specific growth of bakers yeast in a feed batch system

8/3/2019 Specific Growth of Bakers Yeast in a Feed Batch System

http://slidepdf.com/reader/full/specific-growth-of-bakers-yeast-in-a-feed-batch-system 1/52

3. Fed-Batch Fermentations

3.1. Fixed volume fed-batch

3.2. Variable volume fed-batch

3.3. Advantages and disadvantages of fed-batch culture

3.4. Equipment 3.4.1. Vessels

3.4.2. Pumps

3.5. Control techniques for fed-batch fermentations

3.6. Modelling of fed-batch reactors

3.6.1. Fixed volume fed-batch

3.6.2. Variable volume fed-batch

3.6.3. Models of possible situations that may occur in fed-batch fermentation

3.7. Parameters used to control fed-batch fermentations

3.7.1. Calorimetry

3.7.2. Specific growth rate

3.7.3. Substrate (carbon or nitrogen source) 3.7.4. By-product concentration

3.7.5. Inductive, enhancer or enrichment components

3.7.6. Respiratory quotient 3.7.7. General feeding mode

3.7.8. Proton production rate

3.7.9. Fluorescence

3.8. Parameters to start and finish the feed, and stop the fed-batchfermentation

3.9. Preliminary knowledge required to implement fed-batch

3.10. Algorithms for operating a fed-batch reactor at optimum specificgrowth rate (model independent and applicable to adapting systems)

3.11. Some examples of fed-batch use in industry

Two basic approaches to the fed-batch fermentation can be used: the constant volumefed-batch culture - Fixed Volume Fed-Batch - and the Variable Volume Fed-Batch. Thekinetics of the two types of fed-batch culture will be described in section 3.6.

http://userpages.umbc.edu/~gferre1/fedbatch.html#Fixed%20volume



http://userpages.umbc.edu/~gferre1/fedbatch.html#Variable%20volume



http://userpages.umbc.edu/~gferre1/fedbatch.html#Advantages%20and%20disadvantages%20of%20the%20fed-batch



http://userpages.umbc.edu/~gferre1/fedbatch.html#Equipment



http://userpages.umbc.edu/~gferre1/fedbatch.html#Control%20techniques%20for%20fed-batch






http://userpages.umbc.edu/~gferre1/fedbatch.html#Fixed%20volume%20fed-batch



http://userpages.umbc.edu/~gferre1/fedbatch.html#Variable%20volume%20fed-batch



http://userpages.umbc.edu/~gferre1/fedbatch.html#Models%20of%20possible%20situations%20that%20may%20occur%20in%20a%20fed-batch%20fer



http://userpages.umbc.edu/~gferre1/fedbatch.html#Parameters%20used%20to%20control%20fed-batch%20fermentations



http://userpages.umbc.edu/~gferre1/fedbatch.html#Calorimetry



http://userpages.umbc.edu/~gferre1/fedbatch.html#Specific%20growth%20rate



http://userpages.umbc.edu/~gferre1/fedbatch.html#Substrate%20%28carbon%20and%20nitrogen%20source%29



http://userpages.umbc.edu/~gferre1/fedbatch.html#By%20-product%20concentration



http://userpages.umbc.edu/~gferre1/fedbatch.html#Inductive,%20enhancer%20or%20enrichment%20components



http://userpages.umbc.edu/~gferre1/fedbatch.html#Respiratory%20quotient



http://userpages.umbc.edu/~gferre1/fedbatch.html#General%20feeding%20mode



http://userpages.umbc.edu/~gferre1/fedbatch.html#Proton%20production



http://userpages.umbc.edu/~gferre1/fedbatch.html#Fluorescence







http://userpages.umbc.edu/~gferre1/fedbatch.html#Preliminary%20knowledge%20required%20to%20implement%20fed-batch



http://userpages.umbc.edu/~gferre1/fedbatch.html#Algorithms%20for%20operating




http://userpages.umbc.edu/~gferre1/fedbatch.html#Some%20examples%20of%20fed-batch%20use%20in%20industry


































3.1. Fixed volume fed-batch

In this type of fed-batch, the limiting substrate is fed without diluting the culture.

The culture volume can also be maintained practically constant by feeding the growthlimiting substrate in undiluted form, for example, as a very concentrated liquid or gas(ex. oxygen).

Alternatively, the substrate can be added by dialysis or, in a photosynthetic culture,radiation can be the growth limiting factor without affecting the culture volume5.

A certain type of extended fed-batch - the cyclic fed-batch culture for fixed volume systems -refers to a periodic withdrawal of a portion of the culture and use of the residual culture

as the starting point for a further fed-batch process. Basically, once the fermentationreaches a certain stage, (for example, when aerobic conditions cannot be maintainedanymore) the culture is removed and the biomass is diluted to the original volume withsterile water or medium containing the feed substrate22. The dilution decreases thebiomass concentration and result in an increase in the specific growth rate (seemathematical description in section 3.6). Subsequently, as feeding continues, the growthrate will decline gradually as biomass increases and approaches the maximumsustainable in the vessel once more, at which point the culture may be diluted again26.

| Outline | Top of document |

3.2. Variable volume fed-batch

As the name implies, a variable volume fed-batch is one in which the volume changeswith the fermentation time due to the substrate feed. The way this volume changes it isdependent on the requirements, limitations and objectives of the operator.

The feed can be provided according to one of the following options:

(i) the same medium used in the batch mode is added;

(ii) a solution of the limiting substrate at the same concentration as that in the initialmedium is added; and

(iii) a very concentrated solution of the limiting substrate is added at a rate less than (i),(ii) and (iii) 21.

http://userpages.umbc.edu/~gferre1/references.html







http://www.gl.umbc.edu/~gferre1/outline.html



http://userpages.umbc.edu/~gferre1/fedbatch.html#Top%20of%20document









This type of fed-batch can still be further classified as repeated fed-batch process or cyclic fed-batch culture, and single fed-batch process.

The former means that once the fermentation reached a certain stage after which is noteffective anymore, a quantity of culture is removed from the vessel and replaced by

fresh nutrient medium. The decrease in volume results in an increase in the specificgrowth rate, followed by a gradual decrease as the quasi-steady state is established.

The latter type refers to a type of fed-batch in which supplementary growth medium isadded during the fermentation, but no culture is removed until the end of the batch.This system presents a disadvantage over the fixed volume fed-batch and the repeatedfed-batch process: much of the fermentor volume is not utilized until the end of thebatch and consequently, the duration of the batch is limited by the fermentor volume26.


3.3. Advantages and disadvantages of the fed-batchreactors

Fed-batch fermentation is a production technique in between batch and continuousfermentation12. A proper feed rate, with the right component constitution is required

during the process8.

Fed-batch offers many advantages over batch and continuous cultures. From theconcept of its implementation it can be easily concluded that under controllableconditions and with the required knowledge of the microorganism involved in thefermentation, the feed of the required components for growth and/or other substratesrequired for the production of the product can never be depleted and the nutritionalenvironment can be maintained approximately constant during the course of the batch.The production of by-products that are generally related to the presence of highconcentrations of substrate can also be avoided by limiting its quantity to the amounts

that are required solely for the production of the biochemical. When highconcentrations of substrate are present, the cells get "overloaded", this is, the oxidativecapacity of the cells is exceeded, and due to the Crabtree effect, products other than theone of interest are produced, reducing the efficacy of the carbon flux. Moreover, theseby-products prove to even "contaminate" the product of interest, such as ethanolproduction in baker's yeast production, and to impair the cell growth reducing thefermentation time and its related productivity.




















Sometimes, controlling the substrate is also important due to catabolic repression. Sincethis method usually permits the extension of the operating time, high cellconcentrations can be achieved and thereby, improved productivity [mass ofproduct/(volume.time)]. This aspect is greatly favored in the production of growth-associated products1.

Additionally, this method allows the replacement of water loss by evaporation anddecrease of the viscosity of the broth such as in the production of dextran and xanthangum13, by addition of a water-based feed.

As previously mentioned, fed-batch might be the only option for fermentations dealingwith toxic or low solubility substrates.

When dealing with recombinant strains, fed-batch mode can guarantee the presence ofan antibiotic throughout the course of the fermentation, with the intent of keeping the

presence of an antibiotic-marked plasmid. Since the growth can be regulated by thefeed, and knowing that in many cases a high growth rate can decrease the expression ofencoded products in recombinant products, the possibility of having different feeds andfeed modes makes fed-batch an extremely flexible tool for control in these cases7, 8.

Because the feed can also be multisubstrate, the fermentation environment can still beprovided with required protease inhibitors that might degrade the product of interest,metabolites and precursors that increase the productivity of the fermentation19.

