentration lead to taking more time for the process to reach the final reactor volume. This can be represented by the following

$$\frac{{\rm d}S}{{\rm d}t} < \,0,\,\,\frac{{\rm d}S}{{\rm d}t}\frac{{\rm d}\mu^+}{{\rm d}S}=\frac{{\rm d}\mu ^+}{{\rm d}t} < \,0,\,\,\frac{{\rm d}S}{{\rm d}t}\frac{{\rm d}({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^-)}}} \right. } {\mu ^-)}}{dS}=\frac{d({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^-)}}} \right. } {\mu^-)}}{{\rm d}t} > \,0$$

We can conclude that the substrate concentration decreases with time in the direction of gradually increasing the ratio of specific rates \(({\mu^+} \mathord{\left/ {\vphantom {{\mu^+} {\mu^-)}}} \right. } {\mu^-)}\) at the expense of μ+ and μ on a singular arc.

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Shin, H.S., Lim, H.C. Optimal fed-batch operation of recombinant cells subject to plasmid instability and death. Bioprocess Biosyst Eng 31, 655–665 (2008).

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  • Plasmid-bearing cells
  • Plasmid stability
  • Different cell death rate
  • Fed-batch
  • Singular arc
  • Optimal feed rate profile