Modeling Microbial Populations in the Chemostat

Modeling Microbial Populations in the Chemostat Hal Smith A R I Z O N A S T A T E U N I V E R S I T Y H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 / 34

Outline Why study microbial ecology? 2 Microbial Growth Growth under nutrient limitation, Monod(942) 3 Continuous Culture: The Chemostat Microbial Growth in the Chemostat Competition for Nutrient Competitive Exclusion Principle The Math supporting the CEP 35 year old Open Problem H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 2 / 34

Why study microbial ecology? Why study microbial ecology? The study of the growth of bacterial cultures does not constitute a specialised subject or a branch of research: it is the basic method of microbiology. J. Monod, 949 quantify microbial growth quantify antibiotic efficacy microbial ecology bio-engineering: synthetic bio-fuels waste treatment bio-remediation biofilms, quorum sensing mammalian gut microflora food, beverage (beer,wine) H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 3 / 34

Microbial Growth Growth under nutrient limitation, Monod(942) Growth rate and nutrient concentration J. Monod, The growth of bacterial cultures, Annu. Rev. Microbiol. 949 H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 4 / 34

Microbial Growth Growth under nutrient limitation, Monod(942) Growth under Nutrient Limitation, Monod(942) Monod s experimental data on bacterial growth rate as a function of nutrient concentration led him to propose: specific growth rate dn N varies with S = [glucose] approx. as some r > 0 and a > 0. dn N = rs a+s 2 growth rate and nutrient consumption rate are proportional dn = γds 3 This implies that nutrient depletion is: ds = γ rsn a+s French biologist and nobelist Jacques Monod led an interesting life. Check it out on: http://en.wikipedia.org/wiki/main_page H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 5 / 34

Microbial Growth Growth under nutrient limitation, Monod(942) The anatomy of Monod s function G = r S a+s.8.6 r = maximum rate.4 Growth Rate.2 0.8 0.6 a = half saturation 0.4 0.2 0 0 2 3 4 5 Nutrient S Alternative functions, having the same general shape, have been proposed. Monod s function can be derived from enzyme kinetics: see wikipedia. r and a can be inferred from least squares applied to /G = (a/r)(/s) +(/r). H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 6 / 34

Microbial Growth Growth in Batch Culture* Growth under nutrient limitation, Monod(942) S-Nutrient. N-Bacteria. 2.5 Batch Growth No maintenance S N ds dn = rsn γ a+s = rsn a+s 2.5 0.5 *Batch culture is jargon for a closed-system culture, i.e., a covered Petri dish 0 0 2 3 4 5 6 7 time in hours H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 7 / 34

Microbial Growth Growth under nutrient limitation, Monod(942) The principle of conservation of nutrient ds dn = rsn γ a+s = rsn a+s Total nutrient, nutrient bound up in microbes plus free nutrient, is conserved: ( ) d N γ + S = 0 N(t) + S(t) = N(0) + S(0) γ γ H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 8 / 34

The Chemostat: see Google images H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 9 / 34

enter Chemostat into http://en.wikipedia.org/wiki/main_page 0 DS D(S+x) Substrate V Biomass H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 0 / 34

The Old Tank Problem-No Bacteria V = Volume of chemostat(ml) F = Inflow = Outflow rate (ml/hr) S 0 = Concentration of Substrate in Feed (gm/ml). S = Concentration of Substrate in Chemostat (gm/ml). Rate of change of Substrate (gm/hr)= INFLOW(gm/hr) - OUTFLOW(gm/hr) d (VS) = FS0 FS Let D = F/V be the Dilution Rate. Then ds = D(S 0 S) Solution: S(t) = S(0)e Dt + S 0 ( e Dt ) S 0 H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 / 34

Classical Chemostat Model Novick & Szilard, 950. Microbial Growth in the Chemostat ds dn = D(S 0 S) }{{} γ dilution = rsn a+s }{{} growth DN }{{} dilution rsn a+s }{{} consumption Environmental parameters: dilution rate D = F/V. 2 nutrient concentration in inflow S 0. Biological parameters: maximal growth rate r. 2 half-saturation concentration a. 3 yield γ. H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 2 / 34

Microbial Growth in the Chemostat Break-even nutrient level for survival of microbes when dn N = rs a+s D = 0 S = λ = ad r D.8.6.4 D = Dilution rate.2 Growth Rate 0.8 0.6 0.4 Break even S 0.2 0 0 2 3 4 5 Nutrient S H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 3 / 34

