Resting Stages and the Population Dynamics of Harmful Algae in ...

Resting Stages and the PopulationDynamics of Harmful Algae in BatchCultures and ChemostatsWilber Ventura, Tyler RandolphJayce Rodriguez, Alicia Prieto LangaricaBetty Scarbrough, Hristo KojouharovJames GroverTechnical Report 2011-19http://www.uta.edu/math/preprint/

Resting Stages and the Population Dynamics ofHarmful Algae in Batch Cultures and Chemostats ∗Wilber Ventura † Tyler Randolph † Jayce Rodriguez ‡Alicia Prieto Langarica † Betty Scarbrough ‡Hristo Kojouharov † James Grover ‡September 7, 2011AbstractThe unicellular species Prymnesium parvum, known as golden algae,releases potenttoxinsintoitsenvironment,whichcangreatlyupsetaquaticpopulation dynamics. P. parvum are understood to have a motile statewhich produces toxin and a much less metabolically active state whichdoes not produce much toxin, which may be a cyst. Our research attemptsto provide a mathematical model for the conversion between themotile and non-motile states of P. parvum, in two different cultures. Inour model, population growth is a saturating function of nutrient concentration.For both the batch and chemostat cultures, we used systems ofdifferential equations to further understand the population dynamics ofP. parvum. Specifically, we observed the steady states of both cell populations(motile and non-motile) and the stability of those equilibria toshow how conversion rates influence overall population dynamics.1 IntroductionPrymnesium parvum are flagellates that are ovoid in shape (Green et al., 1982).They havealsobeen observedasamixotrophicspecies, both photosyntheticandheterotrophic (Tillman, 2003). P. parvum are euryhaline and euthermic, thereforetolerant of a wide range of temperatures and salinity (Larsen and Bryant1998). Certain environmental conditions and their effect on bloom dynamicshave also been studied. Results of previous research have provided the optimalgrowth conditions in lab media and established parameters for our research∗ This research was supported by an NSF UBM-Institutional grant DUE#0827136 as partof the UTTER Program at UT Arlington (http://www.uta.edu/math/utter/).† Department of Mathematics, The University of Texas at Arlington, P.O. Box 19408,Arlington, TX 76019-0408‡ Department of Biology, The University of Texas at Arlington, P.O. Box 19498, Arlington,TX 76019-04981

(Baker et al., 2007). It was discoveredthat P. parvum have the ability to deploya poison that paralyzes prey, prevents grazing, and reduces competition especiallyduring nutrient limited environments (Tillman, 2003; Roelke et al., 2007;Graneli et al., 2008; Brooks et al., 2010). This poison was found to be not one,but multiple poisons that are released simultaneously whose chemical structureshave yet to be clearly defined (Igarashi et al., 1999). These toxins can have verypotent effects on non-planktonic species, especially fish, causing massive fishkills numbering over 30 million in total (Southard et al., 2010). Toxic eventsduring winter may be especially severe since temperature and salinity are farfrom optimal for P. parvum, which appears to enhance its toxicity (Baker etal. 2007, 2009). P. parvum have also shown that, during certain stressful conditions,it can assume a markedly reduced metabolic state. In this paper thisstate is referred to as the non-motile state, which may possibly be a cyst formation(Green et al., 1982). Therefore, for analytical purposes, we assume thatthis non-motile state has negligible rates of reproduction, nutrient consumption,toxin release, and meaningful movement compared to those of the active, motilestate. Thus, a better understanding of this non-motile state may shed light onharmful algal bloom cycles and environment-dependent toxin release of the P.parvum algae.It has been established that the success of a newly introduced organismdepends on its ability to reproduce successfully. Since environmental factorsplay a large role in P. parvum bloom dynamics, our study attempts to modelcertain environments to observe motile and non-motile cell transitions. The twoenvironments used for the model are batch and chemostat cultures. The batchculture allows an organism to be isolated in a controlled environment with alimited source of a specific nutrient. Since the batch is a closed system, neworganisms and nutrient cannot be added. The chemostat model was establishedto show the rate of successful multiplication of a newly introduced organismin an open system (Powell, 1965). Further, this chemostat model includes adilution of a vessel in which both cell and nutrient populations are introducedand evacuated in equal amounts (Monod, 1950; Johansson and Graneli, 1999).We have proposed a theoretical model that can demonstrate the impact of cellconversion rates on population dynamics.2 Batch Culture ModelThe batchculture model usesthe variablesand parameterslisted in Table1. Werepresent the population dynamics by the following three differential equations:2