Finally, in a fed-batch fermentation, no special piece of equipment is required inaddition to that one required by a batch fermentation, even considering the operating

procedures for sterilization and the preventing of contamination12.

A cyclic fed-batch culture has an additional advantage: the productive phase of aprocess may be extended under controlled conditions. The controlled periodic shifts ingrowth rate provide an opportunity to optimize product synthesis, particularly if theproduct of interest is a secondary metabolite whose maximum production takes placeduring the deceleration in growth22.

As a summary of what was described above, fed-batch mode of fermentation has thefollowing features:

Advantages:

production of high cell densities due to extension of working time (particularlyimportant in the production of growth-associated products)






















controlled conditions in the provision of substrates during the fermentation,particularly regarding the concentration of specific substrates as for ex. thecarbon source

control over the production of by-products or catabolite repression effects due to

limited provision of substrates solely required for product formation

the mode of operation can overcome and control deviations in the organism'sgrowth pattern1 as found in batch fermentation

allows the replacement of water loss by evaporation

alternative mode of operation for fermentations leading with toxic substrates(cells can only metabolize a certain quantity at a time) or low solubilitycompounds

increase of antibiotic-marked plasmid stability by providing the correspondentantibiotic during the time span of the fermentation

no additional special piece of equipment is required as compared with the batchfermentation mode of operation

Disadvantages:

it requires previous analysis of the microorganism, its requirements and theunderstanding of its physiology with the productivity

it requires a substantial amount of operator skill13 for the set-up, definition anddevelopment of the process

in a cyclic fed-batch culture, care should be taken in the design of the process toensure that toxins do not accumulate to inhibitory levels and that nutrients otherthan those incorporated into the feed medium become limiting, Also, if manycycles are run, the accumulation of non-producing or low-producing variantsmay result22.

the quantities of the components to control must be above the detection limits ofthe available measuring equipment25























3.4. Equipment

No special piece of equipment is required over the equipment required for batch13. However, some considerations should be made over the equipment used for a fed-batch

fermentation.

3.4.1. Vessels

The vessels, particularly those used for the acid and base control, must be constructedfrom a non-toxic, corrosion-resistant material which is capable of withstanding repeatedsterilization cycles13. Figure 3.4.1. Illustrates two methods of assembling vessels for easytransfer of either inoculum or medium to the fermentor.

Figure 3.4.1. Holding vessels. A. Screw-neck borosilicate glass vessel with medium/inoculum additionassembly. (a) Stainless steel rod; (b) Silicon tubing; (c) Silicon disc; (d) Hypodermic needle; (e) Air vent; (f)Screw cap; (g) Magnetic bar. B. Aspirator-type vessel for introducing an inoculum of filamentous fungiinto the fermentor. (a) Cotton-wool plug; (b) Magnetic stirrer bar13.














3.4.2. Pumps

There are two types of pumps which are suitable for the aseptic pumping of smallvolumes of culture media: the peristaltic pump and the diaphragm-dosing pump. Otherpumps are unsuitable because they are difficult to sterilize and cannot be used for

pumping small volumes13.

The peristaltic pump is typically constituted by a main body that comprises both thedrive motor and electrics, and the rotating unit of rollers. This unit of rollers occludesthe tube which, as it recovers to its original size passes to the nest roller until isexpelled, as the unit moves round. The flow rate can be varied by either the speedsetting or by changing the diameter of the tube being used.

The diaphragm-dosing pump consists of a main body and a detachable heat-sterilizablehead. The fluid is sucked in to the pump head. The suction inlet tube then closes and

the pressure discharge tube opens and forces the fluid out. The suction and pressureforces in the pump head are generated by the reciprocating action of both thediaphragm plunger and the return spring. For a more accurate description of thesepumps, reference 13 can be consulted.


3.5. Control techniques for fed-batch fermentation

Adaptive control is the name given to a control system in which the controller learnsabout the process by acquiring data from a certain process and keeps on updating acontrol model. A parameter estimator monitors the process and estimates the processdynamics in terms of the parameters of a previously defined mathematical model of theprocess. A control design algorithm is then used to generate controller coefficients fromthose estimates, and a controller sets up the required control signals to the devicescontrolling the process. An extremely important feature of an adaptive controller is thestructure of the model used by the parameter estimator to analyze estimates of processdynamics. The process can be described by a set of mass balance equations, whosequantities can be measured directly or indirectly26 . Figure 3.5.1. describes schematically

the concept.




















Figure 3.5.1. Adaptive control: the controller compares the estimates from a mathematical model applied to the

system to the readings obtained from the fermentation process. The controller then sends the signal to the device

controlling the fermentation, for example, by increasing or decreasing a flow rate.

The optimal strategy for the fed-batch fermentation of most organisms is to feed thegrowth-limiting substrate at the same rate that the organism utilizes the substrate, thisis, to match the feed rate with demand for the substrate.

Four basic approaches have been used in attempts to balance substrate feed withdemand (listed in order of increasing accuracy and/or complexity):

(i) open-loop control schemes in which feed is added according to historical data orpredicted data;

(ii) indirect control of substrate feed based on non-feed source parameters such as pH,offgas analysis, dissolved O2 or concentrations of organic products;

(iii) indirect control schemes based on mass balance equations, the values of which are

calculated from data obtained by sensors; and

(iv) direct control schemes based on direct, on-line measurements substrates9.

Better and more flexible control may be obtained when there is direct measurement ofsubstrate or an excreted metabolite in the medium, which can be used to influence







feeding rates to the fermentation. This can be done off-line or semi-on-line, but on-linemeasurements are more useful because of

the shorter analysis required, lower personnel requirement and

a reduced chance of fermentor contamination25.

Regardless the type of control, the design is strongly influenced by both mathematicalmodel availabilities and measurement possibilities14.

Control and optimization of bioreactors is strongly influenced by the quality of the sensors

available for crucial response variables4. Of primary importance is the ratio of the dynamic

parameters of the sensor to those of the process. When these variables cannot be measured easily

or quickly enough, a mathematical model must be used in some way in place of feedback

information.

When an exact mathematical model is at disposal, an open-loop process control can beproposed which generally proves to be insufficient14. The advantage of a feedbackcontrol is that a response to unforeseen and unexpected conditions during thefermentation is achieved and the process is controlled within the desired limits 29.

An indirect feedback control utilizes an observable parameter, such as dissolvedoxygen, pH, respiratory quotient, partial pressure of CO2, culture fluorescence or by-product formation, which is closely related to the course of microbial fermentation. Asexamples of fed-batch systems using this concept, one can mention the pH-stat - asystem in which the feed is provided depending on the pH, - and the DO-stat - a system

in which the feed is provided depending on the reading of the dissolved oxygen24, 29.

A direct feedback controller uses the concentration of limiting substrate in the culturemedium as a feedback feed -related parameter for control. A direct feedback control canhave the disadvantage of not being very feasible due to the difficulty associated withobtaining accurate on-line measurements of substrate concentrations or even by theabsence of on-line sensors for the important compound to control14. The advantage of afeedback control is that a response to unforeseen and unexpected conditions during thefermentation is achieved and the process is controlled within the desired limits 29.

A feedback control can be implemented accordingly to not only a single measurement,but also to obtain a finer control action in a dual-level system. Turner at al. 25, describesa control method applied to a fed-batch culture of recombinant Escherichia coli in whicha two-level control was preferred because it provided much greater flexibility andbetter control over the substrate concentration in the medium and the production of by-products 25, 29.








































As compared with the batch fermentation, two more parameters need to be specified todetermine the operating conditions of a fed-batch fermentation: feed and initial feedingtime. These parameters are usually process and/or microorganism specific and the

parameters commonly used to define them are explained in section 3.7.


3.6. Modeling fed-batch fermentations

3.6.1. Fixed volume fed-batch

The mathematical development that is going to be presented here has the followingassumptions22:

o The feed is provided at a constant rate

o The production of mass of biomass per mass of substrate is constantduring the fermentation time and

o A very concentrated feed is being provided to the fermentor in such a waythat the change in volume is negligible.

The equations that describe the system in terms of specific growth rate, biomass and product

concentration (for both growth and non-growth associated products) with time are the following:

Table 3.6.1.1. Mathematical modelling of fixed volume fed-batch.

