Fatty Acids Released by Chlorella vulgaris and Their Role in Interference with Pseudokirchneriella subcapitata: Experiments and Modelling

Abstract

The role of extracellular fatty acids in the interference between two algae, Chlorella vulgaris Beijerink and Pseudokirchneriella subcapitata (Korshikov) Hindak, was assessed by the co-cultivation of the two selected strains, as well as by the chemical analysis of exudates from the culture media of single strain cultures. The effect of culture age and phosphate limitation was evaluated. The experiments showed that the composition and amount of fatty acids, released by C. vulgaris and by P. subcapitata, both in a batch and in a continuous monoculture, depend on the culture age and on the phosphate concentration in the culture medium. We also found that the amount of chlorellin generated in the two algae co-culture increased and was almost exclusively constituted by a mixture of C18 fatty acids. By using the evaluated concentrations of these fatty acids, an artificial chlorellin was prepared. The toxicity of this mixture to P. subcapitata appears to be similar to that of the natural chlorellin. For both algae, a stimulation of growth was observed at low concentrations of the natural chlorellin, whereas higher concentrations produced inhibitory effects on both species. However, P. subcapitata was much more sensitive than C. vulgaris. By using some of these new experimental results, two new mathematical models have been used to describe the toxicity of chlorellin to C. vulgaris and to the interference between C. vulgaris and P. subcapitata, respectively.

Appendix

Proof (Lemma 2)

If we set

$$ z(t) = S(t) + N(t) + p(t) $$

(11)

due to system (7), we get

$$ \dot{z}(t) = 1 - S - N - p = 1 - z(t), $$

(12)

and by integrating from 0 to t, we obtain

$$ z(t) = z(0){e^{ - t}} + 1 - {e^{ - t}}, $$

and then

$$ \mathop {{\lim }}\limits_{t \to + \infty } z(t) = 1 $$

(13)

thus proving, for any initial condition, the boundedness of the solutions of (7).

Proof( Theorem 2)

In order to prove the theorem, we look for positive steady state solutions (S* > 0, N* > 0, p* > 0) and we solve the system obtained by setting the right-hand side of system (7) equal to zero. In this way, we get

$$ {N^* } = \frac{{1 - {S^* }}}{{k\left( {1 - {S^* }} \right) + 1}}{,}\quad \quad {p^* } = \frac{{k{{\left( {1 - {S^* }} \right)}^2}}}{{k\left( {1 - {S^* }} \right) + 1}} $$

(14)

and

$$ \left( {1 - {S^* }} \right)\left[ {1 - \frac{{f\left( {{S^* }} \right)}}{{k\left( {1 - {S^* }} \right) + 1}}{e^{ - \alpha {{\left( {\frac{{k{{\left( {1 - {S^* }} \right)}^2}}}{{k\left( {1 - {S^* }} \right) + 1}}} \right)}^2}}}} \right] = 0 $$

(15)

with a few calculations from (14) we find that N* > 0, p* > 0, if and only if S* < 1. We have to exclude the value S* = 1 because it makes both p* and N* equal to zero. Therefore (by omitting the star), we study the solutions of the equation

$$ k\left( {1 - S} \right) + 1 - f(S){e^{ - \alpha {{\left( {\frac{{k{{\left( {1 - S} \right)}^2}}}{{k\left( {1 - S} \right) + 1}}} \right)}^2}}} = 0 $$

(16)

if we define

$$ h(S) = k\left( {1 - S} \right) + 1\quad {\hbox{and}}\quad g(S) = f(S){e^{ - \alpha {{\left( {\frac{{k{{\left( {1 - S} \right)}^2}}}{{k\left( {1 - S} \right) + 1}}} \right)}^2}}} $$

(17)

then the problem of finding the solution of (15) is changed into finding the solution of h(S) = g(S). Remembering that S < 1, it is easy to show that in the interval [0, 1], h(S) is strictly decreasing whereas g(S) is strictly increasing.

Therefore, because h(1) = 1 and $g(1) = \frac{m}{{a + 1}}$ the two curves admit only one intersection point with abscissa S* < 1 (Fig. 5).

