Modelling and optimal control of an electro-fermentation process within a batch culture

: The electro-fermentation is a novel process that consists in coupling a microbial fer- 1 mentative metabolism with an electrochemical system. In such a process the electrodes act either 2 as electron sinks or sources modifying the fermentation balance of a microbial fermentative 3 metabolism and provide new options for the control of microbial activity. A theoretical framework 4 for the analysis and control of fermentations using electro-fermentation is currently lacking. In 5 this paper, we propose a simple electro-fermentation model in which a population of fermentative 6 bacteria switch between two metabolic behaviors in response to different electrode potentials. We 7 then mathematically analyze optimal strategies to maximize the production of one of the rising 8 products in a batch fermentation using Pontryagin’s Maximum Principle. The obtained results 9 show that, in some experimental conﬁgurations, a dynamic control of the electrode potential is 10 required for the maximization of the desired product. Consequences of the obtained optimal strat- 11 egy for driving electro-fermentation experiments are discussed through a realistic example. This 12 analysis also highlights that the transition rates between fermentation and electro-fermentation 13 behaviors are currently unknown and would be crucial to quantify in order to apply such a control 14 approach. 15


Introduction
The knowledge of electroactive microorganisms has rapidly grown over the last 20 19 years with the development of microbial electrochemical technologies such as microbial 20 fuel cells (MFC) and microbial electrolysis cells (MEC) for the production of electricity or 21 hydrogen from organic matter [1]. The development of electromicrobiology has also led 22 to several promising applications such as electro-fermentation (EF) in which reducing 23 power provided by means of an electrode can redirect fermentation pathways [2]. In that 24 case, the electrodes act as either electron sinks or sources modifying the fermentation 25 balance of a microbial fermentative metabolism and providing new options for the 26 control of microbial communities [3,4]. EF has been successfully applied on mixed culture 27 fermentations and pure culture fermentations and has proven efficient for increasing 28 yields in various products such as hydrogen, acetate, propionate, butyrate, lactate, 29 3-hydroxypropanoic acid, ethanol, 1.3-propanediol, 2,3-butandiol, butanol or acetone [5-30 16]. For example, a metabolic shift occurred in Clostridium pasteurianum when taking up 31 electrons from an electrode poised at +0.045 V vs. SHE (Standard Hydrogen Electrode) 32 with an increased production of reduced products such as butanol from glucose and 33 1.3-propanediol from glycerol [12]. From a biotechnological point of view, EF could lead 34 to significant improvements of industrial fermentations using only a small amount of 35 electrical power. Moreover, the use of an electrode for the triggering of the EF effect in the 36 fermentation system introduce the possibility of a dynamic control of the fermentation. 37 However, mathematical models describing the EF effect are currently lacking. In EF, 38 the voltage applied between electrodes is a variable that could be adjusted dynamically, 39 typically with the objective of maximizing the total production over a time interval. 40 We shall state and study such an optimization problem with the help of the theory of 41 optimal control, for which several analytical and numerical tools are available. Optimal 42 control has been successfully applied to various bioprocesses, providing practical control 43 strategies, mostly in terms of feedback control (e.g. [17]), as we shall consider here. culture are simply used to establish the model [18]. Based on this model, an optimal 65 control problem for the maximization of the production of s 2 is formulated. The Pon-66 tryagin's Maximum Principle [17,19] is applied for the design of the optimal control 67 strategy. The obtained results show that the optimal strategy is not trivial, in the sens 68 that the control is not always constant (equal to that which correspond to f 2 ), where in 69 some cases the metabolic behavior f 1 should be visited by the fermentative bacteria. The 70 present mathematical analysis of the optimal strategies will serve as an experimental 71 guide to design relevant experiments for testing if these strategies are eventually better, 72 and thus contribute to the validation of the model.

