Cell Quota Based Population Models and their Applications Aaron - - PowerPoint PPT Presentation

Cell Quota Based Population Models and their Applications

Aaron Packer

School of Mathematical & Statistical Sciences Arizona State University

November 17, 2014

• A. Packer

Cell Quota Based Models + Applications Nov 17, 2014 1 / 72

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 2 / 72

Introduction

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 3 / 72

Introduction

Cell Quota Model

In 1968, phycologist M.R. Droop published his famous discovery on the functional relationship between growth rate and internal nutrient status of algae in chemostat culture. µ (Q) = µm

• 1 − q

Q

• Q: cell quota (nutrient/biomass)

q: subsistence quota µm: maximum specific growth rate

• A. Packer

Cell Quota Based Models + Applications : — 4 / 72

Introduction

Cell Quota Model, cont’d

µ (Q) = µm

• 1 − q

Q

• Leadbeater, B., “The ‘Droop Equation’–Michael Droop and the legacy of the ‘Cell- Quota Model’
• f phytoplankton growth”, Protist 157, 3, 345 (2006).
• A. Packer

Cell Quota Based Models + Applications : — 5 / 72

Introduction

Summary

Problems

Why do certain oleaginous algae produce so many neutral lipids, and how can their cultivation for biofuels be improved? What role does ammonia- (and to lesser extend nitrite-) induced toxicity play in the dynamics of producer-grazer systems for aquaculture? Can cell-quota based population models be applied to prostate cancer in a mechanistic way?

Solutions

The nitrogen cell quota quantifies the metabolic shift to neutral lipids in green microalgae and gives rise to a mechanistic modeling framework. Nitrogen toxicity can be modeled by adding new feedback into producer-grazer systems with nutrient recycling. A mechanistically derived model follows naturally via application of the “cell quota” concept to androgens.

• A. Packer

Cell Quota Based Models + Applications : — 6 / 72

Introduction

Summary

Problems

Why do certain oleaginous algae produce so many neutral lipids, and how can their cultivation for biofuels be improved? What role does ammonia- (and to lesser extend nitrite-) induced toxicity play in the dynamics of producer-grazer systems for aquaculture? Can cell-quota based population models be applied to prostate cancer in a mechanistic way?

Solutions

The nitrogen cell quota quantifies the metabolic shift to neutral lipids in green microalgae and gives rise to a mechanistic modeling framework. Nitrogen toxicity can be modeled by adding new feedback into producer-grazer systems with nutrient recycling. A mechanistically derived model follows naturally via application of the “cell quota” concept to androgens.

• A. Packer

Cell Quota Based Models + Applications : — 6 / 72

Introduction

Summary

Problems

Why do certain oleaginous algae produce so many neutral lipids, and how can their cultivation for biofuels be improved? What role does ammonia- (and to lesser extend nitrite-) induced toxicity play in the dynamics of producer-grazer systems for aquaculture? Can cell-quota based population models be applied to prostate cancer in a mechanistic way?

Solutions

The nitrogen cell quota quantifies the metabolic shift to neutral lipids in green microalgae and gives rise to a mechanistic modeling framework. Nitrogen toxicity can be modeled by adding new feedback into producer-grazer systems with nutrient recycling. A mechanistically derived model follows naturally via application of the “cell quota” concept to androgens.

• A. Packer

Cell Quota Based Models + Applications : — 6 / 72

Neutral Lipid Synthesis in Green Microalgae

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 7 / 72

Neutral Lipid Synthesis in Green Microalgae Introduction

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 8 / 72

Neutral Lipid Synthesis in Green Microalgae Introduction

Motivation

Theoretical Why do certain species of green microalgae produce such large quantities of neutral lipids, particularly triacylglycerols, during stressed conditions? Mathematical Is a mechanistic model possible, and what insight can be gained?

• A. Packer

Cell Quota Based Models + Applications : — 9 / 72

Neutral Lipid Synthesis in Green Microalgae Introduction

Motivation

Theoretical Why do certain species of green microalgae produce such large quantities of neutral lipids, particularly triacylglycerols, during stressed conditions? Compensate for lack of electron/carbon sink during uncoupling of photosynthesis from growth. Transient energy storage during stressful times Mathematical Is a mechanistic model possible, and what insight can be gained?

