Microbial Growth Kinetics
Motivation for studying cell growth kinetics • Cells Cells are the biocatal biocatalyst. yst. Differe Different nt than enzymes, enzymes, the catalyst amount (cell) is changing during the catalytic process. • Growth Growth is a result result of of replicat replication ion and and changes changes in cell cell size. size. Cells extract nutrients from the environment and convert them into biological products. • Most produ products cts of intere interest st will will be associat associated ed with the the growth growth and activity of cells • Design Design of system systems s (e.g. (e.g. system system size size and durati duration) on) requir requires es knowledge of – Accumulation Accumulation rate of cells – Depletion Depletion rate of nutrients – Accumulation Accumulation rate of desired desired product product
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Methods used to measure microbial growth • Dire Direct ct Met Method hod – Cell number • Microsco Microscopic pic counts counts • Parti Particl cle e counte counter r • Plat Plate e cou count nt
– Cell mass • Direc Directt method method • Turb Turbid idit ity y • Dry Dry wei weigh ghtt
• Indi Indire rect ct meth method od 3
Viable counts (Fig. 6.21, 6.22)
• Each Each colony colony on plate plate or filte filterr arises arises from from single single live live cell • Only Only cou counti nting ng live live cell cells s • Need Need to caref carefull ully y select select growth growth mediu medium m
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Microscopic counts (Fig. 6.23)
• Need Need a micros microscope cope,, special special slides, slides, high powe power r objective lens • Stainin Staining g can be used used to to disting distinguis uish h dead and and live live cells • Suit Suits s for for bacte bacteri ria a and yeas yeasts ts
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Particle counter • Cell Cell pass pass thro throug ugh h the the orifice and generate a electric pulse • Weig Weight ht of puls pulse e corresponding to cell size • Suit Suitabl able e for for discr discrete ete cells in a particle free medium
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Cell mass: Turbidity • Meas Measur urin ing g cell cell mass mass • Cell Cells s act act like like larg large e particles that scatter visible light • A spec spectr trop opho hoto tome mete ter r sends a beam of visible light through a culture and measures how much light is scattered • Meas Measur ures es both both live live and and dead cells • Blan Blanki king ng agai agains nstt medi medium um 7
Cell dry weight method • Need Need to have have solid solids s free free medi medium um • Sample Sample centr centrifuge ifuged d or or filtere filtered, d, washed washed with buffer • Dry at 80C for 24 hours, then measure the cell dry weight
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Indirect Method Quantifying cell concentration: - indirect: indirect: direct method is inapplicable. inapplicable. (mold solid state state fermentation) by measuring intracellular components such as protein, DNA etc. by measuring products or substrate of metabolism by measuring physical properties of the broth such as viscosity
Microbial Cell Growth Kinetics - Intr Introd oduc ucttion ion - Growth Growth patt pattern erns s and kine kinetic tics s in batch batch culture - growth growth phases phases - effect effect of factors: factors: oxygen oxygen supply supply - heat heat gener generati ation on - Growth Growth kine kinetic tics s (Monod (Monod Equa Equatio tion) n) - Growth Growth in in contin continuou uous s cultur culture e (ideal (ideal chemostat)
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Growth Kinetics Introduction - Autocatalytic Autocatalytic reaction: reaction: The rate of growth is directly related to cell concentration substrates + cells → extracellular products + more cells ∑S + X → ∑P + nX
S: substrate concentration (g/L); X: cell mass concentration (g/L); P: product concentration (g/L); n: increased number of biomass. Net specific specific growth growth rate (1/time): net
1 dX X dt
t: the time
Growth Kinetics Introduction Net specific growth rate (1/time):
net
g k d
g : Gross specific growth rate (1/time)
k d :
The rate of loss of cell mass due to cell death or endogenous metabolism
Endogenous metabolism: during the stationary phase, the cell catabolizes cellular reserves for new building blocks and for energy-producing monomers.
