Escuela de Ingeniería Pontificia Universidad Católica de Chile Casilla 306,
Escuela de Ingeniería Pontificia Universidad Católica de Chile Casilla 306, Fax: 562-354-5803 Email: perez@ing.puc.cl
E-mail: lb01@andrew.cmu.edu
Escuela de Ingeniería Pontificia Universidad Católica de Chile Casilla 306, Email: jorquera@ing.puc.cl Webpage: http://www.ing.puc.cl
Solid substrate fermentation (SSF) can be defined as the cultivation of microorganisms on solid substrates devoid of or deficient in free water (Pandey, 2003). SSF has several advantages (Hölker and Lenz, 2005) over more conventional submerged fermentation, and many promising lab-scale SSF processes are periodically reported in the literature (John et al. 2006; Krasniewski et al. 2006; Lechner and Papinutti, 2006; Sabu et al. 2006). Unfortunately, very few of these processes enter commercial production (Hölker and Lenz, 2005) due to the magnitude of the technical difficulties in operating and optimizing large scale SSF bioreactors. Since modern process control and optimization engineering techniques are model based, mathematical modelling should significantly improve the chances of successfully transforming an SSF process from laboratory to commercial production. Nevertheless, a number of factors make modelling SSF processes particularly trying: the absence of reliable on-line measurements of relevant cultivation variables (like biomass and nutrient concentration) and the system's inherent complexity, considering that microorganism interaction with the environment and its growth and production kinetics are still not well understood on a micro scale (Mitchell et al. 2004). In addition, the more useful mechanistic dynamic models proposed are highly complex and have many parameters that need to be estimated from extensive good quality experimental data. Acquiring such data is costly and time consuming and yet, even when this data is available, attaining reliable parameter estimates is far from trivial (Gelmi et al. 2002). Therefore, in most current SSF lab-scale studies that include modelling, only simple black box or empirical kinetics models are used (Machado et al. 2004; Corona et al. 2005; Jian et al. 2005). However, these models can only reproduce process behaviour encountered in controlled conditions that are never found in large scale SSF bioreactors and, as such, more often than not commercial production yields are disappointingly low compared to lab-scale performance. Parameters in dynamic fermentation models are commonly estimated using the sequential solution/optimization (SeqSO) procedure (Rivera et al. 2006). Though simple, this procedure may be severely limited when fitting complex models with many parameters or constraints that violate standard numerical regularity conditions, as is the case of more mechanistic SSF kinetic models. A high degree of heuristics is therefore required to overcome the method's slow convergence and unreliable estimation (Gelmi et al. 2002). Alternatively, the simultaneous solution/optimization (SimSO) approach (Biegler et al. 2002) is fast, robust and reliable, and have shown its suitability for fitting a variety of complex dynamic models. In this work a SimSO procedure is developed to estimate model parameters in an SSF kinetic model and the results obtained are compared, in terms of fit quality and numerical performance, with those obtained with the commonly used SeqSO approach. The SimSO procedure developed was coded in AMPL and the resulting non-linear program (NLP) was solved using IPOPT (Biegler et al. 2002), a robust interior point NLP solver specially designed for large scale optimization problems. First, the model is described in brief and calibration details are provided. Then, results are shown and discussed, and finally the main conclusions of this work are presented. We
have used a slightly modified version of the lumped parameter model
proposed in (Gelmi et al. 2002) to describe cultivation
in glass columns of .