Parameter Equation Equation #

Specific Growth Rate u = (F . Yx/s) / x (3.6.1.1)

Biomass (as a function of time) xt = xo + F . Y x/s . t (3.6.1.2)

Product Concentration(non-growth associated) P= Pi + qp . xo . t + qp . F . Y x/s . t2 /2 (3.6.1.3)

Product Concentration(non-growth associated)

P= Pi + rp . t (3.6.1.4)

x is the biomass [mass biomass/volume]














xo is the biomass in the beginning of the fermentation [mass biomass/volume]

t is time

F is the substrate feed rate [mass substrate/(volume.time)] and

Y x/s is the yield factor [mass biomass/mass substrate]

u is the specific growth rate [time-1]

P is the product concentration {mass product/volume] and

qp is the specific production rate of product [mass product/(mass biomass . time)

rpis the product formation rate [mass product/(volume . time)]

From equations (3.6.1.1) and (3.6.1.2), it can be observed that

(i) the specific growth rate decreases with time because the biomass (in the denominator) is

increasing with time and

(ii) the biomass increases linearly with time.

The product variation with time will depend on its being growth or non-growth associated, thisis, will depend on qp (specific product formation defined as the product formation rate divided by

the biomass) being dependent on the specific growth rate or not, respectively.

To obtain the derivations that yielded these equations, please click here.

Figure 3.6.1.1. depicts the typical behavior of a fixed-volume fed-batch culture.

Figure 3.6.1.1. Time profiles for a fixed-volume fed-batch culture. u= specific growth rate, x = biomassconcentration, S(GLS) = growth limiting substrate, SN = any other substrate other than the S(GLS), P(nga)

http://userpages.umbc.edu/~gferre1/derivations.html#Fixed%20volume%20fed-batch






is the non-growth associated product and P(ga) is the growth associated profile for productconcentration.


3.6.2.Variable volume fed-batch

In a variable volume fed-batch fermentation, an additional element should be considered: the

feed. Consequently, the volume of the medium in the fermenter varies because there is an inflowand no outflow. Again, it is going to be considered that the growth of the microorganism is

limited by the concentration of one substrate.

For the mathematical developments that will be presented, the assumptions are

o Specific growth rate is uniquely dependent on the concentration of the limitingsubstrate

o The concentration of the limiting substrate in the feed is constant

o The feed is sterile

o The yields are constant during the fermentation time

Table 3.6.2.1. summarizes the equations that apply to this situation. These relations arethe base for all further calculations and specific cases of a variable volume fed-batchfermentation.

Table 3.6.2.1. Mass balances for the main components for a fed-batch reaction.

Component Mass Balance Equation Equation #

Overall F = dV/dt (3.6.2.1)

Biomass dx/dt = x . (u. V -– Kd . V -– F) / V (3.6.2.2)

Substrate ds/dt = F . (so -– s)/V -– u. x/ Yx/s (3.6.2.3)Product dP/dt = qp . x -–P . F / V (3.6.2.4)

V is the volume of the fermentor

t is the time










F is the feed rate [volume/time]. x is the biomass concentration [mass biomass/volume]

u is the specific growth rate [time-1] Kd is the specific death rate [time-1] s is the substrate concentration in the fermentor [mass substrate/volume]

so is the substrate concentration in the feed [mass substrate/volume] Y x/s is the yield factor [mass biomass/mass substrate] P is the product concentration {mass product/volume] and



For a non-growth associated product

In this case it is desirable to have a high cell density. The process can be then divided intwo stages: the first stage of the process would therefore be to grow up a high cellconcentration, followed by a phase where growth is suppressed and only sufficient ofthe substrate is supplied for maintenance and product formation the batch feeding phase.The first stage can be translated by the equations in table 3.6.2.1. For the second stage, u

should be zero and the production formation rate is defined as rp = x15 .

s = 0; ds/dt = 0 and rx= 0 (3.6.2.5)

x = K1 . Kd . exp(-Kd . t)/(1 – K1 . K . exp(-Kd . t)) &nb sp; (3.6.2.6)

Where K1 is defined as the ratio xo / ( Kd + K . x) ; (3.6.2.7)

F/V = K . x ; (3.6.2.8)

where K = { ms + / Yp/s’} . 1/so (3.6.2.9)

P= / K . [ 1 -– exp (-K . f(t)] (3.6.2.10)

A similar and a much simpler development would be implemented if Kd would benegligible. In this situation, the solutions for biomass concentration, flow rate, productconcentration and volume variations with time would be given by

http://userpages.umbc.edu/~gferre1/derivations.html#Variable%20volume%20fed-batch









x = xo / (1 + K . xo . t) &nbs p; (3.6.2.11)

F = K . xo. Vi &nb sp; (3.6.2.12)

(where Vi is the volume in the beginning of the fed-batch phase)

P = . x . t / (1 + K . x . t) (3.6.2.13)

V = Vi . (1 + K . xo.t) (3.6.2.14)

Figure 3.6.2.1. depicts the change in volume, product concentration, feed and biomass

concentration with time for a fermentation as described above.

Figure 3.6.2.1. Time profiles for a variable-volume fed-batch culture for a process involving non-growthassociated production. V=volume of the fermentor, P=product concentration, u=specific growth rate,X=biomass and S(gls)=growth limiting substrate concentration. The feed follows a similar profile to thatone of the specific growth rate.


For a growth associated product13

http://userpages.umbc.edu/~gferre1/derivations.html#For%20a%20non-growth%20associated%20product










Nomenclature:

Y'p/s’ is the mass of product formed per fraction of substrate mass

Y'x/s’ is the mass of cells formed per fraction of substrate mass

S is the difference (so–- s)

In this case, substrate is provided in such a way that maximizes the specific growth rate,assuming that the substrate is not growth or product formation inhibitory for thatconcentration. The substrate is supplied here not only for maintenance and productformation but also for biomass production. The product formation is such that rp =

. u. x15 , being alpha a specific constant of the bioprocess.

For a matter of simplicity, Kd is going to be considered to be approximately zero and

s = constant; ds/dt = 0 ; &n

bsp; (3.6.2.15)

x = K1 . u. exp(-u. t)/( K1 . K . exp (-u . t) - 1) (3.6.2.16)

K = - { u/ Y'x/s’ + ms + . u/ Y'p/s’} . 1/S (3.6.2.17)

and K1 is defined as the ratio xo / (u + K . x) (3.6.2.18)

For product variation with time

P = . u / K . { 1 -– exp (-K . F(t)} (3.6.2.19)




































Figure 3.6.2.2. Time profiles for a variable-volume fed-batch culture for a process involving growth

associated production. V=volume of the fermentor, P=product concentration, u=specific growth rate,X=biomass and S(gls)=growth limiting substrate concentration.


Microorganisms growing exponentially29

Another approach to a situation in which the specific growth rate is maintainedconstant goes as following.

Nomenclature:

rs is the consumption rate of substrate (mass substrate/(volume. time) xo is the initial concentration of biomass inside the fermentor for time zero [mass

biomass/volume] Vo is the initial volume of the fermentor for time zero

V is the volume of the fermentor for time t

F = u . xo . Vo . exp(u. t)/[(so -– s) . Y x/s] (3.6.2.20)

V = Vo (u + A . xo . exp(u. t) – A . xo)/where

http://userpages.umbc.edu/~gferre1/derivations.html#For%20a%20growth%20associated%20product









A = u / (s . Y x/s) &nbs p; (3.6.2.22)

x = u . xo . exp(u. t)/ (u + A . xo . exp(u. t) -– A . xo) (3.6.2.23)


Another alternative would be to maintain the concentration of biomass constant withtime – the quasi-steady stat e6. In this case,

V = Vo + F . t (3.6.2.24)

s = F. t/ (Vo + F . t) . (so -– x/ Y x/s) (3.6.2.25)

Which, for small times, approximates zero and for large times approximates

S = so -– x/ Y x/s &nb sp; (3.6.2.26)

The growth rate varies as following

For small times: du/dt = -F2/Vo2 ; &n bsp; (3.6.2.27)

For large times du/dt = -1/t2 &nb sp; (3.6.2.28)

http://userpages.umbc.edu/~gferre1/derivations.html#Another%20approach










Which shows that the specific growth rate decreases more in the beginning of thefermentation but decreases less as the time passes by6.



3.6.3. Models of possible situations that may occur in afed-batch fermentation

The models that are presented here can be directly used in the equations summarized intable 3.6.3.1.

Table 3.6.3.1. List of growth models that can be found in biotransformations1

Model Form

Monod

Constant yield

u = umax s/(K m + s)

Y x/s = Y0

Substrate inhibition

Constant yield

u = umax s/(Km + s + s2/Ki)

Y x/s = Y0

Substrate inhibition

Variable yield

u = umax s (1 - – T. s)/(K m + s + s2/Ki)

Y x/s = Y0 (1 -– T. s)/(1 + R. s + G. s2)

Substrate and product inhibition

Inhibitions

Constants yields

u = umax s/(Km + s + s2/Ki)

u = umaxo

(1 -– P/P m )

q p = alpha.+ beta

and Y p/s



http://userpages.umbc.edu/~gferre1/derivations.html#Another%20alternative


















Ki refers to inhibition constants and have the same units as the substrate concentration(mass of substrate/volume). Tand Rare constants with units that are volume/masssubstrate and G has (volume/substrate)2 units. These constants are defined byexperimental data analysis1.


