Chapter 6: How Cells Grow
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1 Chapter 6: How Cells Grow David Shonnard Department of Chemical Engineering 1 Presentation Outline: Introduction Batch Growth Characteristics Growth Stages, Effects of Environmental Conditions, Product Formation, Mathematical Models Continuous Growth Characteristics Dilution Rate, Optimum Operation 2 1
2 Introduction Cell growth is the primary response of viable cells to substrates and nutrients. Substrates/nutrients + cells products + more cells specific growth rate (h -1 ), µ 1 dx X dt X = cell mass concentration (g / L) t = time (h) Product formation is a secondary response. 3 Determining Cell Concentration 1. Cell number concentration a) hemocytometer (Petroff-Hausser slide) b) viable cell counts (petri dish) c) electronic particle counter Bioprocess Engineering: Basic Concepts Shuler and Kargi, Prentice Hall,
3 Determining Cell Concentration (cont.) 2. Cell mass concentration a) direct methods dry weight (filtration or centrifugation) packed cell volume (centrifugation) optical density (light scattering, nm) 5 Determining Cell Concentration (cont.) 2. Cell mass concentration (cont.) a) indirect methods measure biomolecule concentration and correlate to dry cell mass concentration. (DNA, protein, ATP, NADH, product formation) Example 1. NH 4+ utilization during growth releases H +, amount of OH - added is proportional to growth. Example 2. Luciferin + O 2 + ATP luciferase light 6 3
4 Batch Growth Curve Inoculum [X o ] [S o ] growth medium (substrate + nutrients) Batch Reactor X = cell Concentration (cells/ml) Bioprocess Engineering: Basic Concepts Shuler and Kargi, Prentice Hall, Lag Phase no increase in cell numbers induction of enzymes to utilized substrate(s) very important to decrease lag period to productivity i. Inoculate with exponential phase cells ii. Pre-acclimate inoculum in growth media iii. Use high cell inoculum size (5-10% by volume) 8 4
5 Exponential Growth Phase 1. Nutrient and substrate concentrations are large 2. Growth rate is independent of nutrient and substrate conc. 3. Cell number and mass concentrations increase exponentially dx dt = µ max X, X = X o at t = 0 ln X X o o o o X = X o e µ max t or ln X X o = µ max t o Slope = µ max t doubling time of cells (t d ), X X o = 2 ln(2) = µ max t d t d = ln 2 µ max or µ max = ln 2 t d 4. Balanced growth occurs cell composition constant 9 Deceleration Phase depletion of one or more nutrients accumulation of toxic byproducts of growth unbalanced growth and metabolism shifts for survival 10 5
6 Stationary Phase no net growth of cell numbers or cell mass (no cell division) cell growth rate = cell death rate secondary metabolites (products) produced endogenous metabolism of energy stores can result in maintaining cell viability removal of inhibitory compounds will result in further growth if additional substrate is provided 11 Death Phase 1. Cell lysis (spillage) may occur 2. Rate of cell decline is first-order dx dt = -k d t, X=X S at t = 0 X = X S e -k d t or ln X X o = -k d t 3. Growth can be re-established by transferring to fresh media 12 6
7 Effects of Temperature on Cell Growth µ max doubles for each 10Û&ÃLQFUHDVHÃQHDUÃT opt (µ max -k d ) µ max = A e -Ea / RT and k d = A e -E d /RT E a = activation energy for growth 10 to 20 kcal / mole E d = activation energy for death 60 to 80 kcal / mole As T, k d faster than µ max, (µ max -k d ) Bioprocess Engineering: Basic Concepts, Shuler and Kargi, Prentice Hall, ph Effects acceptable ph is ± 1 to 2 ph units ph range varies by organism bacteria (most) ph = 3 to 8 yeast ph = 3 to 6 plants ph = 5 to 6 animals ph = 6.5 to 7.5 microorganism have the ability to control ph inside the cell, but this requires maintainance energy ph can change due to utilization of substrates; NH 4+ releases H +, NO 3- consumes H + production of organic acids, amino acids, CO 2, bases 14 7
