Licensed Unlicensed Requires Authentication Published by De Gruyter December 16, 2017

A Mathematical Model for End-Product Toxicity

Mark Ian Nelson
From the journal Chemical Product and Process Modeling
https://doi.org/10.1515/cppm-2017-0061

Abstract

Alcohol based biofuels, such as bio-butanol, have considerable potential to reduce the demand for petrochemical fuels. However, one of the main obstacles to the commercial development of biological based production processes of biofuels is end-product toxicity to the biocatalyst. We investigate the effect of end-product toxicity upon the steady-state production of a biofuel produced through the growth of microorganisms in a continuous flow bioreactor. The novelty of the model formulation is that the product is assumed to be toxic to the biomass. The increase in the per-capita decay rate due to the presence of the product is assumed to be proportional to the the concentration of the product. The steady-state solutions for the model are obtained, and their stability determined as a function of the residence time. These solutions are used to investigate how the maximum yield and the reactor productivity depend upon system parameters. Unlike systems which do not exhibit toxicity there is a value of the feed concentration which maximises the product yield. The maximum reactor productivity is shown to be a sharply decreasing function of both the feed concentration and the toxicity parameter. In conclusion, alternative reactor configurations are required to reduce the effects of highly toxic products.

Keywords: biofuel; bioreactor; end-product toxicity; fermentation; stress tolerance

Appendix

A Attracting Region

In this appendix we show that the region R is both positively invariant and attracting.

A.1 Solution components may not become negative

We first show that if the initial conditions are non-negative that the solution remains non-negative. In order for the solution to become negative it’s value must reduce in value to zero. We have

dS∗dt∗S∗=0=S0∗τ∗≥0,dX∗dt∗X∗=0=0,dP∗dt∗P∗=0=S∗X∗1+S∗≥0.

Note that second of these derivatives shows that the line X∗=0 is itself (positively) invariant.

A.2 The substrate component is bounded

We now show that the region 0≤S∗≤S0∗ is both (positively) invariant and exponentially attracting. We have

dS∗dt∗≤1τ∗S0∗−S∗,(as X∗≥0 and S∗≥0).

Let Z1 be the solution of the differential equation

dZ1dt∗=1τ∗S0∗−Z1

with initial condition Z10=S0. It follows from the classical scalar comparison theorem for ordinary differential equations that S∗t∗≤Z∗t∗. Hence

S∗t∗≤S0∗−S0∗−S∗0exp−t∗τ∗.

This inequality shows that the region 0≤S∗≤S0∗ is invariant, because if the initial condition is within the invariant region, i.e. S∗0≤S0∗, then the solution remains within the invariant region, i.e. S∗≤S0∗. Furthermore, if the initial condition is outside the invariant region, S∗0>S0∗, then the solution must eventually enter the invariant region, i.e. S∗≤S0∗.

A.3 The biomass component is bounded

Let Z=S∗+X∗ with initial condition Z0=S∗0+X∗0. Then adding eqs (5) we have and (6)

dZdt∗≤S0∗τ∗−βτ∗Z,(as X∗≥0, P∗≥0, and 0≤β≤1).

Applying the scalar comparison theorem, as in A.2, we have

Z=X∗t∗+S∗t∗≤S0∗β−S0∗β−S∗(0−X∗0)exp−βτ∗t∗.

Combining this result with our earlier bound on the scaled substrate concentration it follows that the region

0≤S∗≤S0∗,0≤X∗≤S0∗β−S∗

is bounded and exponentially attracting.

A.4 The product component is bounded

Let Z=S∗+P∗ with initial condition Z0=S∗0+P∗0. Then adding eqs (5) we have and (7)

dZdt∗=S0∗τ∗−Zτ∗,

Applying the scalar comparison theorem, as in A.2, we have

Z=P∗t∗+S∗t∗=S0∗−S0∗−S∗(0−P∗0exp−t∗τ∗.

Combining this result with our earlier bound on the scaled substrate concentration it follows that the region

0≤S∗≤S0∗,0≤P∗≤S0∗−S∗

is bounded and exponentially attracting.

Furthermore, taking the limit t→∞ we obtain

P∗=S0∗+P0∗−S∗.