Equilibria Continuous Culture: The Chemostat Microbial Growth in the Chemostat 0 = D(S 0 S) rsn γ a+s ( ) rs 0 = a+s D N Washout Equilibrium: N = 0 and S = S 0. Survival Equilibrium: rs a+s = D, i.e., S = λ and N = γ(s0 λ). Positive survival equilibrium exists iff rs 0 a+s 0 > D. H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 4 / 34

Microbial Growth in the Chemostat Phase Plane: Survival Equilibrium does not exist S = D ( S) (2 S N)/(0.3 + S) N = (2 S N)/(0.3 + S) D N D =.65 2.8.6.4.2 N 0.8 0.6 0.4 0.2 0 0 0.5.5 S H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 5 / 34

Microbial Growth in the Chemostat Phase Plane: Survival Equilibrium exists S = D ( S) (2 S N)/(0.3 + S) N = (2 S N)/(0.3 + S) D N D = 2.8.6.4.2 N 0.8 0.6 0.4 0.2 0 0 0.5.5 S H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 6 / 34

Stability of Washout Equilibrium Microbial Growth in the Chemostat Small perturbations from the washout equilibrium obey the linearized system : (S, N) = (S 0, 0)+(y, y 2 ), y = (y, y 2 ) small perturbation y = Jy where J is the jacobian matrix at the washout equilibrium: ( ) D f(s J = 0 )/γ 0 f(s 0 ) D The eigenvalues are D and f(s 0 ) D, where f(s) = rs a+s. The washout equilibrium is stable if rs 0 a+s 0 < D and unstable if rs 0 a+s 0 > D. H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 7 / 34

Survival or Washout Continuous Culture: The Chemostat Microbial Growth in the Chemostat If the inflow nutrient supply is sufficient for the microbe to grow: rs 0 a+s 0 > D, 5 4.5 4 Operating Diagram then the bacteria survive: 3.5 3 N(t) γ(s 0 λ), S(t) λ, S 0 2.5 Survival Indeed, they are reproducing at the exponential rate set by dilution rate D. 2.5 Extinction 0.5 Otherwise, they are washed out: N(t) 0, S(t) S 0 0 0 0.5.5 2 2.5 D survival boundary: D = rs0 a+s 0. H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 8 / 34

Phase Plane Continuous Culture: The Chemostat Microbial Growth in the Chemostat S = S m S x/(a + S) x = x (m S/(a + S) ) m = 2 a = 0.5 2.8.6.4.2 x 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8.2.4.6.8 2 S H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 9 / 34

The conservation principle Microbial Growth in the Chemostat ds dn = D(S 0 S) γ = rsn a+s DN rsn a+s Multiply N-eqn. by γ and add to S eqn.: [ d S(t)+ N(t) ] ( [ = D S 0 S(t)+ N(t) ]) γ γ which implies that [ S(t)+ N(t) ] [ = S(0)+ N(0) ] e Dt + S 0 ( e Dt ) γ γ Solution trajectory approaches line S + N γ = S0. H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 20 / 34

Competing Strains of Bacteria Competition for Nutrient ds dn dn 2 = D(S 0 S) r N S γ a + S r 2 N 2 S γ 2 a 2 + S ( ) r S = a + S D N ( ) r2 S = a 2 + S D N 2 Exploitative Competition: each organism consumes a common resource. H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 2 / 34

Break-even concentrations Competition for Nutrient dn N dn 2 N 2 = = r S a + S D = 0 S = λ = a D r D r 2 S a 2 + S D = 0 S = λ 2 = a 2D r 2 D.8 Break even values.6 D.4.2 Growth Rate 0.8 0.6 lambda2 0.4 lambda 0.2 0 0 2 3 4 5 Nutrient S λ = λ 2? Coexistence at Equilibrium is Extremely Unlikely! H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 22 / 34

Competition for Nutrient No Coexistence Equilibria if λ λ 2 N wins: S = λ, N = γ (S 0 λ ), N 2 = 0 N 2 wins: S = λ 2, N = 0, N 2 = γ 2 (S 0 λ 2 ) H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 23 / 34

Competitive Exclusion Principle Competitive Exclusion Principle: check it out on wikipedia Assume each species can survive alone in the chemostat ( r i S 0 With no loss in generality, assume: a i +S 0 > D.) a D r D = λ < λ 2 = a 2D r 2 D Then N wins: N (t) γ (S 0 λ ), N 2 (t) 0, S(t) λ Winner is the organism that can grow at the lowest nutrient level. The winner of competition for nutrient is determined by quantities which may be measured by growing each organism separately in the chemostat Mathematical Proof: Hsu,Hubbell,Waltman (977); Experimental Test: Hansen & Hubbell (980) H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 24 / 34