• A. Packer

Cell Quota Based Models + Applications : — 9 / 72

Neutral Lipid Synthesis in Green Microalgae Introduction

Motivation

Theoretical Why do certain species of green microalgae produce such large quantities of neutral lipids, particularly triacylglycerols, during stressed conditions? Compensate for lack of electron/carbon sink during uncoupling of photosynthesis from growth. Transient energy storage during stressful times Mathematical Is a mechanistic model possible, and what insight can be gained? Yes! Mechanistic modeling helps validate current theory explaining the NL phenomenon in oleaginous algae

• A. Packer

Cell Quota Based Models + Applications : — 9 / 72

Neutral Lipid Synthesis in Green Microalgae Introduction

Key Observations - Nitrogen & Light

Nitrogen Starvation Increases neutral lipid synthesis. Decreases cellular growth and production of non–neutral lipid biomass. Decreases capacity of certain mechanisms that prevent and repair both photoinhibition and photooxidation. Light Intensity Increasing light intensity... Increases neutral lipid synthesis. Increases susceptibility to photoinhibition and photooxidation.

• A. Packer

Cell Quota Based Models + Applications : — 10 / 72

Neutral Lipid Synthesis in Green Microalgae Introduction

Key Observations - Nitrogen & Light

Nitrogen Starvation Increases neutral lipid synthesis. Decreases cellular growth and production of non–neutral lipid biomass. Decreases capacity of certain mechanisms that prevent and repair both photoinhibition and photooxidation. Light Intensity Increasing light intensity... Increases neutral lipid synthesis. Increases susceptibility to photoinhibition and photooxidation.

• A. Packer

Cell Quota Based Models + Applications : — 10 / 72

Neutral Lipid Synthesis in Green Microalgae Introduction

Key Observations - Nitrogen & Light

Nitrogen Starvation Increases neutral lipid synthesis. Decreases cellular growth and production of non–neutral lipid biomass. Decreases capacity of certain mechanisms that prevent and repair both photoinhibition and photooxidation. ց ր⇒ NLs are defense from the dangers of photosynthesis–growth uncoupling Light Intensity Increasing light intensity... Increases neutral lipid synthesis. Increases susceptibility to photoinhibition and photooxidation.

• A. Packer

Cell Quota Based Models + Applications : — 10 / 72

Neutral Lipid Synthesis in Green Microalgae Introduction

Case Study: Pseudochlorococcum sp. (Li et al. 2011)

2 4 6 8 10 0.05 0.1 0.15 0.2 0.25 g N/L days 100% 25% 0%

Figure 1: Extracellular N

2 4 6 8 10 12 1 2 3 4 5 6 7 8 g/L days 100% 25% 0%

Figure 2: Biomass

100% culture: 0.24 g N L-1. 25% culture: 0.06 g N/L. 0% culture: 0.0 g N L-1.

Li, Y., D. Han, M. Sommerfeld, and Q. Hu. “Photosynthetic carbon partitioning and lipid production in the oleaginous microalga Pseudochlorococcum sp. (Chlorophyceae) under nitrogen-limited conditions.” Bioresource Technology 102, 1 (2011): 123–129.

• A. Packer

Cell Quota Based Models + Applications : — 11 / 72

Neutral Lipid Synthesis in Green Microalgae Introduction

Case Study: Pseudochlorococcum sp. (Li et al. 2011)

2 4 6 8 10 0.05 0.1 0.15 0.2 0.25 g N/L days 100% 25% 0%

Figure 1: Extracellular N

2 4 6 8 10 12 10 20 30 40 50 % days 100% 25% 0%

Figure 2: NL % of biomass

100% culture: 0.24 g N L-1. 25% culture: 0.06 g N/L. 0% culture: 0.0 g N L-1.

Li, Y., D. Han, M. Sommerfeld, and Q. Hu. “Photosynthetic carbon partitioning and lipid production in the oleaginous microalga Pseudochlorococcum sp. (Chlorophyceae) under nitrogen-limited conditions.” Bioresource Technology 102, 1 (2011): 123–129.

• A. Packer

Cell Quota Based Models + Applications : — 11 / 72

Neutral Lipid Synthesis in Green Microalgae Introduction

Case Study: Pseudochlorococcum sp. (Li et al. 2011)

2 4 6 8 10 12 1 2 3 4 5 g/L days 100% 25% 0%

Figure 1: Non-NL biomass

2 4 6 8 10 12 0.5 1 1.5 2 2.5 3 3.5 g/L days 100% 25% 0%

Figure 2: NLs

100% culture: 0.24 g N L-1. 25% culture: 0.06 g N/L. 0% culture: 0.0 g N L-1.

Li, Y., D. Han, M. Sommerfeld, and Q. Hu. “Photosynthetic carbon partitioning and lipid production in the oleaginous microalga Pseudochlorococcum sp. (Chlorophyceae) under nitrogen-limited conditions.” Bioresource Technology 102, 1 (2011): 123–129.