Fig. 5

Here we represent the plots of the two functions h(S) and g(S) defined in (12) in the proof of Theorem 4. These curves, as shown, admit only one intersection point with abscissa S* < 1

Proof (Theorem 3)

• (i) The roots of Eq. (9) computed in E ₀ = (1, 0, 0) are

$$ {\rho_1} = - 1,{ }{\rho_2} = f(1) - 1,{ }{\rho_3} = - 1. $$

Therefore, the steady state E₀ turns out asymptotically stable if

$$ {\rho_2} < 0 \Rightarrow f(1) < 1{\hbox{ that is }}m < a + 1. $$

• (ii) In E* = (S*, N*, p*), Eq. (9) can be written as follows

$$ \begin{array}{*{20}{c}} {{\rho^3} + \left( {{a_{11}} + {a_{22}} + {a_{33}}} \right){\rho^2} + \left( {{a_{12}}{a_{21}} + {a_{11}}{a_{22}} - {a_{13}}{a_{31}} + {a_{23}}{a_{32}} + {a_{11}}{a_{33}} + {a_{22}}{a_{33}}} \right)\rho + } \hfill \\{ + {a_{12}}{a_{21}}{a_{33}} + {a_{11}}{a_{22}}{a_{33}} - {a_{12}}{a_{23}}{a_{31}} + {a_{11}}{a_{23}}{a_{32}} - {a_{13}}{a_{22}}{a_{31}} - {a_{13}}{a_{21}}{a_{32}} = 0} \hfill \\\end{array} $$

(18)

where

$$ \begin{array}{*{20}{c}} {{a_{11}} = - \frac{{a + {S^{ * 2}}}}{{{S^* }\left( {a + {S^* }} \right)}},} \hfill & {{a_{12}} = - \frac{{1 - {S^* }}}{{{N^* }}},} \hfill & {{a_{13}} = 2\alpha {p^* }\left( {1 - {S^* }} \right),} \hfill & {{a_{21}} = \frac{{a{N^* }}}{{{S^* }\left( {a + {S^* }} \right)}},} \hfill & {{a_{22}} = - k\left( {1 - {S^* }} \right),} \hfill \\{{a_{23}} = - 2\alpha {p^* }{N^* }{,}} \hfill & {{a_{31}} = \frac{{a{p^* }}}{{{S^* }\left( {a + {S^* }} \right)}},} \hfill & {{a_{32}} = 2k\left( {1 - {S^* }} \right),} \hfill & {{a_{33}} = - 1 - 2\alpha {p^{ * 2}}} \hfill & {} \hfill \\\end{array} $$

are positive constants. In order to study the stability properties of the equilibrium, we can use the Routh–Hurwitz criterion. We observe that, being

• 1. ${a_{11}} + {a_{22}} + {a_{33}} > 0$

• 2. $\begin{array}{*{20}{c}} {{a_{12}}{a_{21}}{a_{33}} + {a_{11}}{a_{22}}{a_{33}} - {a_{12}}{a_{23}}{a_{31}} + {a_{11}}{a_{23}}{a_{32}} - {a_{13}}{a_{22}}{a_{31}} - {a_{13}}{a_{21}}{a_{32}}} \hfill \\ { = \frac{{ - \left( {1 - {S^* }} \right)\left( { - \frac{{a + ak + k{S^{ * 2}}}}{{{S^* }\left( {a + {S^* }} \right)}} - 2k{p^* }\left( {{p^* } + 2{N^* }} \right)} \right)}}{\alpha } > 0} \hfill \\ \end{array}$

• 3. $\begin{array}{*{20}{c}} {\left( {{a_{11}} + {a_{22}} + {a_{33}}} \right)\left( {{a_{12}}{a_{21}} + {a_{11}}{a_{22}} - {a_{13}}{a_{31}} + {a_{23}}{a_{32}} + {a_{11}}{a_{33}} + {a_{22}}{a_{33}}} \right)} \hfill \\ { - \left( {{a_{12}}{a_{21}}{a_{33}} + {a_{11}}{a_{22}}{a_{33}} - {a_{12}}{a_{23}}{a_{31}} + {a_{11}}{a_{23}}{a_{32}} + - {a_{13}}{a_{22}}{a_{31}} - {a_{13}}{a_{21}}{a_{32}}} \right) > 0} \hfill \\ \end{array}$

then all the hypotheses of the Routh–Hurwitz criterion are satisfied and the local stability of the equilibrium E* follows.