Table 1: NotationVariables Meaning UnitsM Motile algae cells/mLN Non-motile algae cells/mLR Nutrient concentration µmol/LParameters MeaningUnitsK Half-saturation constant for algal growth µmol/Lµ max maximal growth rate of motile algae day −1δ Rate of conversion from motile to non-motile day −1γ Rate of conversion from non-motile to motile day −1q Nutrient quota of algae µmol/cellR in Nutrient supply µmol/LdMdt= µ maxRMK +R−δM +γN,dNdtdRdt= δM −γN,= − µ maxRMK +R q. (1)Each differential equation represents the factors we assume to impact thechange in populations and toxicity. We assume that only motile algae (M) cantake up the nutrient (R) and reproduce; therefore there exists a relationshipbetween the growth of the motile population and the nutrient. It has been wellestablished that the growthof the algae is limited by the nutrient concentration,and the Monod function,µ max RMK +R ,has repeatedly been used to represent the interaction between the taking up ofnutrient and the growth of P. parvum [3]. In our model µ max (day −1 ) is themaximal growth rate of the motile algae and K (µmol/L) is the half saturationconstant. The consumption of the nutrient represented in dR is proportional todtthe growth rate of the motile algae in dM with coefficient q (µmol/cell), whichdtconverts the units from cells to nutrients (µmol). We also assume that motilealgae converts to non-motile algae (N) at a constant rate, δ (day −1 ), and thatnon-motile algae converts to motile algae at a constant rate, γ (day −1 ).2.1 Reduction of OrderThe system is changed from three differential equations to a much simpler systemof only two differential equations. T is the total nutrients in the system3

and therefore it is the summation of R, Mq, and Nq. The units convert bothM and N into units of nutrients by multiplying by q. ThusandT = R+Mq +Nq,dTdt = dRdt + dM dt q + dN dt q.Direct substitution in System (1) yields dT = 0, which implies that T is constant.Therefore, usingdtthatR = T −Mq −Nqthe equation for R can be eliminated and System (1) is simplified todMdtdNdt= µ maxM(T −Mq −Nq)K +T −Mq −Nq= δM −γN.−δM +γN,(2)2.2 Equilibrium EvaluationWe set the right-hand sides of all differential equations in System (2) equal tozero and solve for the equilibrium values of M and N. We obtain the solutions:E 0 B = (M0 B ,N0 B ) = (0,0)E ∗ B( ) (3)γT= (M∗ B ,N∗ B ) = (δ +γ)q , δT(δ +γ)qAlthough R is not represented in the system, it is a population of interest,with R = R 0 B = T at E0 B , and R = R∗ B = 0 at E∗ B .2.3 Equilibrium AnalysisIn the batch culture model the equilibria are always feasible. This is becausethe equilibria represent populations, which cannot be negative. The Jacobianmatrix evaluated at EB 0 for the simple batch culture model is:⎛−δ + Tµ ⎞maxγ⎜J =K +T ⎟⎝ ⎠ . (4)δThe determinant of the Jacobian matrix (4) evaluated at E 0 B is −γTµ maxK +T .The outcome of the determinant alone, because it is negative, implies that E 0 Bis unstable.−γE 0 B4

76Batchmotil cells (M)non−motil cells (N)nutrients (R)Population densities5432100 50 100 150 200 250 300Time (t)Figure 1: Batch culture: stable non-trivial equilibrium E ∗ B .The Jacobian matrix evaluated at EB ∗ for the simple batch culture model is:⎛−δ − γTµ max(δ +γ)K γ − γTµ ⎞max(δ +γ)KJ =⎜⎝δ−γ⎟⎠E ∗ B. (5)The determinant for the Jacobian matrix (5) is γTµ max, and the trace is −δ−KγTµ max− γ. Thus the determinant is positive and the trace is negative.(δ +γ)KTherefore the equilibrium is stable. Figure 1 shows that EB 0 is an unstableequilibrium and that EB ∗ is a stable equilibrium.3 Chemostat Culture ModelThe same dynamics from the batch culture model are at work in the chemostatsetting. The only difference in the models is the inflows and outflows of thesystem. The rate of the dilution D, represents the inflows and outflows of thesystem. In the inflow only nutrients come into the system thus we include apositive term DR in in dR . We assume that populations and nutrients are welldtmixed so that the outflow of each population is according to its density in thesystem. From the biological assumptions mentioned we develop the system5