3.7. Parameters used to control fed-batch fermentations

All the parameters that are about to be described have as an ultimate aim, to control afed-batch fermentation, more specifically the growth rate and/or the flow through the

central carbon metabolism and/or to reduce the overflow of carbon source to metabolicby-products. The control can be a one-step type in which only one of the parameters isused or it can be dual-level control in which two parameters are used, yielding a more"refined" control25, 29, 4. For example, some processes require the control of differentparameters at different stages of the fermentation. Such is the example of high densitybacterial fermentation like the production of baker's yeast and penicillin. In these cases,as an example, two phases can be distinguished: (i) a phase in which the substrate needsto be controlled so as to avoid by-products formation and (ii) a second stage in which,due to the high cell density, oxygen transfer is limiting and so,this parameter is the oneto be controlled above a critical value, under which the cellular metabolism changes.

The control constraints switch from specific growth rate to oxygen concentration aftersome critical period of time in the fermentation process4.

The choice of each parameter is system dependent and the decision should be based onconvenience and experimental data25

. The mathematical model is used in two ways: firstto estimate the controlling variable and second to calculate the control action4.

A very important fact that should be considered is the quality of the sensors available for thevariables that are whished to be controlled. Temperature, speed of agitation, flow rates, pH,

dissolved oxygen, redox potential of broth are examples of commonly controlled variables.

These sensors usually respond quickly enough (faster than the change of the variable itself in the

system) and so, proportional feedback controllers are often suitable for these variables. Of particular interest, the dead time - time for the sensor to start responding - and the overall time

constant - time required to yield the final reading. The crucial design factor is the ratio of the

first order time constant to the dead time4. Cohen-Coon derived the following expression to

determine the proportionality constant of proportional controllers:

Kc = Tp / (Kp . td) . [1 + td / (3Tp)] &nbs p; 3.7.1.

































where

Kc is the constant of the proportional controller

Tp is the overall time constant Kp is the slope of the response curve, or the process gain

The steady stater error for this controller can be defined by being equal to -1/(1+K cKp).

Proportional - integral (PI) and proportional - integral - derivative (PID) controllers14

can also be

used, yielding a more refined control and eliminating the set-point offset usually related withproportional controllers. Even when these "more effective" controllers are used, some problems

can still occur due to nonlinearities inherent to the process 4.

To design a feedback controller, a certain parameter to be maintained within certain limits is

analyzed as far as requirements to keep its value within the desired range or level. The process

response curve, an open loop response of the pH to a step change is generated by changing the

value of the parameter by small increments.

Computer technology allows the handling of mathematical models which are solved on-line andpredict and evaluate the future performance.

Because of some difficulties with measurement of some variables, some linear estimation of statecan be used such as the Kalman filter .The Kalman filter uses past measurements for a weighted

least square estimate of the current variable as reflected through the dynamic mode l4. Another

alternative is the use of a Dynamic Matrix Control, which is basically a linear dynamic

mathematical modelof the process that calculates the response resulting from initial conditions,disturbances, manipulated variable inputs and setpoint changes. The compensation is then

applied in such a way that minimizes the sum of squares of deviations from the setpoint,subjectto constraints on the manipulated variable4.

3.7.1. Calorimetry

Calorimetry is an an excellent tool for monitoring and controlling microbialfermentations11. Its main advantage is the generality of this parameter, since microbialgrowth is always accompanied by heat production, and the measurements areperformed continuously on-line without introducing any disturbances to the culture.Moreover, the rate of heat production is stoichiometrically related to the rate ofsubstrate consumption and product, including biomass formation. In many cases it can

be replaced by exhaust gas analysis, although this approach can not be considered inanaerobic processes which proceed without formation of gaseous products.

This technique has been proved successful to indirectly determine the substrate andproduct concentrations continuously during aerobic batch growth of Saccharomycescerevisiae with glucose as the carbon and energy source. In the presence of this substrate,this yeast shows diauxic growth by initially consuming the glucose with concomitant























production of ethanol and then, once glucose is depleted, using the produced ethanol asan energy source. Calorimetry can then be used to control the feed rate in such a waythat ethanol formation is avoided.

Another interesting description of a temperature-based controlled reactor follows a

stability criterion that predicts that the range of operation is controlled by the reactantfeed, being the flow rate of the cooling medium the control variable23. Although thestudy has been performed in a chemical reactor, the concepts can be easily extended to abiotransformation process.

3.7.2. Specific growth rate

For the production of a growth-associated product, the production of a certain productis related with the specific growth rate of the producing microorganism. Consequently,it is of interest to feed the fermentor in such a way that the specific growth rate remains

constant. Such is the case of the production of hepatitis B surface antigen bySaccharomyces cerevisiae 1, 2. The yield of the antigen is ten times more than that of thefed-batch cultivation for the same volume and total substrate added. Care should begiven to the value of the chosen specific growth rate, because cells may not be"activated" easily19, stress proteases can be produced that may degrade the product19 and also there might be a threshold value of specific growth rate above which there isproduction of by-products29.

3.7.3. Substrate (carbon and nitrogen source)

Substrate is a particularly important parameter to control due to eventual associatedgrowth inhibitions and to increase the effectiveness of the carbon flux, by reducing theamount of by-products formed and the amount of carbon dioxide evolved.

An example of adaptive feedback control of glucose concentration is found in the fed-batch production of thuringiensin, –an exotoxin that shows efficacious control againstflies, beetles, bugs and mites – by Bacillus thuringiensis8. Glucose concentration was

estimated by using empirical correlation equations between the consumed glucose andthe values of "Oxygen Uptake Rate" (OUR) and "Carbon Dioxide Production Rate"(CPR). Then, the glucose concentration G(t) in the fermentor was expressed as

G(t) = [{initial mass amount} -– {consumed mass} + {mass infeed}]/V(t) (3.7.3.1.)

























G(t) = [Gi . Vi - Gcons(t) + Gf . Va]/V(t) (3.7.3.2.)

Where

Gi is the initial glucose concentration in the fermentor

Vi is the initial working volume

Gcons(t) is the mass of glucose consumed

Gf is the glucose concentration of the feed medium

Va is the added volume of substrate during fed-batch culture and

V(t) is the volume of the fermentor for time t

The equation for adaptive control of substrate addition is

dG(t)/dt = a . G(t) + b . u(t) &nbs p; (3.7.3.3.)

Where

a and b are estimated and

u(t) is the feeding rate (volume/time)

Now, the model strategy is given by

dG'’(t)/dt = A . [G'’(t) -– Gs] (3.7.3.4.)



being A a negative constant, G’(t) the value of glucose concentration that is estimatedand Gs the setpoint of G(t). An error signal E(t) can be defined as

E(t) = G(t) -– G'(t) ; (3.7.3.5.)

Combining equations 3.7.3.3., 3.7.3.4. and 3.7.3.5., gives

dE(t)/dt = A . E(t) + (a –- A) . G(t) + b . u(t) + A . Gs ; (3.7.3.6.)

By making the error signal approach zero,

dE(t)/dt = A . E(t) (3.7.3.7.)

Then, the feeding rate is determined by combination of equations (3.7.3.6.) and (3.7.3.7.)

u(t) = - 1/b . (a -A) . G(t) - 1/b . A . Gs (3.7.3.8.)

By using this adaptive type of control, the production of thuringiensin was significantly

improved, with readings that were ten times higher for fed-batch (according to the feedmedium) as compared with batch fermentation8.

A predictive and feedback control algorithm can be also set up to form a product orgrow cells such as E. coli 9. The control in this case will be more refined than that of afeedback control. Basically, the control scheme can be divided in two: a feed-forwardcomponent that predicts (according to same statistical method and the previous










collected data) the "need" of a certain substrate measurable on line, and a feed-backcontroller which corrects for minor errors in the predicted "need". These errors canoccur in exponentially growing microorganisms, in which the predicted value will begreater than the effective value9 . The main advantage to this system is that theinvestigator does not need to know the metabolic constants for a given organism prior

to growth of that organism in the system, and it can be applied to any substrate that canbe measured on-line, being particularly valuable in the minimization of by-products.