Oxygen mass transfer resistance is negligible. Next, we present a brief description of the model. The
total amount of measurable dry biomass ( kg),_{d}._{b}.[1] Assuming
a first order death rate, the active biomass ( [2] Here,
The
model assumes that urea, Gibberella
fujikuroi uses this nutrient for biomass growth,[3] As this equation does not satisfy standard numerical regularity conditions we replaced equation (3) with the smooth approximation, [4] which is also used in the nitrogen balance below (5); ε is a small number. The concentration of assimilable nitrogen is given by, [5] In
these equations The microorganism consumes starch for growth and maintenance, [6] The
differential equations for CO [7] [8] where
Yare the mass yield
coefficients between biomass and respiratory gases._{X/O2} GA [9] The
specific growth rate, [10] Here,
k is the substrate inhibition constant. A substrate
inhibition expression describes the specific GA_{N}_{3} production
rate,[11] where
_{3} production
rate and k is the associated substrate inhibition
constant._{i}The above model was calibrated in four culture conditions, i)
Temperature = Further details regarding the experimental set up and the above model are available elsewhere (Gelmi et al. 2000; Gelmi et al. 2002). We
have applied the simultaneous (SimSO) approach to solve the parameter
estimation problem. The set of differential equations, A differential variable is approximated as a polynomial within a finite element, on a monomial basis (Rice and Duong, 1995). [12] where
HD(i) is the length of element
i, dy/dt is the value of the derivative in
element i at the collocation point _{q,i}q, ncol is the
number of collocation points _{} and is a polynomial of
degree ncol, satisfying,[13] Where
[14] and
ncol, i.e., and We have used 69 finite elements and Radau points with two internal collocation points per finite element, since this configuration achieves a good compromise between precision and efficiency and it is easy to add constraints at the end of each finite element (Rice and Duong, 1995). Integrating
equation [15] with Radau points ( T = 0.6449948; _{2}T = 1), and _{3}t
= _{i,q}t + _{i-1}HD(i)·T, leads to values
for the polynomial coefficients, W_{q}(t._{i,q})[16] with
The system [15] approximates the differential variable over the respective finite element. To solve the differential equations over the entire time domain, we expand these equations for each finite element and collocation point. This large set of nonlinear algebraic equations represents a high-order implicit Runge-Kutta (IRK) approximation to the differential equations. The model parameters were estimated by weighted least squares,
Subject to, [18] where
Δt, _{i}ŷ and _{i}y
are the number of measured values, the sampling interval, the solution
of the differential equation and the normalization value for variable
y, respectively. The vector of estimated parameters
is represented by _{i}_{}.Because
parameter k
in equation [9] are correlated, since many combinations of these parameter
values achieved the same data fit. Thus, we obtained the values of
_{N}m from curves of the accumulated respiratory gases
(Saucedo-Castañeda et al. 1994), and estimated _{M}k
using the least squares procedure described above. The values obtained
of both rates (_{N}k and m) for all cultivation
conditions are given in Table 1._{M}The
vector of estimated parameters, therefore, through least squares using
equations 17 and 18 is θ = ( Results obtained with the procedure described above were compared with those obtained with the SeqSO approach, as described in Gelmi et al. (2002). Once
the optimization is carried out, the solution of (17, 18) can be formally
written as ŷ [19] We have found that using ∆θ = 0.001 θ is sufficient to obtain accurate estimates of the Jacobian. Now,
since the number (n) of experimental points is large enough (O(10 [20] Where
θ* is the true parameter vector, We compare our calibration results for each cultivation condition with the fit obtained using the SeqSO approach. We were specifically interested in fit quality, the performance of the optimization procedure and the value and accuracy of the estimated parameters.
In
Table 2 we can see that the SimSO estimation
of the GA _{3} degradation rate constant,
k, was unreliable (t value is almost zero). Therefore,
it is pointless to compare the two methods' estimation of these parameters.