dMdt= µ maxRMK +R−DM −δM +γN,dNdtdRdt= δM −γN −DN,= D(R in −R)− µ maxRMqK +R . (6)3.1 Reduction of OrderWe obtain a reduction of order of the system, evaluated as time goes towardsinfinity. Simplification of the model, by reducing the system of three differentialequations to a system of two differential equations, helps ease the stabilityanalysis.As in the batch model, we first define T as the total nutrientsand therefore the rate of change of T isT = R+Mq +Nq,dTdt = dRdt + dM dt q + dN dt q.By direct substitution from System (6) we obtain:dTdt = D(R in −T).After solving the above differential equation we obtain T as a function of time:T(t) = (T(0)−R in )e −Dt +R in . (7)As t goes to infinity, in Equation (7), T approaches the nutrient supply concentrationR in . This implies that as t approaches infinityR in = R+Mq +Nq.We then eliminate the equation for R and replace R with its asymptotic valueR in −Mq −Nq. Thus System (6) is simplified todMdtdNdt= µ maxM(R in −Mq −Nq)K +R in −Mq −Nq= −DN +δM −γN.−δM +γN −DM,(8)6

3.2 Equilibrium EvaluationTo determine the equilibria of System (8) we set both dM dtzero and solve for M and N. Our solutions are:and dN dtequal toEc 0 = (M0 c ,N0 c ) = (0,0), (9)⎧⎪ M ⎨c ∗ = (D(D +δ +γ)(K +R in)(D +γ))−((D +γ) 2 R in µ max ))(D +δ +γ)q(D(D +δ +γ)−(µ max (D +γ)))Ec ∗ = ⎪ ⎩N ∗ c = δ(D(D +δ +γ)(K +R in)−((D +γ)R in µ max ))(D+δ +γ)q(D(D +δ +γ)−(µ max (D +γ)))3.3 Equilibrium Analysis.(10)The equilibria representpopulation sizes thus they must be non-negativevalues.Ec 0 is always feasible because M0 c = 0 and ( N0 c = 0. ) Some algebra is needed toδfind conditions for feasibility of Ec. ∗ Nc ∗ = Mc ∗ thus if Nc ∗ is positiveD +γM ∗ c must be positive becauseδD +γ is positive. If N∗ c is negative M ∗ c must benegativeas well. Thus if N ∗ c is feasible then M ∗ c must also be feasible. Thereforewe will only need to prove that N ∗ c is feasible by proving that it is positive. N∗ cis expressed as a fraction; therefore N ∗ c is feasible if the numerator:and the denominator:δ(D(D +δ +γ)(K +R in )−((D +γ)R in µ max ))(D +δ +γ)q(D(D +δ +γ)−(µ max (D +γ)))arebothpositive(Case1)orbothnegative(Case2). Ifinanycasethenumeratorand denominator are not both positive or both negative then Ec ∗ is not feasible.The two cases of feasibility have the following conditions.(C1) The first condition for Case 1 is a positive numerator. Thuswhich impliesD(D +δ +γ)(K +R in ) > (D+γ)R in µ max ,D(D +δ +γ)D+γ> R inµ maxK +R in. (11)7

The second condition for Case 1 is to have a positive denominator. Thuswhich impliesD(D +δ +γ) > (D +γ)µ max ,D(D +δ +γ)(D +γ)> µ max . (12)Condition (12) implies Condition (11). Therefore if Condition (12) is met thenCase 1 of feasability is obtained.(C2) The first condition for Case 2 is to have a negative numerator. Thuswhich impliesD(D +δ +γ)(K +R in ) < (D+γ)R in µ max ,D(D +δ +γ)D+γ< R inµ maxK +R in. (13)The second condition for Case 2 is to have a negative denominator. Thuswhich impliesD(D +δ +γ) < (D +γ)µ max ,D(D +δ +γ)(D +γ)< µ max . (14)Condition (14) is implied by Condition (13). Therefore if Condition (13) is metCase 2 of feasability is obtained. In both cases Nc ∗ is postive, thus in both casesEc ∗ is feasible.In order to analyze the stability of the equilibria in System (8) we letrhs 1rhs 2= µ maxM(R in −Mq −Nq)K +R in −Mq −Nq= −DN +δM −γN.−δM +γN −DM,(15)We then obtain the partial derivatives, of Equations (15), with respect to Mand N:8