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Growth Kinetics Introduction Net specific replication rate (1/time):
R
1 dN
N dt
R
N
: '
R
k d :
R
' R
k d
Cell number concentration (cell number /L)
: Gross specific replication rate (1/time)
The rate of cell death (1/time)
Growth Kinetics - Growth Growth pattern patterns s and kinet kinetics ics in in batch batch cultu culture re - growt growth h phase phases s In batch culture: - lag lag pha phase se - logrithmic logrithmic or exponentia exponentiall growth phase - deceler decelerati ation on phase phase - statio stationary nary phase phase - death death phase phase
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Typical growth curve for a bacterial population
Batch Growth Kinetics Lag phase A period of adaptation for the cells to their new environment • New enzymes are synthesized. • A slight increase in cell mass and volume, but no increase in cell number • Prolonged by low inoculum volume, poor inoculum condition (high % of dead cells), age of inoculum, nutrient-poor medium • Multiple lag phases: phases: (diauxic (diauxic growth) medium medium contains more than one carbon source
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Batch Growth Kinetics Exponential growth phase In this phase, the cells have adjusted to their new environment and multiply rapidly (exponentially) • Balanced Balanced growth –all components components of a cell cell grow at the same rate. • Growth rate is independent of nutrient concentration, as nutrients are in excess • Cell Cell growth growth rate rate is highest highest in in this phase phase
Batch Growth Kinetics Exponential growth phase net μ m
R
μm
is the maximum specific growth rate (1/time)
Doubling time of cell mass: the time required to double the microbial mass:
d
ln X / X 0
net
ln 2
net
0.693
net
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Batch Growth Kinetics Deceleration growth phase Very short phase, during which growth decelerates due to either: • Depletion of one or more essential nutrients • The accumulation of toxic by-products of growth (e.g. Ethanol in yeast fermentations) • Period Period of unbalanc unbalanced ed growth growth:: Cells Cells underg undergo o internal restructuring to increase their chances of survival
Batch Growth Kinetics Stationary Phase: With the exhaustion of nutrients (S≈ (S ≈0) and build-up of waste and secondary metabolic products - The The growth growth rat rate e equal equals s the death death rate rate.. - There There is no no net grow growth th in the the organ organism ism popula population tion.. - Cells Cells may have have active active metabo metabolism lism to produce produce secon secondary dary metabolites. Primary metabolites metabolites are growth-related: growth-related: ethanol ethanol by S. cerevisae. Secondary Secondary metabolites are non-growth-relat non-growth-related: ed: antibiotics, pigments.
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Batch Growth Kinetics Stationary phase - Cell Cell lysis lysis may occu occurr and viab viable le cell cell mass may drop. drop. A second second growth phase phase may occur and and cells may grow on lysis products of lysed cells (cryptic growth) - Endoge Endogenou nous s metabolis metabolism m occurs occurs by catabol catabolizin izing g cellular cellular reserves for new building blocks and energy-producing monomer (maintenance energy). The rate describing the conversion of cell mass into maintenance energy or the loss of cell mass due to cell lysis:
dX
k d X dt k d is the rate constant for endogenous metabolism.
Batch Growth Kinetics Death Phase: The living organism population decreases with time, due to a lack of nutrients and toxic metabolic byproducts. The rate of death usually follows:
dN dt
'
k d N
'
k d is the first - order death rate constant.
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Kinetic Pattern of Growth and Product Formation
Growth-associated Mixed-growth-associated Non growth-associated
exercise: A simple simple batch fermentation fermentation of an aerobic bacterium growing on substrate, S, gave the results listed below • How would you find maximal net growth rate and YX/S from the data set? time( time(h) h) 0 9 16 23 30 34 36
cell cell (g/L) glucose(g/L) glucose(g/L) 1.25 100 2.45 92 5.1 90.4 10.5 76.9 22 48.1 33 20.6 37.5 9.38
How would you determine the minimum substrate concentration needed to produce 50 kg cells in 100 0 L of medium (assume the inoculums contribution is negligible)?