• A. Packer

Cell Quota Based Models + Applications : — 11 / 72

Neutral Lipid Synthesis in Green Microalgae Model

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 12 / 72

Neutral Lipid Synthesis in Green Microalgae Model

Model State Variables

A(t) = algae biomass density, excluding neutral lipids (g d.w. · m−3), L(t) = neutral lipid density (g NL · m−3), Q(t) = N-quota of A(t) (g N · g−1 d.w.), H(t) = chl a content of A (g chl · g−1 d.w.), N(t) = extracellular nitrogen concentration (g N · m−3). A and L

Biomass is divided into two compartments: non-NL biomass A(t) and NLs L(t). Therefore, total biomass density is the sum of the two compartments, A(t) + L(t).

Q and H

The N-quota, Q(t), is the intracellular N per unit A(t). Q(t)A(t) = total intracellular N. Similarly, H(t) is the intracellular chl a per unit A(t). H(t)A(t) = total chl a density.

• A. Packer

Cell Quota Based Models + Applications : — 13 / 72

Neutral Lipid Synthesis in Green Microalgae Model

Model

dA dt = µA(t),

cellular growth

(1) dL dt = (p − cµ) A(t),

• NL synthesis

(2) dQ dt = v

• N uptake

− µQ(t),

growth dilution

(3) dH dt = µ p/cθmv

N uptake devoted to chl a synthesis

− µH(t),

growth dilution

(4)

• dN

dt = −vA.

N uptake

• (5)
• A. Packer

Cell Quota Based Models + Applications : — 14 / 72

Neutral Lipid Synthesis in Green Microalgae Model

Growth rate

µ = min

• µm
• 1 −

q Q(t)

• , p

c

• q1

minimum/subsistence N quota g N g-1 d.w. c C subsistence quota g C g−1 d.w. µm maximum N-limited growth rate s−1 p dw-specific photosynthesis rate (g C g-1 dw s-1) Growth is either N or light limited. N limited growth follows the cell quota model, µm

• 1 −

q Q(t)

• .

Light limited growth is p

c.

c is how much carbon is required per unit increase in dry weight. p is the photosynthesis rate.

• A. Packer

Cell Quota Based Models + Applications : — 15 / 72

Neutral Lipid Synthesis in Green Microalgae Model

Photosynthesis rate

p = H(t)pm

• 1 − exp

−aΦI pm

• a

absorption efficiency normalized to chl a m2 g−1 chl Φ quantum efficiency g C (µmol photons)−1 pm light-saturated photosynthesis rate g C g−1 chl s−1 The photosynthesis rate is modeled using the general Poisson model. Light-limited rate is governed by aΦI. Light-saturated rate, pm, is a function.

• A. Packer

Cell Quota Based Models + Applications : — 16 / 72

Neutral Lipid Synthesis in Green Microalgae Model

Light-saturated photosynthesis rate, pm

pm = p0

• Q2
• Q2 + q2 =

(AQ)2p0 (AQ)2 + q2 (A(t) + L(t))2 q minimum/subsistence N quota g N g−1 d.w. p0 maximum photosynthesis rate g C g−1 chl s−1

• Q

N content relative to A + L g N g−1 d.w. Previous models have assumed pm = 0 for Q = q, which does not work for the NL model here. pm > 0 for Q = q in this model indicates decoupled photosynthesis from growth. pm decreases as Q decreases.

• A. Packer

Cell Quota Based Models + Applications : — 17 / 72

Neutral Lipid Synthesis in Green Microalgae Model

NL synthesis

dL dt = (p − cµ) A(t) c C subsistence quota g C g−1 d.w. p photosynthesis rate (g C g-1 dw s-1) µ cellular (non-NL) growth rate s−1 NL synthesis results from an excess of C-fixation relative to the C requirements for growth. cµ is C required for growth. For Q(t) = q ( ˜ Q(t) ≤ q), all increases in total biomass are due to de novo NL synthesis.

• A. Packer

Cell Quota Based Models + Applications : — 18 / 72

Neutral Lipid Synthesis in Green Microalgae Model

Self shading and reactor depth

I = I0 aH(t)A(t)z (1 − exp (−aH(t)A(t)z))

I0 incident irradiance µmol photons m−2 s−1 z light path m a absorption efficiency normalized to chl a m2 g−1 chl

I is average irradiance in the reactor. Derived using Lambert-Beer law of light attenuation. Enables model to incorporate self shading and to make qualitatively accurate predictions of biomass and NL dependence on z.