∂rhs 1∂M= µ max(R in −2Mq−Nq)K +R in −Mq −Nq+ qµ maxM(R in −Mq −Nq)(K +R in −Mq −Nq) 2 −D −δ,∂rhs 1∂N=µ max MqK +R in −Mq −Nq + qµ maxM(R in −Mq −Nq)(K +R in −Mq −Nq) 2 +γ,∂rhs 2∂M = δ,∂rhs 2∂N= −D −γ.After setting the Jacobian matrix⎛∂rhs 1∂MJ = ⎜⎝ ∂rhs 2∂M∂rhs 1∂N∂rhs 2∂N⎞⎟⎠(16)(17)we evaluate it at E 0 c and E ∗ c. Then we analyze its determinant and trace todetermine stability of the equilibria.3.3.1 Stability Analysis - Case 1The Jacobian matrix (17) evaluated at E 0 cJ =⎛⎜⎝−D−δ + R inµ maxK +R inThe determinant of the Jacobian matrix (18) isδfor the chemostat model isγ−D−γ⎞⎟⎠D 2 +Dδ +Dγ − DR inµ maxK +R in− γR inµ maxK +R in.Therefore the condition for the determinant to be positive isD 2 +Dδ +Dγ > DR inµ maxK +R in+ γR inµ maxK +R in.E 0 c. (18)By doing some algebraic simplifications and rearranging terms the condition issimplified toD(D +δ +γ)(D +γ)> R inµ maxK +R in. (19)9

So if we are in Case 1 of feasibility then Condition (12) is met which impliesCondition (19) is also met. Therefore in Case 1 the determinant of the Jacobianmatrix (18) must be positive. The trace of the Jacobian matrix (18) is−2D−δ + R inµ maxK +R in−γ.Thus the condition for the trace to be negative is2D+δ +γ > R inµ maxK +R in. (20)Using the fact thatD(D+δ +γ)2D+δ +γ > ,(D +γ)Condition (11), and the transitivity property we know that if we are in Case1 then Condition (20) has been met. Therefore we can say that in Case 1 offeasibility the trace of the Jacobian matrix (18) is negative. Furthermore wecan say that Ec 0 is stable in Case 1.The Jacobianmatrix (17) is evaluated at Ec ∗ for the chemostat model. Againwe want to find conditions for a positive determinant and a negative trace. Thecondition for a positive determinant isD(D +γ) . (23)K +R in D+γSo for Case 1 of feasibility the equilibrium Ec ∗ is unstable, because Condition(11) is directly contradicted by Condition (23), meaning that the determinant isnotpositive andthe traceis notnegative. In Case1offeasibility the equilibriumEc 0 is stable and the equilibrium E∗ c is unstable. This is shown in Figure 2.10

0.250.2Case 1 of Chemostatmotil cells (N)non−motil cells (M)nutrients (R)Population densities0.150.10.05010 −2 10 0 10 2Time (t)10 4 10 6Figure 2: Chemostat culture (Case 1): stable trivial equilibrium E 0 c .3.3.2 Stability Analysis - Case 2In the previous section we established the conditions for stability of both equilibriaEc 0 and Ec. ∗ Using those same conditions we can determine the stabilityof Ec 0 and E∗ c in Case 2. Condition (19) for stability of E0 c directly contradictsCondition (13), which is needed for feasibilty in Case 2. So for Case 2 of feasibilitythe equilibrium Ec 0 is unstable. For Case 2 of feasibility the equilibriumEc ∗ is stable, because Condition (13) implies Condition (23). This is shown inFigure 3.4 Discussion and ConclusionThe aim of this research was to establish a relationship between motile andnon-motile states. These transitions are characterized as certain rates (γ andδ) in which one state converts to the other. In addition, our model includedthe rate at which P. parvum cells are reproduced. Together these rates give abetter understanding of the total cell population. Both batch and chemostatmathematical models produced meaningful results and are discussed below.4.1 Batch CultureThe batch culture model was analyzed and gave two equilibrium solutions. Thetrivial equilibrium, which is unstable according to our stability analysis, had littlebiological significance because it represents a non-existing initial population.Having a zero initial population is not useful for our research. Our non-trivial11