In some other situations such as -amylase production in recombinant Bacillusbrevis, the fermentation kinetics are mainly driven by the nitrogen source, since thismicroorganism responds very slowly to glucose depletion and it prefers a nitrogensource over a carbon nutrient for growth and production of recombinant protein16. However, when a large amount of nitrogen source is fed, the larger the cell mass isproduced but the lower the recombinant gene production is. Therefore, the feednitrogenous sources need to be maintained at low levels and can be provided in optimal

manner using L-amino acids concentrations control at various levels on-line17.

As a final comment the operator should be aware of the analytical equipment that willbe using so as to guarantee that the readings from the substrate concentrations fallwithin the limits of detection of the assay and/or analytical assay. At the same time, theoperator should make sure that the concentrations are low enough to prevent by-product formation21. Another aspect to consider are eventual interferences in thereadings of this due to some other component that might be present in the growthmedium.

3.7.4. By -product concentration

The production of by-products is undesirable because reduces the efficacy of the carbonflux in a fermentation. The production of these components take place whenever thesubstrate is provided in quantities that exceed the oxidative capacity of the cells. Thisapproach has been used in the fermentation of Saccharomyces cerevisiae, in which acidproduction rate is used to provide on-line estimates of the specific growth rate1. Also, in

modern fed-batch processes for yeast production, the feed is under strictly controlbased on the measurement of traces of ethanol in the exhaust gas of the fermenter22.

3.7.5. Inductive, enhancer or enrichment components























In certain fermentations it is of interest to continuously add either an inductive or fastconsumed components and not only a limiting substrate. An example is the continuousaddition of an antibiotic in recombinant microorganims bearing an antibiotic markedplasmid29. Another example is given by the production of gluthathione by high-gluthathione-accumulating Saccharomyces cerevisiae, the commonly microorganims used

for commercial production. Cysteine was found to be the only amino acid thatenhanced gluthathione formation. However, the growth inhibition occurred and it wasrelated to the concentration of cysteine. This problem was then resolved by an adequateaddition of cysteine in exponential fed-batch culture without growth inhibition11.

Fed-batch proves to be an appropriate mode of fermentation in microorganisms that areproducing heterologous proteins and whose elevated protein expression results inproduct degradation by activation of proteases. A general insight on this subject wasthe study of a recombinant E. coli for production of chloramphenicol acetyltransferase.A gradual induction with IPTG and phenylalanine (rate limiting precursor) addition

strategies were able to reduce the physiological burden imposed on the bacterium,thereby avoiding cellular stress responses and enhancing bioreactor productivity19. Inthis case, IPTG and phenylalanine were the driving parameters that dominated the feed.

As a final note, the addition of precursors or inducers should take into account if theproduct of interest is growth associated or not. For example, the use of a tyrosine-deficient strain of E. coli in the production of phenylalanine requires a balance feed oftyrosine that, if not provided in low quantities is used as carbon source with subsequentproduction of excessive biomass synthesis at the expense of phenylalanine synthesis.This limitation on biomass production is possible because the phenylalanine productionwas not growth associated19.

3.7.6. Respiratory quotient (RQ)

Gas analyzers,especially mass spectrophotometers are relatively fast4. Respiratoryquotient, the ratio between the moles of carbon evolved per moles of oxygen consumed,has been a general method used to determine indirectly the lack of substrate in the

growth medium4

. It is a fairly rapid method of measurement that is useful because thegas analyses can be related to crucial process variables. The method is not "universal"toall bioprocesses since some biosystems can produce by-products that affect theproductivity of the process without affecting the RQ, such as the production of aceticacid by E.coli4.

Usually, the signal is characterized by a sharp rise in dissolved oxygen13. The processresponse of RQ can be represented very closely by a first order transfer function defined





























by RQ/ (feed of substrate). Equation 3.7.1. can be still used but, any steady state offsetfor a proportional controller in this case is not desirable. Instead, a PI can be considered.

Based on the concept of respiratory quotient, there are the so called DO -stats, in whichthe feed is regulated in accordance with the dissolved oxygen29. The analysis of the

dissolved oxygen or carbon dioxide evolution rate can also be used to control orprevent the production of by-products, 1, 10. The respiratory quotient is often analyzedto study the carbon flux, this is, the feed should be conditioned in such a way that itshould prevent excess of carbon dioxide evolution caused by unnecessarily severesubstrate limitation4,9.

Care should be given to the mathematical model so that equations are explanatory for certainsubstrates limits or ranges for which below or above the system behaves differently due to

different metabolic reactions and,consequently, different metabolism. Suppose that the desirable

point of operation is at RQ=1,corresponding to zero by-product formation (such is the case with

Baker's yeast). However, if the substrate feed rate is further reduced, the RQ will remain at 1 butsuboptimal conditions will occur. Conversely,RQ can be still equal to 1 if that is the

steicheometry of the consumption of the by-product as alternative energy source4. The objective

thus is slightly modified to control RQ as near to 1 as possible, but just slightly larger like 1.02 4.

For most processes,amore reliable control system is required4.

3.7.7.General feeding mode

The feeding mode influence a fed-batch fermentation by defining the growth rate of themicroorganisms and the effectiveness of the carbon cycle for product formation andminimization of by-product formation. Inherently related with the concept of fed-batch,the feeding mode allows many variances in substrate or other componentsconstitution24 and provision modes and consequently, better control over inhibitoryeffects of the substrate and/or product. The feed mode can be defined based on anopen-loop, if an exact mathematical model is at disposal (not very common and usuallyinsufficient)14, a feedback control (ex. pH - stat or dissolved oxygen (DO) – stat) or inany other way depending on the specific kinetics of each fermentation and even withinthe time frame of the fermentation process. In fact, the feed can be modified accordinglyto the different phases of the microbial growth, as a consequence of physiologicalalterations that the cells undergo upon transfer through eventual consecutive stages ofthe fed-batch cultivation10.

Usually, a fed-batch starts as a batch mode and after a certain biomass concentration orsubstrate consumption, the fermentor is fed with the limiting susbtrate solution.However, that approach does not need to be the absolute rule. Some cases happen in

































which the rate of production of a certain product is limited not only by the susbtrate butalso by a primary product, associated with the growth of the microorganism. That is thecase of streptokinase formation. Streptokinase is a vital and effective drug for thetreatment of myocardinal infection, that is currently produced in industry by mainlynatural or mutated strains of streptococci. The specific growth rate is inhibited by the

susbtrate and by lactic acid. A near optimal feed policy based on a chemotaxisalgorithm has been established that defines an initial decreasing feeding phase,followed by a batch fermentation with no more added substrate in the medium. Thestarting point was the data provided by the batch fermentations and the feed wasdefined as being a polynomial function of time. By iterative calculations and having thebatch time fermentation or the maximum allowable volume of the fermentor as timelimits, a feed strategy was defined yielding a 12% increase in streptokinase activity overbatch fermentations16. This type of approach has been previously suggested also forethanol production by Saccharomyces cerevisiae that follows the same kinetics16.

Finally, the feed can be continuous, can be provided in pulses24

, as a shot feeding11

, single or multisubstrate24, increasing linearly19, be exponential 7,19 or constant with time.The design of the feed solution may follow a conventional approach – in which thenutrients are more concentrated as compared with the growing medium in thefermentor - or follows a quantitative design in such a way that depletion oraccumulation of nutrients can be avoided or reduced28. An optimization problem for afeed to the production of a non-growth associated product is given by Meszaros14.

3.7.8.Proton production

An "unusual" type of controlling process parameters is the proton production toestimate on-line the specific growth rate in a fed-batch culture and indirectly, thesubstrate concentration. In an anaerobic alcohol fermentation, Won et al.27 definedspecific growth rate (u) as being

u= dln(proton production)/dt ; (3.7.8.1.)

The measured amount of proton produced during the fermentation was calculatedbased on the volume of base added to the fermentor to control the pH at a pre-set value.

The control based on pH usually uses a on-off mode because the magnitude of the acid and alkali

feeds is so low that implemententing a proportional controller is difficult. On -off operation is






































similar to proportional control with a very high gain because a small amount of acid/alkali feed

raises the pH above the set point. Flow must be discontinued to await the reaction to bring thepH back down to the desired value. A variation on the on-off control valve system is frequency

modulation. If the pH is very far from the set point, the valve will be open for a long period of

time and closed for a shorter period of time. The reversal happen when the pH is in the vicinity

of the setpont value. The frequency modulated on-off controller offers more accurate control if that is needed4.

3.7.9. Fluorescence

A linear relationship exists between the culture fluorescence (as a function of theintracellular NAD(P)H pool) and the dry cell weight concentration up to 30g dry cellweight/liter29. Thus, fluorescence can be used to estimate on-line the biomassconcentration and be a controlling parameter in the feed provision29.