Other parameter estimates were reliable (t values above 2) and accurate
(relatively small σ). We don't have the estimation of σ
for the parameters obtained with the SeqSO calibration method. Therefore,
if we assume that both methods yield the same σ it is reasonable,
as a first approximation, to consider that both parameter estimates
are similar when their difference is smaller than 3 times the standard
deviation computed for the SimSO estimate (Wild and
Seber, 2000). Then, in this cultivation condition four parameter
estimates differed significantly using the two fitting methods,
the Monod's constant in equation [10], _{p}k, the biomass/oxygen
yield coefficient, _{N}Y, the biomass/nitrogen yield
coefficient, _{X/O2}Y, and the maximum specific GA_{X/N}_{3}
production rate, β._{M}We should expect therefore model simulations with both parameter sets to be similar. This supposition is supported by a marginal improvement in the objective function (equation 17) (see Table 3). Hence, the SimSO method shows only a slightly better fit for the respiratory gases (Figure 2) and the other model variables are almost indistinguishable for the two methods (not shown). For the particular conditions of cultivation 1, the main advantage of the SimSO calibration procedure is its efficiency and robustness. We started from two different guesses and achieved the same estimation parameters in less than 30 CPU s on an Athlon XP 2K PC, running Windows XP. Here, deviation errors in equation [17] were normalized by the maximum measured value.
Under
these conditions there was an unusual delay of 20 hrs in microorganism
growth, which the model does not consider. Hence, calibration only
included the data after 20 hrs. Estimates for k were not significant (t value below 2); most
of the other parameter estimates were very significant (t above 10),
as shown in Table 4. The SimSO fit presented
estimated parameter values different from the SeqSO method, except
for the death rate constant, _{P}K, the biomass/nitrogen
yield coefficient, _{D}Y, and the oxygen maintenance
coefficient, _{X/N}m._{O2}Despite
the differences in parameter values observed in Table
4, only oxygen consumption and GA Here, deviation errors in the objective function were also normalized by the maximum measured value. Since starting from two different guesses produced the same set of estimates in less than 30 sec, just like in the fitting of condition 1, the SimSO calibration procedure for this data was efficient and robust.
Table
5 shows that under these conditions, the k estimate was significant
and more accurate; at least here, contrary to cultivations 1 and 2,
σ for this parameter is smaller than the estimate values allowing
us to compare both methods. Again, the rest of the parameter estimates
were significant and accurate. Table 5 shows
that both methods yielded different estimates for all model parameters,
except for the death rate constant, _{i}K. Moreover,
a much better fit was achieved with SimSO parameter values for biomass,
GA_{D}_{3} and starch (Figure 4) that, in
turn, is reflected in the sharp reduction in the objective function
(Table 3). Here, again, deviation errors were
normalized by the maximum measured value and optimization was started
from two different initial guesses. Convergence was more difficult
than for the previous cultivation conditions, requiring more iterations
and CPU time (Table 3).
Here, k estimates were not significant, while the
remaining SimSO parameter estimates were accurate and significant
(Table 6). Except for _{P}m,
_{CO2}Y and _{X/CO2}Y, SimSO parameter
estimates did not stray much from the SeqSO estimates. Nevertheless,
the SimSO procedure achieved a better fit for biomass and GA_{X/N}_{3}
(Figure 5), and a significantly lower objective
function value (Table 3). Convergence for these
conditions was the most difficult of the 4 cases, requiring a 5-fold
increase in CPU time than in cultivation conditions 1 and 2 (Table
3) and twice as much as cultivation 3. We had to normalize by
the average measured value in equation [17] to get convergence here,
too.In
a comparison of cultivation conditions 1 and 4 only the death rate
was unaffected by temperature (Table 2 and β and the maintenance coefficients increased
with temperature, while the yield coefficients and _{M}k
decreased with temperature (Table 1, Table
2 and Table 6). The net result was that
at _{N}_{3} was produced at _{2} and consumption of O_{2} was
a bit higher (results not shown).Comparing
cultivation conditions 1 and 2, we observe that, except for μ and _{M}β
were higher for _{M}a = 0.992, while _{w}k
and _{N}K were lower (Table 1,
Table 2_{D} and Table 4).
The net result was that for a =
0.992 around 30% more biomass and twice as much GA_{w}_{3} were
produced (results not shown). Comparing cultivations 3 and 4, on the
other hand, we observe that for a = 0.985 the death
rate was unaffected, _{w}k, _{N}β
and maintenance coefficients were higher, while the yield coefficients
and μ_{M} were lower (Table
1, Table 5 and Table 6).