• A. Packer

Cell Quota Based Models + Applications : — 19 / 72

Neutral Lipid Synthesis in Green Microalgae Model

N uptake and Chl a synthesis

v = qM − Q(t) qM − q vmN(t) N(t) + vh

• dH

dt = cµ p θmv − H(t)µ

q minimum/subsistence N quota g N g−1 d.w. qM maximum N quota g N g−1 d.w. c C subsistence quota g C g−1 d.w. vm maximum uptake rate of nitrogen g N g d.w.−1 s−1 vh half-saturation coefficient g N m−3 θm maximum chl:N g chl a g−1 N

Chl a synthesis is coupled to N uptake/assimilation. Proportion of N uptake devoted to chl a synthesis is (cµ/p)θm. cµ/p represents the utilization ratio of fixed carbon.

• A. Packer

Cell Quota Based Models + Applications : — 20 / 72

Neutral Lipid Synthesis in Green Microalgae Results

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 21 / 72

Neutral Lipid Synthesis in Green Microalgae Results

Results

2 4 6 8 10 0.05 0.1 0.15 0.2 0.25 g N/L d 100% 25% 0%

Figure 3: Extracellular N

2 4 6 8 10 12 1 2 3 4 5 6 7 d g/L 25% 0%

Figure 4: Total biomass A + L

Model fitted to data from Li et al. (2011). (100% culture excluded from Figure 4.)

• A. Packer

Cell Quota Based Models + Applications : — 22 / 72

Neutral Lipid Synthesis in Green Microalgae Results

Results

2 4 6 8 10 12 20 40 60 80 d % 25% 0%

Figure 3: NL % of biomass

2 4 6 8 10 12 1 2 3 4 d g NL/L 25% 0%

Figure 4: Neutral lipids L

Model fitted to data from Li et al. (2011). (100% culture excluded.)

• A. Packer

Cell Quota Based Models + Applications : — 22 / 72

Neutral Lipid Synthesis in Green Microalgae Results

Results

2 4 6 8 10 12 0.02 0.04 0.06 0.08 g N/g dw d 25% Q 0% Q

Figure 3: Q and Q

2 4 6 8 10 12 0.005 0.01 0.015 0.02 0.025 g Chl/g dw d 100% 25% 0%

Figure 4: H(t), the chl a content of A

Model fitted to data from Li et al. (2011). (Chl a data not reported.)

• A. Packer

Cell Quota Based Models + Applications : — 22 / 72

Neutral Lipid Synthesis in Green Microalgae Conclusion

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 23 / 72

Neutral Lipid Synthesis in Green Microalgae Conclusion

Conclusion

NL synthesis in oleaginous microalgae Decoupling of photosynthesis from growth may explain NL synthesis. Minimum cell quota of limiting nutrient may represent a threshold for NL synthesis. 100% (high-N) culture Was N not the limiting resource for the 100% culture? Most likely given that a 4-fold increase in N resulted in 36% biomass increase. Future work Splitting biomass into separate compartments for functional biomass and neutral lipids is a useful framework, and has since been adopted in later models. Modeling neutral lipid synthesis is an active research area which has since used ideas from the model presented here.

• A. Packer

Cell Quota Based Models + Applications : — 24 / 72

Neutral Lipid Synthesis in Green Microalgae Conclusion

Conclusion

NL synthesis in oleaginous microalgae Decoupling of photosynthesis from growth may explain NL synthesis. Minimum cell quota of limiting nutrient may represent a threshold for NL synthesis. 100% (high-N) culture Was N not the limiting resource for the 100% culture? Most likely given that a 4-fold increase in N resulted in 36% biomass increase. Future work Splitting biomass into separate compartments for functional biomass and neutral lipids is a useful framework, and has since been adopted in later models. Modeling neutral lipid synthesis is an active research area which has since used ideas from the model presented here.

• A. Packer

Cell Quota Based Models + Applications : — 24 / 72

Neutral Lipid Synthesis in Green Microalgae Conclusion

Conclusion

NL synthesis in oleaginous microalgae Decoupling of photosynthesis from growth may explain NL synthesis. Minimum cell quota of limiting nutrient may represent a threshold for NL synthesis. 100% (high-N) culture Was N not the limiting resource for the 100% culture? Most likely given that a 4-fold increase in N resulted in 36% biomass increase. Future work Splitting biomass into separate compartments for functional biomass and neutral lipids is a useful framework, and has since been adopted in later models. Modeling neutral lipid synthesis is an active research area which has since used ideas from the model presented here.