73
The paper is organized as follows. In Section 2, we give the electro-fermentation   81 In order to describe the switching between the two metabolic pathways described 82 in the introduction, we suppose that the fermentative population is splitted into two 83 sub-populations x 1 and x 2 in a commensal relationship to consume a substrate s. The 84 sub-population x 1 with microbial growth rate µ 1 gives rise to a product s 1 and the sub-85 population x 2 with microbial growth rate µ 2 gives rise to a product s 2 . We suppose that in 86 the absence of polarized electrodes the fermentation is mainly guided by the population 87 x 1 and when the external voltage is sufficiently large the metabolic function switches to 88 a metabolism guided by x 2 . This electro-fermentation process can be described by the 89 following system of ordinary differential equations:

Model description and optimization problem
where Y 1 , Y 2 are the yields coefficients, r 1 , r 2 > 0 are positive constants and α ∈ {0, 1} is 91 a control variable which is directly related to the external potential V and satisfies the 92 following property: where V 0 > 0 is a threshold on the external potential over which the metabolic pathway 94 is guided by x 2 . The value of the threshold potential V 0 depends on the microorganisms 95 x 1 and x 2 . We shall assume that growths do not present inhibition, which amounts to 96 consider the following hypothesis.
for some positive real numbers k i , K i , for i = 1, 2.

102
Observe that, due to the migration phenomenon between the two sub-populations, 103 the relation between x 1 and x 2 is not simply reduced to a competition phenomenon.

105
We shall assume that the two sub-populations have the same conversion factor.
Note that at the price of change of units of x 1 and x 2 , one can without loss of 108 generality assume that Y = 1; this is conventional when dealing with chemostat type 109 systems [20]. Therefore, we shall consider the simpler model Therefore, we can consider 112 the reduced dynamics in the plane The objective is to maximize the total production of the sub-population x 2 over an where T > 0 is a fixed finite time horizon. (2), the control variable α is constrained to take values 118 in the non-convex set {0, 1}. In this case, one can not a priori guarantee the existence of optimal 119 solutions for the problem (5)- (6). However, a technical approach consists into first considering 120 the convexified problem, i.e., solve the problem with α taking values in the whole interval [0, 1], 121 for which the existence of solutions is guaranteed (see for instance [21]). Then, the optimal 122 solution can be approached with an arbitrary precision via chattering controls [22], which consist 123 in commuting rapidly between the values 0 and 1 so that the averaged dynamics behave close 124 from that one with α different from 0 and 1.

125
We shall use the Maximum Principle of Pontryagin (PMP) [19] to obtain necessary 126 optimality conditions. Defining the Hamiltonian where p = (p 1 , p 2 ) is the adjoint vector, the Pontryagin Maximum Principle claims that 128 for any optimal solution, there exists an absolutely continuous function p : solution of the adjoint equation with the terminal condition 131 p(T) = 0 (9) and an optimal control α that satisfy the maximization of the Hamiltonian In addition, the map t →H( be the switching function. From the maximization of the Hamiltonian, one gets the 134 property for an optimal control 3. Sufficient condition for the absence of singular arc 136 In this section we give a sufficient condition under which any optimal solution 137 of problem (5)-(6) does not admit singular arc. Recall that a singular arc is a piece of 138 an optimal trajectory for which the switching function φ given by (11) vanishes in an Suppose that the growth rate functions satisfy the following assumption.
is valid for relatively small values of s.

152
One has the following proposition.
Therefore, p 1 , p 2 have to be constant on this time interval, given by Then,ṗ 1 =ṗ 2 = 0 gives the equations from which we obtain that p 1 µ 1 (s) = (p 2 + 1)µ 2 (s). Replacing p 1 , p 2 by their expression 160 and using Assumption 3 we obtain that a = 1 and thus a contradiction.