8 ph Effects (cont.) Bioprocess Engineering: Basic Concepts Shuler and Kargi, Prentice Hall, Dissolved O 2 Effects O 2 may be a limiting substrate for aerobic fermentation, since O 2 is sparingly soluble in water critical O 2 concentration 5 to 10% of saturation ( 7 mg O 2 /L) for bacteria/yeast growth exhibits saturation kinetics with respect to O 2 concentration (see next page) 16 8
9 Dissolved O 2 Effects (cont.) Obligate aerobic cells Saturation kinetics Saturation kinetics Facultative aerobic cells µ * = anaerobic µ - µ max µ aerobic anaerobic max - µ max Bioprocess Engineering: Basic Concepts Shuler and Kargi, Prentice Hall, Other Effects on Cell Growth dissolved CO 2 (DCO 2 ); to high or low DCO 2 is toxic ionic strength (I); too high dissolved salts is inhibitory to membrane function (membrane transport of nutrients, osmotic pressure) I = 1/2 Σ C i Z i 2 C i = molar concentration of ion i Z i = ion charge maximum non-inhibitory concentrations of substrates, products glucose (100 g/l), ethanol (10 g/l), NH 4+ (5 g/l),
10 Stoichiometric Coefficients for Growth Yield coefficients, Y, are defined based on the amount of consumption of another material Growth Yield Y X/ S = - X S S = S assimilated into biomass + S assimilated into extracellular products + S growth energy + S maintenance energy Because S changes with growth condition, is not a constant 19 Stoichiometric Coefficients for Growth (cont.) Typical range of yield coefficients Growth Yield Y X/ S = - X S 0.4 to 0.6 g dry cells/g substrate consumed, oxidized S (0.4 to 0.6 ) <, reduced S (0.6 to 1.0) Other Yield Coefficients: Y X/ O2 = - X O 2 Y X/O2, reduced S (0.15 to 0.3 ) < Y X/ O2, oxidized S (0.3 to 1.5) 20 10
11 Stoichiometric Coefficients for Growth (cont.) Other Yield Coefficients: Y X/ ATP = X ATP ( g dry cells mole ATP generated ) Y X/ATP > complex medium (a.a.s and nucleic acids available) Y X/ATP < minimal medium (only inorganic salts and substrate) Anaerobic Fermentations: Y X/ATP 10.5±2 (g dry cell wt./mole ATP) C 6 H 6 O 6 2 C 2 H 5 OH + 2 CO 2 : 2 moles ATP/mole glucose Y X/ATP (2)/MW glu = (10.5)(2)/180 = 0.12 g dcw/g glucose 21 Product Yield Coefficients 1. Growth associated products: products appear simultaneously with cells in culture 1 dp q P = = Y P/ X µ X dt 2. Non-growth associated products: products appear during stationary phase of batch growth q P = β 3. Mixed-growth associated products: products appear during slow growth and stationary phase q P = αµ +β 22 11
12 Product Yield Coefficients (cont.) 1. q P = 1 X dp dt = Y P/ X µ 3. q = α µ +β 2. q = β P P Bioprocess Engineering: Basic Concepts Shuler and Kargi, Prentice Hall, Heat Generation by Growth Only 40 to 50% of the energy stored in a carbon substrate is converted to biological energy (ATP) during aerobic metabolism. The remainder is released as heat upon conversion to CO 2 and H 2 O Energy Balance: I Total Available Energy of Substrate = II Energy Released by Growth + III Energy Available in Biomass Substrate + O 2 1/Y H (kj/g dcw) II CO 2 + H 2 O + Cells I Combustion H S (kj/g S) III CO 2 + H 2 O Combustion of Cells H C (kj/g dcw) 24 12
13 Heat Generation by Growth (cont.) On a per gram of substrate basis (1 g S) H S = (1 g S) /Y H + (1 g S) H C Solving for Y H Y H = ( H S - H C ) Typical H C = 20 to 25 kj/g dcw 25 Heat Generation by Growth (cont.) For Substrates: S H S (kj/g S) Y H (g dcw/kj) Glucose Methanol Ethanol n-decane Methane The oxidation state of S has a large effect on 1/ Y H 26 13