• A. Packer

Cell Quota Based Models + Applications : — 24 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 25 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 26 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction

Aquaculture

Farming of aquatic organisms. Fish (Salmon, Carp, Grouper, Tilapia) Crustaceans (Shrimp, crab, prawn) Molluscs (Oyster, mussel) Aquatic plants (algae, seaweed) Methods Ponds Tanks Raceways Cages

• A. Packer

Cell Quota Based Models + Applications : — 27 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction

Aquaculture

Farming of aquatic organisms. Fish (Salmon, Carp, Grouper, Tilapia) Crustaceans (Shrimp, crab, prawn) Molluscs (Oyster, mussel) Aquatic plants (algae, seaweed) Methods Ponds Tanks Raceways Cages

Perfect for applied mathematical ecology.

• A. Packer

Cell Quota Based Models + Applications : — 27 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction

Nitrogen & toxicity

Nitrogen cycle An important part of aquaculture systems, with much existing research and even mathematical models in the literature. Toxicity Accumulation of waste in culture. High levels of inorganic N can be toxic (ammonia, nitrite). Even low levels of ammonia or nitrite can have inhibitory effect on some species. Producer-grazer modeling What implications does N-induced toxicity have for N recycling and dynamical behavior in producer-grazer systems?

• A. Packer

Cell Quota Based Models + Applications : — 28 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction

Nitrogen & toxicity

Nitrogen cycle An important part of aquaculture systems, with much existing research and even mathematical models in the literature. Toxicity Accumulation of waste in culture. High levels of inorganic N can be toxic (ammonia, nitrite). Even low levels of ammonia or nitrite can have inhibitory effect on some species. Producer-grazer modeling What implications does N-induced toxicity have for N recycling and dynamical behavior in producer-grazer systems?

• A. Packer

Cell Quota Based Models + Applications : — 28 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Model

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 29 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Model

Model

State variables y(t) = grazer biomass density (g C · L−1) x(t) = producer (phytoplankton) biomass density (g C · L−1) Q(t) = N:C of producer (g N · g−1 C)

• N(t) = external N concentration (g N · L−1)

∗ N(t) *System is assumed to be closed under nitrogen, so N(t) can be decoupled from the system.

• A. Packer

Cell Quota Based Models + Applications : — 30 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Model

Toxicity Model

x′ = m

• 1 − q

Q

• x
• N-limited growth

− f(x)y

• grazing

y′ = r min

• 1, Q

θ

• f(x)y
• growth

− dy

• natural death

− h (N) y

intoxication

Q′ = v(N)

• uptake

− m (Q − q)

• growth dilution
• N′ = −v(N)x

uptake

+

• Q − r min {θ, Q}
• f(x)y
• grazer waste

+ dθy + h (N) θy

• grazer death
• A. Packer

Cell Quota Based Models + Applications : — 31 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Model

Toxicity Model

x′ = m

• 1 − q

Q

• x
• N-limited growth

− f(x)y

• grazing

y′ = r min

• 1, Q

θ

• f(x)y
• growth

− dy

• natural death

− h (T − Qx − θy) y

• intoxication

Q′ = v(T − Qx − θy)

• uptake

− m (Q − q)

• growth dilution
• A. Packer

Cell Quota Based Models + Applications : — 31 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Model

Parameter Description Units T total system nitrogen g N L-1 m producer maximum growth rate d-1 q producer minimum N:C quota g N g-1 C θ grazer homeostatic N:C g N g-1 C r grazing/digestion efficiency scalar d grazer natural death rate d-1 Function Description Units f(x) functional response g C g-1 C d-1 v(N) producer-specific N uptake rate g N g-1 C d-1 h(N) grazer toxicity death rate d-1 Model parameters and generalized functions. f(0) = 0, f ′ > 0, f ′′ ≤ 0, and similarly for v and h.

• A. Packer

Cell Quota Based Models + Applications : — 32 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Equilibria

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 33 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Equilibria

Equilibria

Boundary equilibria There are two unique boundary equilibria: E0 = (0, 0, q + v(T)/m) (Extinction) E1 = (T/q, 0, q) (Grazer-only extinction) Interal equilibria Depending on f and the parameter values, there may be zero, one unique, or multiple internal equilibria. It is not possible to find explicit formulas for the internal equilibria in general. E2 = (x∗, y∗, Q∗) (Coexistence)

• A. Packer

Cell Quota Based Models + Applications : — 34 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Equilibria

Boundary Stability

Total extinction E0 is always unstable. Grazer-only extinction E1 is locally asymptotically stable if and only if rf(T/q) < dθ q . E1 is globally asymptotically stable if rf(T/q) d < 1 or rTf ′(0) dθ < 1.

• A. Packer

Cell Quota Based Models + Applications : — 35 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Equilibria

Boundary Stability

Total extinction E0 is always unstable. Grazer-only extinction E1 is locally asymptotically stable if and only if rf(T/q) < dθ q . E1 is globally asymptotically stable if rf(T/q) d < 1 or rTf ′(0) dθ < 1.