Therefore, for_{M} a = 0.985 less biomass and a little
more GA_{w}_{3} were obtained (results not shown).Overall,
estimation of GA _{3} degradation
in these experiments was most probably negligible. Moreover, estimations
of k were unreliable or inaccurate in almost all
instances. GA_{i}_{3} is a secondary metabolite that Gibberella
starts producing when available nitrogen in the medium is almost exhausted.
Therefore, many measurements close to the point of nitrogen exhaustion
are required to obtain accurate estimations of k.
In the range studied, we also verified that_{i} β
increases with temperature yet it decreases with water activity. Regarding
variations of growth kinetic parameters, we found the death rate was
unaffected by temperature and reached a maximum for _{M}a
= 0.999. In addition, _{w}μ increased with temperature
and water activity, while _{M}k decreased with temperature
and it appeared to reach a minimum at _{N}a = 0.992.
Maintenance coefficients increased with temperature and took their
highest values at _{w}a = 0.985. In turn, yield coefficients
decreased with temperature and appeared to reach their maximum at
_{w}a = 0.992._{w}The
importance of this work is the significant reduction in convergence
time the SimSO approach achieved. As a consequence it drastically
simplified the parameter estimation problem. A typical calibration
with the SeqSO approach, as described in Gelmi et al.
(2002), required many runs of the optimization program, each taking
several hours to converge and requiring a high degree of heuristics
and, in all, the entire SeqSO procedure took over a week to complete.
Another important consideration is that for most conditions the SimSO
calibration strategy achieved a significantly better fit for biomass,
GA The authors thank Alex Crawford for his assistance in improving the style of the text. BIEGLER,
Lorenz T.; CERVANTES, Arturo M. and WÄCHTER, Andreas. Advances in
simultaneous strategies for dynamic process optimization.
HÖLKER,
Udo and LENZ, Jürgen. Solid-state fermentation - are there any biotechnological
advantages? JIAN,
Xu; SHOUWEN, Chen and ZINIU, Yu. Optimization of process parameters
for poly γ glutamate production under solid state fermentation
from JOHN,
Rojan P.; NAMPOOTHIRI, K. Madhavan and PANDEY, Ashok. Solid-state
fermentation for L-lactic acid production from agro wastes using
KRASNIEWSKI,
Isabelle; MOLIMARD, Pascal; FERON, Gilles; VERGOIGNAN, Catherine;
DURAND, Alain; CAVIN, Jean-François and COTTON, Pascale. Impact
of solid medium composition on the conidiation in LECHNER,
B.E. and PAPINUTTI, V.L. Production of lignocellulosic enzymes during
growth and fruiting of the edible fungus MACHADO,
Cristina M.M.; OISHI, Bruno O.; PANDEY, Ashok and SOCCOL, Carlos
R. Kinetics of MITCHELL,
David A.; VON MEIEN, Oscar F.; KRIEGER, Nadia and DALSENTER, Farah
Diba H. A review of recent developments in modeling of microbial
growth kinetics and intraparticle phenomena in solid-state fermentation.
PANDEY,
Ashok. Solid-state fermentation. RICE,
Richard G. and DUONG, Do D. RIVERA,
Elmer Copa; COSTA, Aline C.; ATALA, Daniel I.P.; MAUGERI, Francisco;
WOLF MACIEL, Maria R. and MACIEL FILHO, Rubens. Evaluation of optimization
techniques for parameter estimation: Application to ethanol fermentation
considering the effect of temperature. SABU,
A.; AUGUR, C.; SWATI, C. and PANDEY, Ashok. Tannase production by
SAUCEDO-CASTAÑEDA,
G.; TREJO-HERNÁNDEZ, M.R.; LONSANE, B.K.; NAVARRO, J.M.; ROUSSOS,
S.; DUFOUR, D. and RAIMBAULT, M. On-line automated monitoring and
control systems for CO SEBER,
George A.F. and WILD, C.J. WILD,
C.J and SEBER, George A.F. |