161
Thanks to Proposition 1, the optimal control of problem (5)-(6) is a sequence of 162 commutation between α = 0 and α = 1.  164 We investigate here the simple case for which one can assume that the two sub-

Optimal synthesis with identical growth functions
whatever is the control α(·). Then, the variable Note that at any time t,Ẋ 2 is maximal for α = 1, whatever is X 2 ∈ [0, 1]. LetX 2 (·) be the 174 solution for the control α identically equal to 1. From standard results of comparison of 175 solutions of scalar ordinary equations, one has X 2 (t) ≤X 2 (t) at any t, whatever is the 176 control α(·). Therefore, one has and we conclude that the constant control α = 1 is optimal.
178 Proposition 2 shows that, in the case of identical growth functions, the optimal 179 strategy in order to maximize (6) is by keeping α constantly equal to one, i.e., by keeping 180 an external potential sufficiently larger than V 0 . 182 We consider here distinct growth rates, but that we assume to be constant for

188
The following definition will be useful in the following.

193
On has the following result.
Note that system (18) is decoupled from the dynamics of x 1 , x 2 and can be studied 201 independently to the initial condition (x 1 (0), x 2 (0)). The switching function satisfies At  (20) Note that p 2 (T) = 0 implies that one has necessarily p 2 > 0 on [t 1 , T).

224
Finally, let us consider the dynamics (20) in the backward time τ = T − t: Note that the map ϕ : τ → p 2 (τ) defines a diffeomorphism from R + to R + . The solution 226 φ can then be parameterized by p 2 , as solution of the non-autonomous scalar differential 227 equation 228 dφ dp 2 where L and K are given by Definition 1. The solution of (23) can be made explicit as 229 given by (16). As it is underlined by Remark 2, when L > 1 (that is when a 1 > a 2 ) the 230 numberτ is well-defined. Then, we have that φ(τ) = 0 with φ(τ) > 0 for τ ∈ (0,τ).

231
Therefore, one getst 1 = max(0, T −τ) and the control 232 233 Remark 3. Let us underline that when µ i are constant functions, the optimal synthesis does 234 not depend on the initial state. Also, it is worth noting that in this case the optimal control α 235 does not depend on r 2 , the migration rate constant from population x 2 to population x 1 . This is 236 clearly recognizable from the statement of Proposition 3 and Definition 1.   (17) is given byτ = log(4) and the optimal control is given by As underlined in Remark 2, the optimal control α does not depend on the migration 244 rate constant r 2 . On Figure 2, we plot (with Matlab) the optimal control given by 245 Proposition 3 together with the optimal trajectories in the two cases r 2 = 0.1 (bottom-left) 246 and r 2 = 0.5 (bottom-right). As a verification, these plots are also compared with the 247 ones obtained with Bocop [23], which is a numerical optimization software dedicated to 248 optimal control problems (using direct method) on Figure 2 (top).  Using the numerical solver Bocop, we simulate the optimal control together with 254 the optimal trajectories in similar conditions than before with model parameters given 255 in Table 2, and for different values of initial control c = s(0) + x 1 (0) + x 2 (0). Observe 256 that, in this example, the growth kinetic functions µ 1 and µ 2 satisfy Assumption 3. Thus, thanks to Proposition 1, no singular arcs are present in this case. We observe in Figure 4 258 that the optimal trajectories have the same structure than for constant growth rates given 259 by Proposition 3, but here the optimal switching time depends on the initial condition.  control in the case of proportional growth functions is given. An optimal control is syn-296 thesised in two particular cases: similar and constant (but different) growth rate kinetics.

297
The obtained results show that the optimal control strategy is far from being trivial, in the 298 sense that undesirable metabolic pathways may be visited by the fermentative bacteria 299 for the maximization of a desired fermentation product. Consequences of the obtained 300 optimal strategy for driving electro-fermentation experiments are discussed through a 301 realistic example. This study is elaborated under the hypothesis of identical yield factors.

302
The optimality for the case of different yield factors with more general growth kinetics, 303 is an open problem, that will be explored in a near future. This theoretical approach 304 also underscores the importance of evaluating biological parameters such as transition 305 rates between fermentation mode and electro-fermentation mode for the application of 306 the optimal control. The proposed model as well as the optimal control law need to be 307 confronted with experimental data, and this will be the subject of future experimental 308 investigation within the LBE laboratory.