• A. Packer

Cell Quota Based Models + Applications : — 35 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Equilibria

Internal equilibira

The coexistence equilibria E2 = (x, y, Q) are given by values Q which satisfy T = N(Q) +

• Q + r

Am(Q − q) d + h (N(Q))

• f −1

d + h (N(Q)) rA

• ,

where N(Q) = v−1(m (Q − q)), and the corresponding values x, y given by x = f −1 d + h(N(Q)) rA

• ,

y = rAm(Q − q)/Q d + h(N(Q)) x, where for notational convenience A = min

• 1, Q

θ

• and

A = θ

QA = min

• θ

Q, 1

• .
• A. Packer

Cell Quota Based Models + Applications : — 36 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Linear Functional Response

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 37 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Linear Functional Response

f(x) = ax

If the functional response f is the linear function f(x) = ax, the model dynamics are greatly simplified and less interesting. Theorem: E1 LAS = GAS If f(x) = ax then grazer-only extinction E1 is globally asymptotically stable if and only if raT < dθ, which is the same as the necessary and sufficient condition for LAS when f is defined only generally.

• A. Packer

Cell Quota Based Models + Applications : — 38 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Linear Functional Response

f(x) = ax, cont’d

For the general f not defined explicitly, it is not feasible to find explicit conditions for the existence of any coexistence equilibria E2. However, with f(x) = ax: Theorem: E2 existence and uniqueness If f(x) = ax then the coexistence equilibrium E2 exists if and only if E1 is unstable, i.e., raT > dθ. Further, there is only one unique E2.

• A. Packer

Cell Quota Based Models + Applications : — 39 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Linear Functional Response

f(x) = ax, cont’d

0.5 1 1.5 0.5 1 1.5 2 2.5 3 ←dθ=qrf(T/q) y (mg C/L) d (days−1)

Figure 5: y

0.5 1 1.5 2 4 6 8 10 12 14 ←dθ=qrf(T/q) x (mg C/L) d (days−1)

Figure 6: x

Bifurcation on d for the model with linear functional response f(x) = ax.

• A. Packer

Cell Quota Based Models + Applications : — 40 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Linear Functional Response

f(x) = ax, cont’d

0.5 1 1.5 0.1 0.2 0.3 0.4 ←dθ=qrf(T/q) Q (mg N/mg C) d (days−1)

Figure 5: Q

0.5 1 1.5 2 4 6 8 10 12 14 ←dθ=qrf(T/q) x (mg C/L) d (days−1)

Figure 6: x

Bifurcation on d for the model with linear functional response f(x) = ax.

• A. Packer

Cell Quota Based Models + Applications : — 40 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Holling Type II Functional Response

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 41 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Holling Type II Functional Response

f(x) = ax/(x + s)

If the functional response f is the Holling type II function f(x) = ax/(x + s), the model dynamics are more complicated. Differences from linear f LAS and GAS of E1 are not equivalent. Multiple E2 can coexist. E2 can exist even if E1 is LAS. Bistability: both LAS E1 and either LAS E2 or a stable periodic orbit about E2 can exist simultaneously.

• A. Packer

Cell Quota Based Models + Applications : — 42 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Holling Type II Functional Response

Holling Type II

0.2 0.4 0.6 0.8 1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 dθ / [rqf(T/q)] y (mg C/L) LAS unstable

Figure 7: y

0.2 0.4 0.6 0.8 1 1.2 0.5 1 1.5 2 dθ / [rqf(T/q)] x (mg C/L) LAS unstable

Figure 8: x

Bifurcation on d for the model with Holling type II functional response f(x) = ax/(x + s).

• A. Packer

Cell Quota Based Models + Applications : — 43 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Holling Type II Functional Response

Holling Type II, cont’d

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 X Y ←T=qX+θY E2 (unstable) E2 (LAS) E1 (LAS) E0 (unstable)

Figure 9: E1 and an E2 are LAS

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 X Y ←T=qX+θY E2 (unstable) E1 (LAS) E0 (unstable)

Figure 10: E1 is LAS; stable periodic orbit about an E2

Orbits projected into the xy-plane illustrating bistability.

• A. Packer

Cell Quota Based Models + Applications : — 44 / 72

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Holling Type II Functional Response

Holling Type II, cont’d

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 X Y ←T=qX+θY E2 (unstable) E1 (LAS) E0 (unstable)

Figure 9: E1 is LAS; stable periodic orbit about an E2

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 X Y ←T=qX+θY E1 (LAS) E0 (unstable)

Figure 10: E1 is GAS; no E2 exist.