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Maintenance coefficient: describe the specific rate of substrate uptake for cellular maintenance
dS ]m m dt X [
S S assimilation S assimilation S growth energy S maintenance into biomass
into an extracellular product
energy
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Mathematical models Mathematical models can be used to assist in predicting cell growth, substrate utilization and product formation. Models provide valuable tools for process design and control. Types of model for cell growth include:
Segregated model • All All cel cells ls are are tre treat ated ed as individuals in a population • Need Need to be able able to measure individual traits for this type of model to be useful
Structured model • Inclu clude a detail taile ed description of intracellular events
Non-segregated model • All All cel cells ls are are beh behav ave e in in the same way in some “average sense” • Math Mathem emat atic ical ally ly much much simper Unstructured model • No de detai tails in te terms of intracellular reactions, only biomass is considered
Quantifying Growth Kinetics Monod equation: equation: unstructured and nonsegrega nonsegregation tion model • Unstruc Unstructure tured d model: model: assumin assuming g fixed cell cell compo compositi sition. on. Applicable Applicable to balanced-gro balanced-growth wth condition: condition: - exponential exponential growth phase phase in batch culture culture - single-stage, single-stage, steady state state continuous continuous culture - cell response response is fast compared to external external changes changes - the magnitude magnitude of the external changes changes is not too large (e.g. 10%-20% variation from initial conditions). • Nonseg Nonsegrega regation tion model: model: assum assuming ing all cells cells are are the the same same in the culture.
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Quantifying Growth Kinetics • Mono Monod d equ equat atio ion: n: unstructured and nonsegre nonsegregatio gation n model model Assumption: - a single enzyme system system with Michaelis-Menten Michaelis-Menten kinetics is responsible for uptake of substrate S, and the amount of that enzyme or its activity is sufficient low to be growth-rate limiting.
- the relationship of specific growth rate to substrate concentration assumes the form of saturation kinetics. - a single chemical chemical species is growth-rate growth-rate limiting while while changes in other nutrient concentrations have no effect.
Monod type saturation growth kinetics
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Quantifying Growth Kinetics • Monod equation equation:: when when applie applied d to cellular cellular systems, the gross specific growth rate g (1/time) g
m
m S
K S S
is the maximum maximum specific growth growth rate (1/time). (1/time).
Ks is the saturation saturation constant constant or half-velocit half-velocity y constant constant (g/l) when the substrate concentration S>> Ks g m (exponential growth phase), When endogenous metabolism is unimportant,
g net
Monod vs. Michaelis-Menten: Michaelis-Menten: recap of differences • Monod
• Mich Michae aeli lis s Ment Menten en
– Growth
– No growth; growth; constant constant E
– Empirical Empirical
– Derived from from theory theory
– Ks
– Km
– , 1/t
– v, mg/L-t
Simlarities are shape of curves, form of function, parameter estimation techniques.
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Determination of Monod Parameters Batc Batch: h: X, S, S, t → lnX ~ t , get µ m (slope) from data in exponential phase.
ln dX Xdt 1
g
K S 1 S
m
K S
μnet t μmt
X 0
g
g
S
X
m
m S
K S S 1
, K d 0
( Lineweaver - Burk)
m
S
(Hanes - Woolf)
m
Classical Growth Kinetics Empirical Approximation Approximation Monod Growth Kinetic • Monod Growth μ g
μ m
• Tessier Growth Tessier Growth Kinetic g
S Κ s
S
m 1 e
• Moser Growth Moser Growth Kinetic g
S K s
m 1 K s s
n 1
• Contois Growth Kinetic g
S m K sx X S
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Substrate Inhibition • Non-competitive inhibition g
m
(1
S K I
)(1
K s S
)
if _ K I K s g
m S
K s S S 2 / K I
• Competitive inhibition g
S S ) S K S (1 K I m
Product Inhibition • Non-competitive inhibition m S /(1 g
P K I
)
K s S
• Competitive inhibition g
S P ) S K S (1 K I m
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Presence of Toxic compounds • Non-competitive inhibition m S g I (1 )( K s S ) K I
• Un-competitive inhibition m S /(1
g
• Competitive inhibition g
S I
K S (1
K I
)
K I I Ks /(1 ) S K I
• Lead to cell death
m
I
) S μ g
μ m
S Κ s
S
'
k d
Batch Growth Kinetics Effect of factors: aerobic growth is more efficient. - Diss Dissol olve ved d oxy oxyge gen n (DO (DO)) - aerobic fermentation fermentation requires requires oxygen oxygen - oxygen gas gas is sparingly sparingly soluble in water water - specific growth growth rate may may be limited by DO DO if DO is below a critical oxygen concentration. Critical oxygen concentration: the growth rate becomes independent of DO concentration. bacteria and yeast: 5%-10% of the saturated DO mold: 10%-50% of the saturated DO The saturated DO in aqueous solution is 7 ppm at 25 oC and 1 atm.
