Periodic Fermentor Yield and Enhanced Product Enrichment from Autonomous Oscillations

Abstract

Four decades of work have clearly established the existence of autonomous oscillations in budding yeast culture across a range of operational parameters and in a few strains. Autonomous oscillations impact substrate conversion to biomass and products. Relatively little work has been done to quantify yield in this case. We have analyzed the yield of autonomously oscillating systems, grown under different conditions, and demonstrate that it too oscillates. Using experimental data and mathematical models of yeast growth and division, we demonstrate strategies to increase the efficient recovery of products. The analysis makes advantage of the population structure and synchrony of the system and our ability to target production within the cell cycle. While oscillatory phenomena in culture have generally been regarded with trepidation in the engineering art of bioprocess control, our results provide further evidence that autonomously oscillating systems can be a powerful tool, rather than an obstruction.

Appendices

Appendix 1: Asymptotic Periodicity

Let the population density be periodic with period T. This means that P(t + T,s) = p(t) for all s. From this, it follows that I(t) is periodic with period T:

$$I\left( {t + T} \right) = \left\langle {p\left( {t + T,s} \right),f\left( s \right)} \right\rangle = \left\langle {p\left( {t,s} \right),f\left( s \right)} \right\rangle = I\left( t \right)$$

where we have used the inner product notation to represent the integrals from Eq. 4. Now consider the function F(t). Using the well-known variation of parameters solution of Eq. 5, we have that

$$F\left( t \right) = e^{ - Dt} \left[ {\int\limits_0^t {e^{Ds} I\left( s \right){\text{d}}s + C} } \right]\quad {\text{where}}\;C\;{\text{is}}\;{\text{an}}\;{\text{integration}}\;{\text{constant}}{\text{.}}$$

Using the periodicity of I, we have:

$${\eqalign{ & F{\left( {t + T} \right)} = e^{{ - D\,t}} {\left[ {{\int\limits_0^{t + T} {e^{{D{\left( {s - T} \right)}}} I{\left( s \right)}{\mathop{\rm d}\nolimits} s + C} }} \right]} \cr & = e^{{ - D\,t}} {\left[ {{\int\limits_0^{t + T} {e^{{D{\left( {s - T} \right)}}} I{\left( {s - T} \right)}{\mathop{\rm d}\nolimits} s + C} }} \right]} \cr & = e^{{ - D\,t}} {\left[ {{\int\limits_{ - T}^t {e^{{D\,s}} I{\left( s \right)}{\mathop{\rm d}\nolimits} s + C} }} \right]}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,. \cr & = e^{{ - D\,t}} {\left[ {{\int\limits_{ - T}^t {e^{{D\,s}} I{\left( s \right)}{\mathop{\rm d}\nolimits} s + {\int\limits_{ - T}^0 {e^{{Ds}} I{\left( s \right)}{\mathop{\rm d}\nolimits} s + } }C} }} \right]} \cr & = F{\left( t \right)} + e^{{ - D\,t}} {\int\limits_{ - T}^t {e^{{D\,s}} I{\left( s \right)}{\mathop{\rm d}\nolimits} s} } \cr} }$$

Since the last integrand is finite and e ^−Dt limits to 0 exponentially as t goes to ∞, F(t) is an asymptotically periodic with period T.

Appendix 2: Leslie Model of Yeast Growth and Division

$${\left[ {\begin{array}{*{20}c} {{f^{0}_{0} {\left( {t + \tau _{0} } \right)}}} \\ {{f^{0}_{1} {\left( {t + \tau _{1} } \right)}}} \\ {{f^{0}_{2} {\left( {t + \tau _{2} } \right)}}} \\ { \vdots } \\ {{f^{0}_{n} {\left( {t + \tau _{n} } \right)}}} \\ \end{array} } \right]} = {\left[ {\begin{array}{*{20}c} {1} & {1} & {1} & { \cdots } & {1} & {1} \\ {1} & {0} & {0} & { \cdots } & {0} & {0} \\ {0} & {1} & {0} & { \cdots } & {0} & {0} \\ { \vdots } & { \vdots } & { \vdots } & { \ddots } & { \vdots } & { \vdots } \\ {0} & {0} & {0} & { \cdots } & {1} & {0} \\ \end{array} } \right]}\;{\left[ {\begin{array}{*{20}c} {{f^{0}_{0} {\left( t \right)}}} \\ {{f^{0}_{1} {\left( t \right)}}} \\ {{f^{0}_{2} {\left( t \right)}}} \\ { \vdots } \\ {{f^{0}_{n} {\left( t \right)}}} \\ \end{array} } \right]}$$

where:

$$\left\{ {\matrix {f_{\operatorname{in} }^i \left( t \right) = \operatorname{Total} \;flux\;entering\;the\;i - th\;\operatorname{generation} \;at\;time\;t} \\ {f_i^0 \left( t \right) = \operatorname{Daughter} \;cells\;produced\;from\;a\;division\;in\;age\;\operatorname{class} \;i\;\operatorname{at} \;time\;t} \\ {\tau _i = \operatorname{Flux} \;residence\;time,\;i.e.\;cell\;cycle\;length} \ } \right\}.$$

The Leslie model [16] used to calculate yeast growth and division encodes as a mathematical object the process flow depicted at the top of Fig. 10. The process flow has age descending from daughter cells with no scars, P ₀, at the top to cells with arbitrarily many scars, P _n, at the bottom. Cells belonging to each of these age classes traverse their separate cell cycles, shown as the horizontal thick black lines. The time it takes a cell of age, k, to traverse their cell cycle is τ _k. The values of τ used are shown in Table 3 along with the temporal position of bud emergence. The thin black lines with arrows indicate the directions of possible flux due to cell divisions. Each arrow corresponds to a 1 in the matrix representation of the process at the bottom of Fig. 10.

Fig. 10

The top schematic represents the process of yeast growth and division. Each age class is represented by a bold horizontal line and corresponding residence time, τ. The age classes are organized by replicative age, with the daughter cells, denoted P₀, as the first grid. Higher generation parent cells are listed below. Each cell division can be characterized by producing a daughter cell back at the beginning of the P₀ grid and a parent of the next highest age class. This process can be represented by a Leslie matrix as shown by the bottom panel

Table 3 Parameters used in the Leslie model simulations.

Given an initial population distribution, the matrix model is iterated to produce the dynamics. Typically, such a system would produce asynchronous exponential growth. Autonomous oscillations result from an additional feedback. We have introduced a delay model to produce population oscillation. In the delay model, a threshold, T _R, is introduced. Once the density of cell in the S-phase of the cell cycle reaches T _R or above, any cells in a 10% strip proximal to S become delayed in their cell cycle progression. In the simulations described in Fig. 8, T _R was 20% of the total culture density. The S-phase is delimited on the left by bud emergence (BE) as described in Table 3. The algorithm used to implement this model is freely available from the authors upon request.