Orbits projected into the xy-plane illustrating bistability.

• A. Packer

Cell Quota Based Models + Applications : — 44 / 72

Prostate Cancer and Androgen Deprivation Therapy

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 45 / 72

Prostate Cancer and Androgen Deprivation Therapy

Application to PCA

Applications to prostate cancer? What if we consider androgen (testosterone) as a limiting nutrient?

Portz, T., Y. Kuang and J. D. Nagy, “A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy”, AIP Advances 2, 1, 011002 (2012).

• A. Packer

Cell Quota Based Models + Applications : — 46 / 72

Prostate Cancer and Androgen Deprivation Therapy

Portz et al. (2012)

Xi: prostate cancer cells (cells×109), Qi: intracellular androgen concentration (nM), P: serum PSA concentration (ng/mL), Ts: serum testosterone concentration (nM). X′

1 = µm

• 1 − q1

Q1

• X1 − c1

(K1)n (Q1)n + (K1)n X1 + c2 (Q2)n (Q2)n + (K2)n X2, X′

2 = µm

• 1 − q2

Q2

• X2 + c1

(K1)n (Q1)n + (K1)n X1 − c2 (Q2)n (Q2)n + (K2)n X2, Q′

i = vm

qm − Qi qm − qi Ts Ts + vh − µm (Qi − qi) − bQi, i = 1, 2, P′ = σ0 (X1 + X2) + σ1X1 (Q1)m (Q1)m + (ρ1)m + σ2X2 (Q2)m (Q2)m + (ρ2)m − δP.

• A. Packer

Cell Quota Based Models + Applications : — 47 / 72

Prostate Cancer and Androgen Deprivation Therapy

It works!

• Ex. result from Portz et al. (2012). Patient data from Akakura et al. (1993).

Akakura, K., N. Bruchovsky, S. L. Goldenberg, P . S. Rennie, A. R. Buckley, and L. D. Sullivan. “Effects of intermittent androgen suppression on androgen-dependent tumors. Apoptosis and serum prostate-specific antigen.” Cancer 71, 9 (1993): 2782–2790.

• A. Packer

Cell Quota Based Models + Applications : — 48 / 72

Prostate Cancer and Androgen Deprivation Therapy

It works!

It works! But why, and how should it be interpreted? X′

1 = µm

• 1 − q1

Q1

• X1 − c1

(K1)n (Q1)n + (K1)n X1 + c2 (Q2)n (Q2)n + (K2)n X2, X′

2 = µm

• 1 − q2

Q2

• X2 + c1

(K1)n (Q1)n + (K1)n X1 − c2 (Q2)n (Q2)n + (K2)n X2, Q′

i = vm

qm − Qi qm − qi Ts Ts + vh − µm (Qi − qi) − bQi, i = 1, 2, P′ = σ0 (X1 + X2) + σ1X1 (Q1)m (Q1)m + (ρ1)m + σ2X2 (Q2)m (Q2)m + (ρ2)m − δP.

• A. Packer

Cell Quota Based Models + Applications : — 49 / 72

Prostate Cancer and Androgen Deprivation Therapy

It works!

Problems Interpretation of Q So-called uptake Use of Droop model Mechanism of treatment resistance “Mutation rate”

• A. Packer

Cell Quota Based Models + Applications : — 50 / 72

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 51 / 72

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation

Return to basics

Modeling AR kinetics Testosterone exchange between serum and prostate is proportional to the blood flow rate to the prostate and the concentration gradient. Prostatic testosterone is uniformly distributed amongst the prostate cells. Free testosterone is enzymatically converted to DHT by 5α-reductase. Free testosterone and DHT bind to free AR in the cytoplasm by second

• rder reaction kinetics.

Free AR, T, and DHT degrade by first order kinetics. A fixed total AR concentration, Rt, is maintained at homeostasis.

• A. Packer

Cell Quota Based Models + Applications : — 52 / 72

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation

AR kinetics model (Eikenberry et al. 2010)

CT(t): T:AR complex concentration (nM), CD(t): DHT:AR complex concentration (nM), R(t): intracellular free AR concentration (nM), D(t): intracellular free DHT concentration (nM), T(t): intracellular free T concentration (nM). C′

T = kT a TR − kT d CT,

C′

D = kD a DR − kD d CD,

R′ = λ − kT

a TR + kT d CT − kD a DR + kD d CD − βRR,

D′ = αkcat T T + KM − kD

a DR + kD d CD − βDD,

T′ = K(Ts − T) − kT

a TR + kT d CT − αkcat

T T + KM − βTT.