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Batch Growth Kinetics Effect of factors: - Diss Dissol olve ved d oxy oxyge gen n (DO (DO)) The rate of oxygen oxygen transfer (OTR) from the gas gas to liquid phase is given by: OTR = No 2 = KLa (C*-CL) KL is the oxygen transfer coefficient (cm/h), a is the gas-liquid interfacial area (cm 2/cm3) KLa is the volumetric oxygen oxygen transfer coefficie coefficient nt (h -1) C* is saturated DO concentration (mg/l); CL is the actual DO concentration (mg/l); No2 is the rate of oxygen transfer (OTR) (mgO 2/l.h)
Batch Growth Kinetics Effect of factors: - Diss Dissol olve ved d oxy oxyge gen n (DO (DO)) Oxygen Uptake Uptake Rate (OUR) is oxygen oxygen consumption consumption rate by microbes. If the maintenance requirement of O 2 is negligible compared to growth, then
OUR qo2 X
X
g
Y X / O2
(mg O 2 /h)
qo2 is the specific rate of O 2 consumption (mg O 2 /g dw cells - h)
When oxygen transfer is the rate-limiting step, at steady state, the rate of oxygen consumption is equal to the rate of oxygen transfer. g X
Y X / O 2
K L a ( C * C )
Sufficient oxygen supply: OTR ≥ OUR
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Batch Growth Kinetics Effect of factors: - Diss Dissol olve ved d oxy oxyge gen n (DO) (DO) Question: Oxygen is to be supplied for yeast production. If oxygen uptake rate (OUR) is 15g/l medium-h for a required yeast growth, and the oxygen transfer rate (OTR) is 10 g/l medium-h. Is such oxygen transfer rate sufficient to maintain the required yeast growth? If the required growth has to be maintained, how to improve the oxygen transfer rate? Answers: Answers: OUR=15g/l OUR=15g/l medium-h medium-h > OTR=10 OTR=10 g/l medium-h medium-h insufficient oxygen supply rate. Oxygen transfer rate is limiting. Increase k La so that
g X
Y X / O2
k L a (C * C )
Cells Growth in Continuous Culture Continuous culture: fresh nutrient medium is continually supplied to a well-stirred culture and products and cells are simultaneously withdrawn. At steady state, concentrations of cells, products and substrates are constant. In batch culture: the culture environment changes continually. growth, product formation and substrate utilization terminate after a certain time interval.
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A continuous-cultur continuous-culture e laboratory laboratory setup
medium
Products, cells
Ideal Chemostat
Same as perfectly mixed continuous-flow, stirred-tank reactor (CFSTR). - Control elements: elements: pH, dissolved dissolved oxygen, temperature temperature - Fresh sterile sterile medium is fed to the completely completely mixed and aerated (if required) reactor. - Suspension Suspension is removed removed at the same rate. rate. - Liquid volume volume in the reactor reactor is kept constant. constant.
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g
g
g
net M M
net M
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Cell Growth in Ideal Chemostat At steady state, X0=0, kd ≈ 0, qp=0 , Monod equation applied, g D
m S
S
S
K S S
M
K D m
M
X Y X / S
( S 0 S ) Y
X / S
( S 0
D
K s D ) m D
Cell Productivity: DX d ( DX )
0 Dopt
dD
Dopt m (1
K s ) K s S 0
M
X opt Y
X / S
( S 0 K s ( K s S 0 ) K s )
Cell Growth in Ideal Chemostat Washed out: If D is set at a value greater than µ m (D > µm), the culture cannot reproduce quickly enough to maintain itself. 4
0.3
µm = 0.2
3.5
DX
hr -1
0.25
3 0.2
) 2.5 L / g (
X
2
S
0.15
X , S1.5
0.1
) r h L / g ( X D
1 0.05
0.5 0
0 0
0.05
0.1
0.15
0.2
0.25
D (1/hr)
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When the endogenous metabolism is not negligible, and product formation is negligible, k d0; qp =0 D g k d
g
substrate S
Cell:
K s ( D k d ) m
D k d
m
K s S
X Y x M / s ( S 0 S )
D k d
Re-organize cell equation D( S 0 S )
Y x M / s
Y X / S 1 AP
Y X / S
1 M
Y X / S 1 M
Y X / S
D k d
cell growth
maintenance