B The optimal value for the feed concentration to maximise the yield

After some algebra we find that

(37)ddS0∗Yτ∗=∞=0,
(38)⇒GS0∗=a1S0∗2+b1S0∗+c1=0,

where the coefficients are given by

(39)a1=1−kd∗kp∗2,
(40)b1=−2kd∗kp∗kp∗−1−kd∗,
(41)c1=−1+kp∗+kd∗2−4kd∗kd∗.

The discriminant of the quadratic eq. (38) is positive

(42)b12−4a1c1=4kd∗kp∗2kd∗−kp∗−12.

Consequently eq. (38) has two solutions. As the coefficient a1 is strictly positive we have

(43)S0,+∗>S0,−∗

where

(44)S0,+∗=−b1+b12−4a1c12a1,
(45)S0,−∗=−b1−b12−4a1c12a1.

The condition for the no-washout branch to be physically meaningful, eq. (15), can be written in the equivalent form

(46)S0∗>S0,cr∗=kd∗1−kd∗,0<kd∗<1.

We demonstrate that the solution S0,−∗ is not physically meaningful because

(47)S0,−∗<S0,cr∗.

We have

(48)S0,−∗=kd∗−kd∗1−kd∗−kd∗+kd∗kp∗,
(49)<kd∗1−kd∗,as 0<kd∗<1 and kp∗>0,
(50)=S0,cr∗.

It similarly follows that

(51)S0,+∗>S0,cr∗.

Thus the yield is maximised when the feed concentration is given by S0∗=S0,max∗=S0,+∗ where

(52)S0,+∗=kd∗+kd∗1−kd∗+1−kd∗kd∗kp∗.

C Symbols used

DSpecific decay rate.hr−1
FFlowrate through the bioreactor.dm3hr−1
GSingularity equation.(—)
JJacobian matrix.(—)
KsMonod constant.gdm−3
PProduct concentration within the bioreactor.gdm−3
P∗Dimensionless product concentration.(—)
P∗=αpP/αsKs
P∗0The scaled product concentration inside the reactor at time t∗=0.(—)
Pr∗Dimensionless product productivity.(—)
Pr∗=P∗τ∗
SSubstrate concentration within the bioreactor.gdm−3
S∗Dimensionless substrate concentration.(—)
S∗=S/Ks
S∗ˆThe dimensionless substrate concentration along the no-washout solution branch.(—)
S∗0The scaled substrate concentration inside the reactor at time t∗=0.(—)
S0Substrate concentration in the feed (S0>0).gdm−3
S0∗Dimensionless substrate concentration in the feed (S0∗>0).(—)
S0∗=S0/Ks
S0,cr∗The no-washout solution is only physically meaningful for S0∗>S0,cr∗.(—)
S0,cr∗=kd∗1−kd∗
S0,max∗The value of the feed concentration which maximises the yield.(—)
Defined by eq. (35).
VVolume of the bioreactor.dm3
XConcentration of microorganisms within the bioreactor.gdm−3
X∗Dimensionless microorganism concentration.(—)
X∗=X/αsKs
YYield.(—)
Y=PS0
Y∗Dimensionless yield.(—)
Y∗=P∗S0∗
kdDecay coefficient, representing a combination of endogenous respiration,hr−1
predation, and cell death followed by subsequent lysis kd∗>0.
kd∗Dimensionless decay coefficient (kd∗>0).(—)
kd∗=kd/μm
kpProduct toxicity constant.hr−1g−1dm3
kp∗Dimensionless product toxicity constant.(—)
kp∗=kpμm⋅αsKsαp
tTime.hr−1
t∗Dimensionless time.(—)
t∗=μmt
αpProduct yield factor, the ratio of the weight of product produced to the(—)
weight of substrate consumed.
αsSubstrate yield factor, the ratio of the weight of product produced to the(–)
weight of substrate consumed.
βReactor parameter model.(—)
μSSpecific growth rate model.hr−1
μmMaximum specific growth rate.hr−1
τresidence time (τ>0).hr
τ=V/F
τ∗Dimensionless residence time (τ∗>0).(—)
τ∗=Vμm/F
τcr*The value of the dimensionless residence time at the transcritical bifurcation.(—)

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Received: 2017-09-12
Revised: 2017-11-07
Accepted: 2017-11-08
Published Online: 2017-12-16

© 2018 Walter de Gruyter GmbH, Berlin/Boston

From the journal
Chemical Product and Process Modeling
Chemical Product and Process Modeling
Volume 13 Issue 3

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