Conservation Principle The Math supporting the CEP [ d S(t)+ N (t) + N ] ( [ 2(t) = D S 0 S(t)+ N(t ) + N ]) 2(t) γ γ 2 γ γ 2 which implies that S(t)+ N (t) γ This suggests setting S(t)+ N (t) γ + N 2(t) γ 2 + N 2(t) γ 2 S 0. = S 0 and x i = N i(t) γ i : where dx dx 2 = ( f (S 0 x (t) x 2 (t)) D ) x = ( f 2 (S 0 x (t) x 2 (t)) D ) x 2 f i (S) = r is a i + S H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 25 / 34

The Math supporting the CEP Stability of Equilibria: 0 < λ < λ 2 < S 0 The Jacobian Matrix evaluated at the non-zero equilibria. At ˆx = S 0 λ, ˆx 2 = 0 J = ( ˆx f ˆx f 0 f 2 (λ ) D f 2 (λ ) D < f 2 (λ 2 ) D = 0 so diagonal entries are negative. Asymptotically Stable. ) At x = 0, x 2 = S 0 λ 2 ( f (λ Jacobian = 2 ) D 0 x 2 f 2 x 2 f 2 f (λ 2 ) D > f (λ ) D = 0 so red entry is positive. Unstable Saddle. ) H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 26 / 34

No Periodic Orbits Continuous Culture: The Chemostat The Math supporting the CEP Dulac s Criterion satisfied: (f D)x + ( ) (f 2 D)x f = + f 2 < 0 x x x 2 x 2 x x 2 x 2 x Learn about Dulac s Criterion at: http://www.scholarpedia.org/article/encyclopedia_of_dynamical_systems H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 27 / 34

The Phase Plane of Competition The Math supporting the CEP x =.2 x ( x x2)/(.0 x x2) x x2 =.5 x2 ( x x2)/(. x x2) x2 0.9 0.8 0.7 0.6 x2 0.5 0.4 0.3 0.2 0. 0 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 28 / 34

The Math supporting the CEP Winner may depend on dilution rate.8 Break even values.6 D.4.2 Growth Rate 0.8 0.6 lambda2 0.4 0.2 lambda 0 0 2 3 4 5 Nutrient S λ 2 < λ so N 2 wins. By increasing D a bit, green line goes up, so λ < λ 2 and N wins! H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 29 / 34

Species Wins when D =.6 The Math supporting the CEP x =.2 x ( x y)/(.0 x y) D x y =.5 y ( x y)/(. x y) D y D =.6 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0. 0 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 30 / 34

Species 2 Wins when D =.7 The Math supporting the CEP x =.2 x ( x y)/(.0 x y) D x y =.5 y ( x y)/(. x y) D y D =.7 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0. 0 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 3 / 34

35 year old Open Problem Famous Open Problem General competition model with species-specific removal rates D j Hypotheses: S S 0, D, D j > 0. 2 f j (0) = 0, f j (S) > 0. = D(S 0 S) n f j (S)x j j= x j = (f j (S) D j )x j, j n. 3 Break-even nutrient values λ j : f j (λ j ) = D j. Conjecture: If λ < λ 2 λ j < S 0, j 2, then: x (t) S 0 λ, x j (t) 0, S(t) λ. H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 32 / 34

35 year old Open Problem Proofs of the Competitive Exclusion Principle. Author(s) and Date Hsu, Hubbel, Waltman 977 Hsu 978 Armstrong & McGehee 980 Butler & Wolkowicz 985 Wolkowicz & Lu 992 Wolkowicz & Xia 997 Li 998,999 Liu et al 203 Your Name, Date Hypotheses D = D i and f i Monod f i Monod D i = D, f i monotone D i = D, f i mixed-monotone D i D, f i mixed-monotone, add. assumptions D i D small, f i monotone D i D small, f i mixed-monotone allows time delays for nutrient assimilation D i D, f i monotone mixed-monotone means one-humped. H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 33 / 34

References Continuous Culture: The Chemostat 35 year old Open Problem The Theory of the Chemostat, Smith and Waltman, Cambridge Studies in Mathematical Biology, 995 Microbial Growth Kinetics, N. Panikov, Chapman& Hall, 995 Resource Competition, J. Grover, Chapman& Hall, 997 H.L. Smith (ASU) Modeling Microbial Populations in the Chemostat MBI, June 3, 204 34 / 34