Experimental S and X data collected at varying D could be used to find constants Y X/S and m
Divide by D AP
D
g X
S S D k d D 0 0 X Y x M / s 1
S
Maintenance:
k d M
Y X / S D
m s
m s D
k d M
Y X / S
Cells Growth in Continuous Culture Ideal Chemostat Endogenous metabolism (X ( X0=0 =0,, kd > 0, qp=0) is considered
1 AP
Y X / S
1 M
Y X / S
m s D
M Y X / S and m s can be obtained from chemostat experiment s AP by plotting 1/ Y X AP / S (Y X / S
X
) against 1/D. S0 S
Then k d can be obtained from
m s
k d M
Y X / S
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If the endogenous metabolism and product formation are all present, kd0; qp 0
D substrate
S
m
S
K s S
k d
K s ( D k d ) m
D k d
D( S 0 S ) (
Cell:
m
net M Y X / S
q Y X / p
k d M Y X / S
D
X Y x M / s ( S 0 S )
D k d q p cell growth
) X 0
maintenance
Y x M / s Y p / s Product formation
Example CSTR problem •
Calculate Calculate the the substrat substrate e concentra concentration tion in the feed feed (g/L) (g/L) of a CSTR CSTR to produce 5 g/L cell biomass. Assume the feed is sterile, there is no product in the feed stream, cell death is negligible, product formation is negligible, maintenance is negligible and the system is at steady state. You are given the following data for the system and microorganism: Volume Volume = 1000L Feed rate = 50 L/hr μm = 0.1 hr -1 Ks = 1 (g/L) Yx/s= 0.5
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Example: A new strain of yeast is being considered for biomass production. The following data were obtained using a chemostat. An influent substrate concentration of 800 mg/L and an excess of oxygen were used at 1 ph of 5.5 and T=35C. Using the following data, calculate u max, ks, Yx/sM, and ms, assuming net= maxS/(k s+S)-k d Dilution rate (hr-1)
Substrate(mg/L)
Cell Conc. (mg/L)
0.1
16.7
366
0.2
33.5
407
0.3
59.4
408
0.4
101
404
0.5
169
371
0.6
298
299
0.7
702
59
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Flow cytometer (FACS) • Flore Floresc scent ent antib antibodi odies es are added to culture and cells in small droplets are sent through a detector single file • Comp Comput uter erss coun countt and and characterize cells as they pass, and deflect cells with desired characteristics • Can Can cou count nt and and kee keep p live live cells cells •
flowflow-cy cytom tometr etry. y.de/ de/img img/fc /fcm. m.gif gif
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Summary of Growth Kinetics - Autocatalytic Autocatalytic reaction: reaction: The rate of growth is directly related to cell concentration Net specific growth rate (1/time): net
1 dX X dt
net
g k d
- Cell concentra concentration tion determinatio determination n - Growth patterns patterns and kinetics kinetics in batch batch culture - lag phase phase X μ net t , d - logrithmic or or exponential exponential growth phase: phase: ln X 0 - decelera deceleration tion phase phase - stationary stationary phase: endogenou endogenous s metabolism dX k d X dt - death death phase phase
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Summary of Growth Kinetics - Effe Effect ct of fact factor ors: s: - Dissolved oxygen : oxygen consumption rate :
g X
Y X / O2
oxygen transfer rate : k L a(C * C )
- Temperature, pH, ionic strength, substrate concentration .
- Heat evolution: H s 1 H c Y Y X / S
H
Summary of Growth Kinetics -Cell growth in continuous culture: qp=0 =0,, kd >0, X0=0, Monod equation is applied applied:: m S net g k d k d Ks S Ks ( D k d ) g D k d ; S m D k d X Y M ( S 0 S ) D X / S D k d Productivities: DP, DP, DX m 1 1 s AP M D Y Y X / S
X / S
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Summary of Growth Kinetics -Cell growth in continuous culture: qp>0 >0,, kd >0, X0=0, Monod equation is applied applied::
DP q p X X Y M ( S 0 S ) X / S
D D k d q p
g D
S
k d
M Y X / S
Y p / s
Ks ( D k d ) m
D k d
Productivities: DP, DP, DX
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