14 Rate of Heat Generation by Growth, Q Gr Q Gr = V L µx 1 Y H ( kj hr ) Liquid Volume Cell Mass Concentration Specific Growth Rate of Cells Heat can be removed by circulating cooling water through an external jacket around the reactor vessel or through a coiled tube within the reactor. 27 Modeling Cell Growth Structured Models: Model divides cell mass into components (by molecule or by element) and predicts how these components change as a result of growth. These models are very complex and not used very often. Unstructured Models: Model assumes balanced growth where cell components do not change with time. Much less complex and much more commonly used. Valid for batch growth during exponential growth stage and also for continuous culture during steady-state operation
15 Monod Equation Similar to Michaelis-Menten Kinetics Assumes that a single enzyme system is responsible for the uptake of substrate (S), and that S is limited (growth-dependent variable). This is the most common kinetic model for cell growth. µ = µ m S K S +S µ = specific cell growth rate (hr -1 ) µ m = maximum specific cell growth rate (hr -1 ) S = substrate concentration (g/l) K S = Saturation constant (g/l) = S when µ = 1/2 µ m. 29 Batch Culture Growth Model µ = 1 X dx dt = µ m S K S +S We relate changes in S to changes in X through X - X o = (S o -S), or S = S o + X o / -X/ = cell mass yield (g dcw/g S consumed) X o, S o = initial concentrations of cells and substrate dx dt = µ m (S o + X o -X) (K S +S o + X o -X) X ; at t = 0, X = X o 30 15
16 Batch Culture Growth Model (cont.) Logistic Equation (K S +S o + X o ) (S o + X o ) ln X X - o K S (S o + X o ) ln { (S o +X o -X)S o }= µ m t Bioprocess Engineering: Basic Concepts, Shuler and Kargi, Prentice Hall, Batch Growth Data and Monod Parameters Though the Logistic Equation qualitatively predicts the shape of batch growth, it is not very useful when attempting to determine K S and µ max from X versus time data. K S is determined differently. K S is equal to S when µ = 1/2 µ max ln X X o X o + S o o o o o o o o Slope = µ max t µ = 1/X dx/dt needs to be determined from available data, especially data at low S concentrations. µ = 1 dx X dt Bioprocess Engineering: Basic Concepts Shuler and Kargi, Prentice Hall, 2002 µ max o o o o µ = 1/2 µ o max o K S S o 32 16
17 6.4 Cell Growth in Continuous Culture Automated Chemostats control of ph, temp. agitation, dissolved oxygen sterilization required Bioprocess Engineering: Basic Concepts Shuler and Kargi, Prentice Hall, Chemostat as a Tool evaluate K S, µ max, and other system parameters study changes in environment and effects on cell physiology select for cells with desired metabolic capabilities (e.g. selection for cells capable of degrading a toxic compound) 34 17
18 Chemostat Mass Balance Why derive mass balance equation? 1. Describe dynamics of cell growth, substrate utilization, and product formation. 2. Useful for control of bioreactors. 3. Evaluate kinetic and yield parameters. 4. Determine the optimum values for bioreactor operating parameters. 35 Ideal Constant-Stirred Tank Reactor Chemostat Bioprocess Engineering: Basic Concepts Shuler and Kargi, Prentice Hall,
19 Mass Balance Statement for Cell Mass mass rate of cells into - bioreactor or mass rate of cell mass rate mass rate growth without of cells out + of cell loss - = endogenous by endogenous of bioreactor metabolism metabolism mass rate of cells accumulation in bioreactor FX o - FX + V R µx - V R k d X = V R dx dt F = in and out volumetric flow rate (L / hr) X = bioreactor and outlet cell mass concentration (g/ L) X o = inlet cell mass concentration (g/ L) µ = specific cell growth rate neglecting endogenous metabolism (hr -1 ) k d = endogenous cell loss rate constant (hr -1 ) 37 Steady-State and Sterile Feed Chemostats are normally operated at steady-state, d/x/dt = 0. Assume a sterile feed (X o = 0), and k d is so small that is neglected, k d = 0. The cell mass balance equations becomes, mass rate of cells out = of bioreactor or mass rate of cell growth without endogenous metabolism FX = V R µx F V R = µ or D = µ where F V R = D, dilution rate 38 19