10.90.8Case 2 of Chemostatmotil cells (N)non−motil cells (M)nutrients (R)Population densities0.70.60.50.40.30.20.1010 −1 10 0 10 1 10 2 10 3Time (t)10 4 10 5 10 6Figure 3: Chemostat culture (Case 2): stable non-trivial equilibrium E ∗ c .equilibrium did provide biologically meaningful results. According to our stabilityanalysis, if there is an initial cell population, it will approach our non-trivialequilibrium over time. We can see that populations N and M monotonicallyapproached EB. ∗ We observed that both cell populations approached equilibriumas nutrient was depleted to near zero values. In a typical batch culture,one would observe a plateau where the sample reaches its unique maximumconsumption rate, maintains the rate, and then begins the death phase of thegrowth cycle. Because our model does not account for death, it shows only thetime at which both P. parvum cell populations reach EB.∗The ratio of these equilibrium populations gives us an idea as to how thealgae will react in a limited nutrient environment. Adjusting the amount oftotal nutrient will only affect the total population size ofequilibrium(MB+N ∗ B).∗BecauseMB∗NB∗ = γ δ ,total nutrient has no effect on the M ∗ B and N ∗ B proportion. This means thatonly the conversion rates, δ and γ, govern the M and Nsteady state populationproportion. Our model demonstrates the impact of δ and γ on populationdynamics despite being given an elementary constant value. We make thisassumption for simplicity of the model although it may not represent the actualdynamics of the conversion rates.12

4.2 Chemostat CultureThe chemostat model represents a somewhat novel approach for analysis ofalgal species with transitions to resting stages. From the mathematical modelwe deduced thatMB∗NB∗ = 1 δ D + γ δ .Thus our model suggests that if the transition rates are constant, then they canbe calculated by knowing the steady state population values and the dilutionrate.References[1] Clodong, S. and Blasius, B., 2004.Chaos in a periodically forced chemostatwith algal mortality. Proceedings of the Royal Society of London B 271,1617-1624.[2] Green, J. C., Hibberd, D. J. and Pienaar, R. N., 1982. The taxonomyof Prymnesium (Prymnesiophyceae) including a description of a new cosmopolitanspecies, P. patellifera sp. nov., and further observations on P.parvum (N. Carter). British Phycological Journal 17:4, 363-382.[3] Grover, J.P., Crane, K.W., Baker, J.W., Brooks, B. W., and Roelke, D.L., 2011. Spatial variation of harmful algae and their toxins in flowingwaterhabitats: a theoretical exploration. Journal of Plankton Research,33, 211-227.[4] Jordan, R.W. and Chamberlain, A.H.L., 1997. Biodiversity among haptophytealgae. Biodiversity and Conservation 6:1, 131-152.[5] Southard, G.M., Fries, L.T. and Barkoh, A., 2010. Prymnesium parvum:the Texas experience. Journal of American Water Resources Association46:1, 14-23.[6] Larsen, A., and S. Bryant. 1998. Growth rate and toxicity of Prymnesiumparvum and Prymnesium patelliferum (Haptophyta) in response to changesin salinity, light and temperature. Sarsia 83: 409-418.[7] Roelke, D. L., R. M. Errera, R. Kiesling, B. W. Brooks, J. P. Grover, L.Schwierzke, F. Urena-Boeck, J. Baker, and J. L. Pinckney. 2007. Effectsof nutrient enrichment on Prymnesium parvum population dynamics andtoxicity: Results from field experiments, Lake Possum Kingdom, USA.Aquatic Microbial Ecology 46: 125-140.[8] Igarashi, T., Satake, M., Yasumoto, T., 1999.Structural and partial stereochemicalassignmentsfromprymnesin-1andprymnesin-2:potenthemolyticand ichthyotoxic glycosides isolated from the red tide alga Prymnesiumparvum. Journal of the American Chemical Society 121, 8499-8511.13

[9] Brooks, B.W., James, S.V., Valenti, T.W., Urena-Boeck, F., Serrano, C.,Schwierzke, L., Mydlarz, L.D., Grover, J.P., Roelke, D.L., 2010. Comparativetoxicity of Prymnesium parvum in inland waters. Journal of the AmericanWater Resources Association 46, 45-62.[10] Tillmann, U., 2003. Kill and eat your predator: a winning strategy of theplanktonic flagellate Prymnesium parvum. Aquatic Microbial Ecology 32,73-84.14