Appendix 1
The model used within the topology optimization routine is presented below. It was
implemented and solved using the commercial software COMSOL. A list of parameters and
corresponding values is given in Table A1.
The total immobilized biomass ( Xim ) is the actuator variable. It is defined based on a design
parameter,  , that will be varied in the routine iterations. (1-  ) represents the fraction of

carrier used by the immobilized
biomass (Eq. A1).
max
Xim  (1  )Xim

(A1)
The plasmid bearing immobilized biomass, i.e., the fraction of immobilized biomass that
possesses the genetic information necessary to the production of the desired protein, can be
determined by multiplying the total immobilized biomass by the plasmid loss factor p, thus
implying a constant factor of plasmid loss (Eq. A2).

Xim
 (1 p)Xim

(A2)
The transport of the suspended (free) biomass, glucose and ethanol is carried out by fluid
convection and diffusion, and is mathematically described by the Equations (A3) to (A6),
X imact
where act
accounts for the fact that not all immobilized cells are actively growing.
X im  X im
 X imact

X im Pc
uX f  (1  2  3 ) act
 X f  DX f 2 X f
X im  X im Vr

(A3)

 X act

X P
uX f  ( 1  2  3 ) act im  im c  X f  DX  2 X f
f
X im  X im Vr

(A4)

 

  X im
X im Pc
uG   F1  O2  act act
 X f  DG  2G
YX /G YX /G X im  X im Vr

(A5)



  X act
X im Pc
uE  YE / X 1  3  act im
 X f  DE 2 E
YX / E X im  X im Vr


(A6)

The growth rates for each of the pathways (Eq. A7 - A9) consist of modified Monod
equations where Ki ' and ki ' are saturation and regulation constants which have been
recalculated in order to obtain explicitly defined functions which are mathematically
equivalent to the implicitly defined expressions proposed in the work of Zhang et al. (1997).
ki ' in Eq. (A7) and (A8) have no physical meaning. The term (1  tanh(G)) in Eq. (A9)
operates as switch function that tends to 0 when the glucose (G) concentration is high and to

1 when G approaches zero.
G 1 ka'G
K1'  G kb'  ka'G
(A7)
2  2,max
G
1 kc'G
K 2'  G 1 k c' k d 'G
(A8)
3  3,max
E
1 tanh(G)
K3  E
(A9)
1  1,max



In a topology optimization routine, the optimization function is, by default, minimized. As in
the case of the presented work, it is desired that the protein product rate (Eq. A10) is
maximized rather than minimized, a negative signal is added when writing the optimization
function Ф (Eq. A11)
X  P

R p  ( 2 2   3 3 ) im c  X f 
 Vr

(A10)
( )     Rp (t)dV
(A11)


Table A1 – List of model parameters
Parameter Description
act
X im
Fraction of active immobilized cells (Brányik et al.,
Value
Unit
0.62
gXim.gC-1
g.L-1
2004)

Pc/Vr
Carrier mass/reactor volume (Brányik et al., 2004)
13.6
p
Plasmid loss factor (Zhang et al., 1997)
0.05
YFX/G
Yield coefficient of biomass on glucose for glucose
0.12
gX.gG-1
0.48
gX.gG-1
3.35
gE.gX-1
0.65
gX.gE-1
32.97
units.gX-1
fermentation (Zhang et al., 1997)
YOX/G
Yield coefficient of biomass on glucose for glucose
oxidation (Zhang et al., 1997)
YE/X
Yield coefficient of ethanol on biomass for glucose
fermentation (Zhang et al., 1997)
YX/E
Yield coefficient of biomass on ethanol for ethanol
oxidation (Zhang et al., 1997)
α2
Protein yield coefficient for glucose oxidation (Zhang et
al., 1997)
α3
Protein yield coefficient for ethanol oxidation (Zhang et
33.80
units.gX-1
0.38
l.h-1
0.25
l.h-1
0.10
l.h-1
al., 1997)
μmax,1
Maximum specific growth rate for glucose fermentation
(Zhang et al., 1997)
μmax,2
Maximum specific growth rate for glucose oxidation
(Zhang et al., 1997)
μmax,3
Maximum specific growth rate for ethanol oxidation
(Zhang et al., 1997)
K1’
Saturation constant
1.8.10-1
g.L-1
K2’
Saturation constant
10-2
g.L-1
ka’
Enzyme pool regulation constant
- 4.0.10-3
-
kb’
Enzyme pool regulation constant
2.3
-
kc’
Enzyme pool regulation constant
20
-
kd’
Enzyme pool regulation constant
2.9
-
Appendix 2
Table A2 - Simulation Parameters
Parameter
Description
Value
Unit
dp
Pressure drop
0.1
Pa
DXf (+)
Diffusion coefficient
1.10-10
m2.s-1
DG
Diffusion coefficient for glucose
1.10-9
m2.s-1
DE
Diffusion coefficient for ethanol
1.10-9
m2.s-1
DP
Diffusion coefficient for the product
1.10-9
m2.s-1
l
Reactor length
1.2e-2
m
w
Reactor width
1.2e-2
m
h
Reactor height
1e-3
m
η
Fluid viscosity
1e-3
Pa.s
Ginlet
Glucose feed concentration
1e-3 – 1
g.L-1
The diffusion coefficient for the suspended biomass (cells) can be estimated by the
Einstein-relation for diffusion, D  kB T /(6a), where a is the radius of the cells
and kB is Boltzmann’s constant. This estimates a coefficient of 4.4·10-14 m2s-1(a = 5
 very small diffusion coefficients may result in very steep
μm, T = 300K). However,
concentration gradients, and for the numerical optimization method to resolve these
gradients, a lot of computer memory is needed. As a consequence, we have had to
increase the diffusion coefficient DXf(+) to 10-10 m2s-1 and DE=DG=DP=10-9 m2s-1
for the optimization to be doable.