3.8.Parameters to start and finish the feed, and stop thefed-batch fermentation

The times at which the feeding should start and finish, as well as the criteria to stop afed-batch fermentation is very much dependent on the specific cultivation kinetics andthe operator’s interest. For example, in substrate limited processes, the feed should startimmediately after all substrate is consumed from the batch phase, otherwise the processmay be difficult to control, for example, because of a lag phase due to previousstarvation13. The most commonly criteria to start the feed is the depletion of substrate24, which can be measured by a multitude of techniques, from specific enzymatic assays,HPLC25 to indirect methods such as the exhaust gas analysis7, 9. Still related with the

amount of substrate in the medium, the operator might not find necessary to reach thecomplete depletion but to be below a predetermined set-point (eventually related withhistorical data, growth models and known yields) 8,19.

The fed-batch fermentation should be halted when the production slows down becauseof cell death13, because the metabolic potential of the culture becomes inadequately lowor because by-product excretion starts at significant levels10. Some other criteria can be













































an increase in viscosity that implies an increased oxygen demand until the oxygenlimitation is achieved, which is the case for penicillin production22.


3.9.Preliminary knowledge required to implement fed-batch

Before starting a fed-batch process, a batch fermentation should be implemented to "getto know" the fermentation of the microorganism. From a batch fermentation, theoperator should have a knowledge of:

Best abiotic conditions such as temperature, light, agitation, pH, growthmedium, etc.

Specific needs of precursors, inducers or other enrichment factors

The different growth phases and the consumed (substrate) and producedcomponents (product of interest and by-product)

The relationship between the biomass and product formation (growth or non-growth associated product) and the oxygen uptake rates

Limiting substrate for growth and the relationship between the specific growthrate and the limiting substrate concentration

Eventual inhibitions from the substrate and/or product

Now, the operator should define the objective functions and the best parameter tocontrol the fermentation, considering both accuracy of data and convenience. Also, theoperator should define if the control that wishes to be implemented is based on anfeedback control (direct or indirect) or an open-loop control based on mathematicalmodels established for the system.


3.10.Algorithms for operating a fed-batch reactor atoptimum specific growth rate (model independent andapplicable to adapting systems) 1
























The control of a fed-batch fermentation can implicate many difficulties: low accuracy ofon-line measurements of substrate concentrations, limited validity of the feed scheduleunder a variety of conditions and prediction of variations due to strain modification or

change in the quality of the nutrient medium. These aspects point to the need of a fed-batch fermentation strategy which is model independent, identifies the optimal stateon-line, incorporates a negative feedback control into the nutrient feeding system andcontemplates saturation kinetic model, variable yield model, variation in feed substrateconcentration and product inhibited fermentation. The following described algorithmsdescribe methods for operating a fed-batch fermentor at the maximum possible rate offermentation (so that the productivity is maximized). The only requirement is theestablishment of a reliable on-line estimate of the specific-fermentation rate1.

3.10.1. Open-loop performance

In an open-loop operation system, a predetermined feed schedule is used1. Thisapproach considers that the system can be exactly translated in a set of mass balanceequations in which the specific growth rate. However, it is easy to assume that due to anon-identified physiological problem of the cells the specific growth rate can be eitherhigher or lower than the one that was previously established. If, for example, thespecific growth rate is higher than the pre-set one, and if the substrate is being fed insuch a way that is assumed that all the substrate is being consumed as soon as it entersthe fermentor, then there will be substrate limitation during the course of thefermentation. Consequently, the open-loop feed policy does not always result in anoptimal operation.

3.10.2. Feed-back control algorithm

This algorithm requires only a reliable on-line estimate of the specific growth rate, thatcan be provided by any of the parameters described in sections 3.7.1. to 3.7.9. Since theobjective of the algorithm is to optimize the cell-mass production by controlling the

specific growth rate (u) at an optimum value uopt, the feedback law can be defined:

Fin(t n+1) = Fin (tn)± Kc [u opt (tn) -u(tn)] (3.10.2.1.)





















This relation can be used to manipulate the feed flow rate (Fin) to the fermentor, whereKc is a controller constant which is assumed to be positive. When u is different fromuopt, either S < Sopt or S > Sopt. Then, the positive sign in equation (3.10.2.1.) applies to the

former case and, similarly, the negative sign applies to the latter case. By analysis ofwhat has just been described, then opt and Sopt need to be identified. Figure3.10.2.1.describes a flow diagram of the simple control algorithm to find those values by

an initial open-loop period. This period continues until starts to decrease. The

maximum value of obtained during this period is set as uopt and the correspondingvalue of S as Sopt.

Figure 3.10.2.1. A flow diagram of the control agorithm1.






In situations where it is difficult to obtain on-line measurements or estimates of S, it is

proposed to estimate S as S’ and Sopt as Sopt ’ during the open-loop period, in which opt value is identified. Figure 3.10.2.2 shows the flow diagram that includes all the features

that has just been described. This algorithm is model independent and therefore can beapplied to many industrial fermentations which utilize complex media. Moreover, since

the values of opt and Sopt are continuously being updated, the control methodology should work

well even when the microbial system undergoes adaptation.



Figure 3.10.2.2. A flow diagram of the control algorithm including estimates of optimum substrate andspecific growth rate1.














3.11. Some examples of fed-batch use in industry

The use of fed-batch culture by the fermentation industry takes advantage of the factthat the concentration of the limiting substrate may be maintained at a very low level,thus

avoiding repressive effects of high substrate concentration

controlling the organism’s growth rate and consequently controlling the oxygendemand of the fermentation.

Saccharomyces cerevisiae is industrially produced using the fed-batch technique so as tomaintain the glucose at very low concentrations, maximizing the biomass yield andminimizing the production of ethanol, the chief by-product13, 15, 22.

Hepatitis B surface antigen (HbsAg) used as a vaccine against type B hepatitis has beenpurified from human plasma and expressed in recombinant yeast, being now producedcommercially. Again, the production of the recombinant protein is achieved using fed-batch culture techniques very similar to that developed for Saccharomyces cerevisiae. Acyclic method is used due to reports of superior productivity22.

Penicillin production is an example for the use of fed-batch in the production of asecondary metabolite. The fermentation is divided in two phases: the rapid-growthphase during which the culture grows at the maximum specific growth rate, and the

slow-growth phase in which penicillin is produced. During the rapid-growth phase, anexcess of glucose causes an accumulation of acid and a biomass oxygen demand greaterthan the aeration capacity of the fermentor, whereas glucose starvation may result inthe organic nitrogen in the medium being used as a carbon source, resulting in a highpH and inadequate biomass formation. During the production phase, the feed ratesutilized should limit the growth rate and oxygen consumption such that a high rate ofpenicillin synthesis is achieved and sufficient dissolved oxygen is available in themedium 15, 22.

Some other examples are the production of thiostrepton from Streptomyces laurentiiand the production of cellulase by Trichoderma reesei. The production of thiostreptonuses pH feedback control and the production of cellulase utilizes carbon dioxideproduction as a control factor15.

3.6.1. Fixed volume fed-batch (derivations)


















The mathematical development that is going to be presented here has the followingassumptions21:

o The feed is provided at a constant rate

o The production of mass of biomass per mass of substrate is constant

during the fermentation time and o A very concentrated feed is being provided to the fermentor in such a way

that the change in volume is negligible.

Consider a batch culture in which the growth of the process organism has depleted thelimiting substrate to a limiting level. If this limiting substrate is fed to the fermentor insuch a way that the volume does not change (as a very concentrated feed, for example),then

dx/dt = F . Y x/s ( 3.6.1.1.)

where

x is the biomass [mass biomass/volume]

t is time

F is the substrate feed rate [mass substrate/(volume.time)] and

Y p/x is the yield factor [mass biomass/mass substrate]

But dx/dt = u x (3.6.1.2)

Where u is the specific growth rate [time-1]

Using equation (3.6.1.2) in (3.6.1.1), then u x = F . Y p/x <(F . Y x/s ) / x (3.6.1.3)

Considering that (F . Y x/s )/x has as a upper limit umax, then the limiting substrate will be

consumed as soon as it enters the fermentor and ds/dt is approximately zero, being "s" the

concentration of substrate inside the fermentor [mass substrate/volume]. However, because cells

are growing in the fermentor and then biomass is increasing with time, dx/dt is not zero.

Integrating equation (3.6.1.1) between the initial time (t=0) and between time t, and between thebiomass concentration at the onset of the fed-batch culture (x o) and the biomass concentration

after operating the fed-batch system after t time (x t), equation (3.6.1.4) is obtained.

xt = xo + F . Y x/s . t (3.6.1.4)

http://userpages.umbc.edu/references.html






Then, from equations (3.6.1.3) and (3.6.1.4), it can be observed that

(i) the specific growth rate decreases with time because the biomass (in the denominator) is

increasing with time and

(ii) the biomass increases linearly with time.