20 Substrate Concentration Using the Monod Equation, we can predict the bioreactor and outlet stream concentration of Substrate. µ = µ max S K S + S = D rearranging, S = K S D µ max - D 39 Bioreactor Washout Condition There is an upper limit on D, or the cells will washed out of the bioreactor. D µ max S o K S + S o 40 20
21 With Endogenous Metabolism If endogenous metabolism is considered, it is left as an exercise for the students to show that (for k d ž 0) D = µ - k d and S = K (D + k ) S d µ max - D - k d Why is S higher than the case when k d = 0? Answer: X is lower! 41 Cell Concentration How is X affected by D? A similar mass balance equation for S in the absence of endogenous metabolism is written to answer this question. FS o - FS - V R µx 1 Y - V M R q px 1 = V ds X/S Y R P/S dt S = bioreactor and outlet substrate concentration (g / L) S o = inlet substrate concentration (g/ L) YM X/S = maximum cell yield coefficient (g cells / g substrate) Y P/ S = product yield coefficient (g product / g substrate) q p = specific rate of extracellular product formation g P g cells hr 42 21
22 Cell Concentration (cont.) For the simple case of no product formation (q p =0), steady-state (ds/dt=0), and no endogenous metabolism, k d =0. D(S o - S) = µx M at steady -state, µ = D, and solving for X, X = Y M X/S (S o - S) or X = Y M X/S (S o - K S D µ max - D ) 43 Effects of Endogenous Metabolism Thus far, the substrate balance eqn. Has been written assuming M that is a constant at. D (S o - S) - (D+k ) d X M = 0 With endogenous metabolism, rearranging, µ = D + k d and with no extracellular product D (S o - S) - D X formation, the substrate mass YM - k d X/S YM = 0 X/S balance is at steady-state, and D where YAP - D X/S YM - k d k X/S YM = 0 m S = d M X/S 1 maintenance coefficient Y = 1 AP X /S Y + m S M D = 0 X/S based on S
23 Measurement of Maximum Cell Yield and Maintenance using a Chemostat From measurements of X, S, S o, and D in a chemostat experiment at different D values, a double reciprocal plot can be made. with m S = k d M M k d = m S Bioprocess Engineering: Basic Concepts Shuler and Kargi, Prentice Hall, 2002 M 45 Using a Chemostat to Determine µ max and K S From data collected using a chemostat, we can obtain the Monod Equation kinetic parameters. Data include S at several Dilution Rates (D), Recall that, D = µ k d D = µ max S K S + S k d rearranging 1 1 = + K S 1 D+k d µ max µ max S 1 D+k d o o o o slope = K S µ max intercept = 1 µ max 1 S 46 23
24 Productivity of a Chemostat Pr X = productivity for cell production = DX Pr P = productivity for product formation = DP The dilution rate (D) which maximizes productivity is found by taking dpr/dd = 0 and solving for D (D optimum ). For example, D optimum for X with k d = 0 and q P = 0 X = Y M X/S (S o K S D µ max -D ) DX = take d(dx) dd = 0 and solve for D (D opt ) D opt = µ max (1- K S K S +S o ) M D(S o K S D µ max -D ) K S is usually << S so D opt ~ µ max (washout point) 47 Product Mass Balance dp FP o - FP + V R q P X = V R dt at steady - state, dp / dt = 0 and for P o =0 DP = q P X or P = q X P D for k d = 0, no endogenous metabolism S = K D S from X mass balance µ max -D X = Y M X/S D (S o -S) M from S mass balance Y (D + q X/S P ) Y P/S 48 24
25 Product Mass Balance for k d 0, with endogenous metabolism S = K S (D + k d ) (µ max -D-k d ) X = Y M X/S from X mass balance D (S o -S) M from S mass balance Y (D + k d +q X/S P ) Y P/S to determine D for optimum P formation, d(dp) dd = 0 solve for D opt 49 A Summary of Chemostat Response to D X S o Conc. of X or S D opt DX (g/l hr) D (hr -1 ) washout at D = µ max S o K S +S o 50 25
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