In terms of a product P, the product balance is

dP/dt = qp . x (3.6.1.5)

where



Note that equations (3.6.1.2) and (3.6.1.5) do not take into consideration the variablevolume because we are treating a fixed-volume fed-batch process.

If equation (3.6.1.4) is substituted into equation (3.6.1.5), then

dP/dt = qp . (xo + F . Y x/s . t) (3.6.1.6)

It is observed that the production of product rises with biomass, if qp is constant (non-growth associated products). If equation (3.6.1.6) is integrated between the initial time (t

= 0) and time t, and initial product concentration Pi and the concentration P for time t

P= Pi + qp . xo . t + qp . F . Y x/s . t2 /2 (3.6.1.7)



If fed-batch mode is established from the beginning of the fermentation, then Pi can bezero or approximated to zero.

If qp is related with u (growth associated product) then the relationship betweenproduct concentration and time will vary according to that relationship. By definition qp

is defined as the ratio of the product production rate over the biomass concentration inaccordance to equation (3.6.1.8)

qp = rp / x (3.6.1.8)

where rp is the product formation rate [product mass/(time * volume)]. Assume that rp

is a constant. Combining equations (3.6.1.4), (3.6.1.6) and (3.6.1.8), it comes to

dp/dt = rp (3.6.1.9)

which reflects a linear relationship between product and time. Figure 3.6.1.1. depictsthese relationships for constant or non-constant qp.

Other relationships between qp and the biomass production can be used in equation(3.6.1.6) so that, by integration, the concentration of product with time is known.

The same conclusions can be withdrawn if the Ludeking Piret (1959)20, 12 model type isused.

For non-growth associated products

rp = .x (3.6.1.10)

Using equation (3.6.1.8), then qp becomes a constant of value , which confirmsequation (3.6.1.7).

For growth-associated products







rp = ..x (3.6.1.11)

If this equation is used in combination with equation (3.6.1.3), then

rp = .F . Y x/s (3.6.1.12)

Once again, using equation (3.6.1.8), it can be seen that qp is not a constant and varieswith time, in accordance with the increase of biomass. Further substitution intoequation (3.6.1.5) yields

dP / dt = .F . Y x/s (3.6.1.1 3)

which, in accordance with was already concluded, reflects a linear relationship betweenproduct and biomass.

Now, if the growth is inhibited by the product such that

u = umax . (1 – - P/Pm) (3.6.1.14)

being Pm the maximum product concentration that can be attained without stopping the specific

growth. Then, the relationships defined by equations (3.6.1.1) – (3.6.1.4) are maintained.

However, if equation (3.6.1.13) is combined with equations (3.6.1.3) and (3.6.1.4), the followingrelationship of P=f(t) becomes

P= Pm . [1 - – F . Y x/s /(umax . (x + F . Y x/s . t))] (3.6.1.15)



The situation in which the substrate is growth inhibitory is not considered here because it was

assumes previously that the growth rate of the organism is defined by F and Y x/s, does not

exceed the umax and the substrate is consumed as soon as it enters the fermentor.

|Back to section 3.6 - modelling fixed-volume fed-batch fermentations | Top of the document |

3.6.2.Variable volume fed-batch

In a variable volume fed-batch fermentation, an additional element should be considered: the

feed. Consequently, the volume of the medium in the fermenter varies because there is an inflowand no outflow. Again, it is going to be considered that the growth of the microorganism is

limited by the concentration of one substrate5,21.

For the mathematical developments that will be presented, the assumptions are

o Specific growth rate is uniquely dependent on the concentration of the limiting

substrate

o The concentration of the limiting substrate in the feed is constant

o The feed is sterile

o The yields are constant during the fermentation time

Considering the overall mass balance

{in} = {out} + {accumulation} (3.6.2.1)

F = 0 + dV/dt <=> F = dV/dt (3.6.2.2)

Where


t is the time

F is the feed rate [volume/time].











Considering now the balance to the biomass

{accumulation} = {in} + {produced}-{lost by cell death} (3.6.2.3)

d(Vx)/dt = F . xo + rx . V -– rd . V (3.6.2.4)

Where

x is the biomass concentration [mass biomass/volume] xo is the biomass concentration in the feed [mass biomass/volume] and

rx is the biomass production rate [mass biomass/(volume. time)] and

rd is the biomass death rate [mass biomass/(volume. time)].

But the feed is considered to be sterile, then the {in} amount equals zero. If the left hand-side term derivative is now developed equation (3.6.2.5) is obtained.

V . dx/dt + x . dV/dt = V . dx/dt + x . F = r x . V - – rd . V (3.6.2.5)

But rx . V =u. X . V and rd . V = Kd . x . V (3.6.2.6)

Where u is the specific growth rate [time-1] and Kd is the specific death rate [time-1].

Substituting in (3.6.2.5) and rearranging the equation



dx/dt = x . (u. V - Kd . V -– F) / V (3.6.2.7)

For a matter of simplicity, the specific death rate is considered to be much smaller than

the specific growth rate and consequently, it can be neglected.

Considering now the balance to the limiting substrate

{accumulation} = {in} + {consumed} (3.6.2.8)

d(V . s)/dt = F . so -– rs . V (3.6.2.9)

Where

s is the substrate concentration in the fermentor [mass substrate/volume] so is the substrate concentration in the feed [mass substrate/volume] rs is the consumption rate of substrate [mass substrate/(volume. time)]

It should be noted that the consumption rate of substrate include the specificconsumption used for biomass production, product formation and maintenance of thecells14. Introducing now the concept of yield Yx/s as being the ratio between the mass ofcells produced per mass of substrate consumed.

rs . V = u. X . V/ Yx/s (3.6.2.10)

If the derivative on the left-hand side of equation (3.6.2.8) is now developed







d(V . s)/dt = V . ds/dt + s . dV/dt = V . ds/dt + s . F (3.6.2.11)

Substituting now in equation (3.6.2.9) and rearranging the equation, one gets

ds/dt = F . (so -– s)/V -– u . x/ Yx/s (3.6.2.12)

Considering a mass balance for the product

{accumulation} = {produced} (3.6.2.13)

d(V . P)/dt = rp . V (3.6.2.14)

Where

Pis the product concentration [mass product/volume] and

rp is the product formation rate [mass product/(volume.time)]

Developing the derivative in the left-hand side of equation (3.6.2.14) as madepreviously, and knowing that

rp = qp . x (3.6.2.15)

being qp the specific product formation [mass product/(volume. time)], equation(3.6.2.14) becomes



dP/dt = qp . x – P. F / V (3.6.2.16)

Table 3.6.2.1. summarizes the equations that have just been developed. These relations

are the base for all further calculations and specific cases of a variable volume fed-batchfermentation.

Table 3.6.2.1. Mass balances for the main components for a fed-batch reaction.

Component Mass Balance Equation Equation #

Overall F = dV/dt (3.6.2.2)

Biomass dx/dt = x . (u. V -– Kd . V -– F) / V (3.6.2.7)

Substrate ds/dt = F . (so -– s)/V -– u. x/ Yx/s (3.6.2.12)

Product dP/dt = qp . x -–P . F / V (3.6.2.16)


For a non-growth associated product12

Nomenclature:


t is the time


u is the specific growth rate [time-1] Kd is the specific death rate [time-1] s is the substrate concentration in the fermentor [mass substrate/volume] so is the substrate concentration in the feed [mass substrate/volume] Y x/s is the yield factor [mass biomass/mass substrate] ms is the substrate consumption rate for cell maintenance [mass

substrate/(volume.time)] Y'p/s’ is the mass of product formed per fraction of substrate mass


qp is the specific production rate of product [mass product/(mass biomass . time) rs' ’ is the substrate consumption rate for maintenance and product formation

[mass of substrate/(volume.time)]

In this case it is desirable to have a high cell density. The process can be then divided intwo stages: the first stage of the process would therefore be to grow up a high cellconcentration, followed by a phase where growth is suppressed and only sufficient of











the substrate is supplied for maintenance and product formation – the batch feeding phase. The first stage can be translated by the equations in table 3.6.2.1. For the secondstage, u should be zero, rp = beta . x14 being beta a specific constant of the bioprocess,and hence:

s = 0; ds/dt = 0 and rx= 0 (3.6.2.17)

Applying these relations to equations (3.6.2.2), (3.6.2.7), (3.6.2.12) and (3.6.2.16), thefollowing system of equations is achieved:

Biomass: dx/dt = -x . (Kd . V + F) / V (3.6.2.18) Substrate: 0 = F . so / V -– rs' (3.6.2.19)

(where rs' ’ is the substrate consumption rate for maintenance and productformation)

rs' = ms . x + rp / Yp/s’ (3.6.2.20)

Product: dP/dt = . x –- P. F / V (3.6.2.21)

o Overall: dV/dt = F (3.6.2.22)

To obtain the optimum operating system, so will be fixed and F is allowed to vary.Equation (3.6.2.19) can be rearranged to give

0 = - K . so . x + F . so /V (3.6.2.23)

where K = { ms + / Yp/s’} . 1/so (3.6.2.24)

From equation (3.6.2.23),

F / V = K . x (3.6.2.25)







Which substituting in equation (3.6.2.18) gives

dx/dt = - x . (Kd + K . x) (3.6.2.26)

By integration, between t = 0 and time t, and xo (the initial cell concentration when thebatch feeding phase begins) and x (the correspondent cell concentration in thefermentor for time t) gives

x = K1 . Kd . exp (-Kd . t) / (1 -– K1 . K . exp (-Kd . t)) (3.6.2.27)

Where K1 is defined as the ratio xo/( Kd + K . x) (3.6.2.28)

The variation of volume with time is given by combination of equations (3.6.2.27) and(3.6.2.22). The variation of the product concentration with time is again given by acombination of equations (3.6.2.27) and (3.6.2.21). Mathematically, the resultingexpression is more difficult to solve although this can be circumvented if

t

F(t) = I[ x dt (3.6.2.29)

0t

where I[ is the integral.

0

Then

P= / K . [ 1 -– exp (-K . f(t)] (3.6.2.30)



A similar and a much simpler development would be implemented if Kd would benegligible. In this situation, the solutions for biomass concentration, flow rate andproduct concentration variations with time would be given by

x = xo / (1 + K . xo . t) (3.6.2.31)

F = K . xo. Vi (3.6.2.32)

(where Vi is the volume in the beginning of the fed-batch phase)

P = . x . t / (1 + K . x . t) (3.6.2.33)

V = Vi . (1 + K . xo.t) (3.6.2.34)


For a growth associated product12

Nomenclature:

V is the volume of the fermentor t is the time


u is the specific growth rate [time-1] Kd is the specific death rate [time-1] s is the substrate concentration in the fermentor [mass substrate/volume] so is the substrate concentration in the feed [mass substrate/volume] Y x/s is the yield factor [mass biomass/mass substrate] ms is the substrate consumption rate for cell maintenance [mass

substrate/(volume.time)] Y'p/s’ is the mass of product formed per fraction of substrate mass


qp is the specific production rate of product [mass product/(mass biomass . time) rs' ’ is the substrate consumption rate for maintenance and product formation

[mass of substrate/(volume.time)]rx is the cell growth rate [masssubstrate/(volume. time)]

Y'x/s’ is the mass of cells formed per fraction of substrate mass











In this case, substrate is provided in such a way that maximizes the specific growth rate,assuming that the substrate is not growth or product formation inhibitory for thatconcentration. The substrate is supplied here not only for maintenance and productformation but also for biomass production. The product formation is such that rp =

. u. x14 , being alpha a specific constant of the bioprocess.

For a matter of simplicity, Kd is going to be considered to be approximately zero and

s = constant; ds/dt = 0 (3.6.2.34)

Applying these relations to equations (3.6.2.2), (3.6.2.7), (3.6.2.12) and (3.6.2.16), thefollowing system of equations is achieved:

o Biomass: dx/dt = x . (u. V–- F) / V (3.6.2.35)

o Substrate: 0 = F . (so–- s) /V -– r's’ (3.6.2.36)

(where r's’ is the substrate consumption rate for growth, maintenance and productformation)

r's = rx / Y'x/s’+ ms . x + rp / Y'p/s’ ( = . x/ Yx/s ) (3.6.2.37)

o Product: dp/dt = . u. x –- p . F / V (3.6.2.38) o Overall: dV/dt = F (3.6.2.39)

Once again, to obtain the optimum operating system, so will be fixed and F is allowed tovary. Moreover, the specific growth rate is a constant and s is a constant also, hence (so–- s) is a constant that will be named S. Equation (3.6.2.36) can be rearranged to give

0 = F . S /V + K . so . x (3.6.2.40)

in which now






K = - { / Y'x/s’ + ms + . u/ Y'p/s’} . 1/S (3.6.2.41)

However, for K to be a constant, u needs to be constant also. Considering that is the

case, then, the solutions for biomass concentration, flow rate and product concentrationvariations with time would be given by

x = K1 . u. exp(-u. t)/( K1 . K . exp (-u . t) - 1) (3.6.2.42)

Where K1 is defined as the ratio xo / (u + K . x) (3.6.2.43)

Again, the variation of the volume with time is given by the combination of equations(3.6.2.39) and (3.6.2.40), using F(t) and by direct integration. For product variation withtime

P = . u / K . { 1 -– exp (-K . F(t)} (3.6.2.44)

Microorganisms growing exponentially28

Another approach to a situation in which the specific growth rate is maintainedconstant goes as following. Taking equation (3.6.2.9)

d(V . s)/dt = F . so – rs . V (3.6.2.9)

Y x/s = rx . V / (rs .V) = u . X . V/(rs . V) (3.6.2.45)

If the growth is exponential






d(x . V)/dt = u. x . V (3.6.2.46)

Integrating equation (3.6.2.46) gives

X . V = xo . Vo . exp (u. t) (3.6.2.47)

Where

rs is the consumption rate of substrate (mass substrate/(volume. time) xo is the initial concentration of biomass inside the fermentor for time zero [mass

biomass/volume] x is the concentration of biomass for time t [biomass/volume] Vo is the initial volume of the fermentor for time zero

V is the volume of the fermentor for time t

Using equation (3.6.2.47) in equation (3.6.2.45), gives

rx . V = u. xo . Vo .exp(u. t) / Y x/s (3.6.2.48)

If now the derivative in the left-hand side of equation (3.6.2.9) is developed

d(V . s)/dt = V . ds/dt + s . dV/dt (3.6.2.49)

But since the specific growth rate is constant, which implies constant substrateconcentration in the medium, then ds/dt equals zero. Equation (3.6.2.49) is resumed to



d(V . s)/dt = s . dV/dt = s . F (3.6.2.49)

being the last relation allowed by equation (3.6.2.2). Finally, substituting equations(3.6.2.48) and (3.6.2.49) in (3.6.2.9), gives

F = u . xo . Vo . exp(u. t)/[(so -– s) . Y x/s] (3.6.2.50)

Which basically reflects the need of an exponential feed to maintain a constant specificgrowth rate in a culture growing exponentially and, consequently the substrateconcentration. To determine how the biomass is increasing with time, take

F = u . x . V/ (s . Y x/s) = u . xo . Vo . exp (u. t) / (s . Y x/s) = dV/dt (3.6.2.51)

By integration of this equation, V comes as a function of time

V = Vo (u + A . xo . exp(u. t) – A . xo)/where

A = u/(s . Y x/s) (3.6.2.52)

Consequently, by substituting equation (3.6.2.52) in equation (3.6.2.47)

X = u . xo . exp(u. t)/ (u + A . xo . exp(u. t) -– A . xo) (3.6.2.53)




Another alternative would be to maintain the concentration of biomass constant withtime – the quasi-steady state. In this case,

dx/dt = 0 (3.6.2.54)

Right from equation (3.6.2.7) and assuming Kd negligible, = F/V, which means that

the specific growth rate will decrease with time since the volume is increasingsimultaneously. Both the substrate concentration and the growth rate are changing withtime. In this case, to solve the system of equations, the feed should be defined. Considera constant feed F (volume/time). Then, equation (3.6.2.2) can be directly integrated and

V = Vo + F . t (3.6.2.55)

Substituting in equation (3.6.2.12), and by integration, it comes

s = F. t/ (Vo + F . t) . (so -– x/ Y x/s) (3.6.2.56)

Which, for small times, approximates zero and for large times approximates

S = so –- x/ Y x/s (3.6.2.57)

To determine how the growth rate varies, then







du/dt = d(F/V)/dt =d[F/( Vo + F.t)]/dt = -F2/V2 = -F2/( Vo + F.t) 2 (3.6.2.58)

For small times: d/dt = -F2/Vo2 (3.6.2.59)

For large times d/dt = -1/t2 (3.6.2.60)

Which shows that the specific growth rate decreases more in the beginning of thefermentation but decreases less as the time passes by5.






specific growth of bakers yeast in a feed batch system

Documents