Eikenberry, S. E., J. D. Nagy, and Y. Kuang. “The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model.” Biology Direct 5, 1 (2010): 1–28.

• A. Packer

Cell Quota Based Models + Applications : — 53 / 72

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation

Population dynamics

Next step Need to translate intracellular AR dynamics to population level: Q Issues with Droop model Usage and meaning not entirely clear. Instead, let’s use hill functions for the growth rate and new androgen-dependent apoptosis rate.

• A. Packer

Cell Quota Based Models + Applications : — 54 / 72

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation

Population dynamics

Next step Need to translate intracellular AR dynamics to population level: Q Issues with Droop model Usage and meaning not entirely clear. Instead, let’s use hill functions for the growth rate and new androgen-dependent apoptosis rate.

• A. Packer

Cell Quota Based Models + Applications : — 54 / 72

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation

Proliferation and apoptosis rates

“Cell quota” Let Q be the sum of intracellular AR:T and AR:DHT complexes, CT + CD (Eikenberry et al. 2010). X′ = µX − δX, P′ = σ Qp Qp + (qσ)p X − βPP, where Q = CT + CD, µ(Q) = µm Qm Qm + (qµ)m , δ(Q) = δm (qδ)n Qn + (qδ)n + δ0.

• A. Packer

Cell Quota Based Models + Applications : — 55 / 72

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation

Yikes!

For each subpopulation (not including the “mutation rates”): X′ = µm Qm Qm + (qµ)m X −

• δm

(qδ)n Qn + (qδ)n + δ0

• X,

C′

T = kT a TR − kT d CT − µCT,

C′

D = kD a DR − kD d CD − µCD,

R′ = λ − kT

a TR + kT d CT − kD a DR + kD d CD − βRR − µR,

D′ = αkcat T T + KM − kD

a DR + kD d CD − βDD − µD,

T′ = K(Ts − T) − αkcat T T + KM − βTT − µT,

• P′ = σ

Qp Qp + (qσ)p X + σ0X − βPP

• A. Packer

Cell Quota Based Models + Applications : — 56 / 72

Prostate Cancer and Androgen Deprivation Therapy Uptake

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 57 / 72

Prostate Cancer and Androgen Deprivation Therapy Uptake

Closer look at uptake

Closer look at uptake Is the model from phycology better after all? Not necessarily Possible to mechanistically derive the “uptake” function from the AR kinetics model. Doing so results in model that better fits data.

• A. Packer

Cell Quota Based Models + Applications : — 58 / 72

Prostate Cancer and Androgen Deprivation Therapy Uptake

Closer look at uptake

Closer look at uptake Is the model from phycology better after all? Not necessarily Possible to mechanistically derive the “uptake” function from the AR kinetics model. Doing so results in model that better fits data.

• A. Packer

Cell Quota Based Models + Applications : — 58 / 72

Prostate Cancer and Androgen Deprivation Therapy Uptake

Closer look at uptake

Quasi steady state Intracellular model works on faster time scale than population dynamics. We can simplify the model using quasi steady states. D = 1 βD aT T + s, T = 1 2 (vTs − αm/h − s) + 1 2

• (vTs − αm/h − αk)2 + 4vTsαk

1/2 .

• A. Packer

Cell Quota Based Models + Applications : — 59 / 72

Prostate Cancer and Androgen Deprivation Therapy Uptake

Closer look at uptake

Quasi steady state Intracellular model works on faster time scale than population dynamics. We can simplify the model using quasi steady states. D = 1 βD aT T + s, T = 1 2 (vTs − αm/h − s) + 1 2

• (vTs − αm/h − αk)2 + 4vTsαk

1/2 .

• A. Packer

Cell Quota Based Models + Applications : — 59 / 72

Prostate Cancer and Androgen Deprivation Therapy Uptake

Closer look at uptake

Rewrite expression for T: T = 1 2 (vTs − αm/h − αk) + 1 2

• (vTs + αm/h + αk)2 − 4vTsαm/h1/2 ,

= 1 2

• (vTs − αm/h − αk) + (vTs + αm/h + αk)
• 1 −

4vTsαm/h (vTs + αm/h + αk)2

1/2

, (Note: The relation 4vTsαm/h (vTs + αm/h + αk)2 < 1 is already established by the fact that the expression under the radical in previous slide is nonnegative.)

• A. Packer

Cell Quota Based Models + Applications : — 60 / 72

Prostate Cancer and Androgen Deprivation Therapy Uptake

Closer look at uptake

Taylor expansion: T(n) = vTs − vTsαm/h vTs + αm/h + αk − (vTsαm/h)2 [vTs + αm/h + αk]3 − . . . − (2n)! 2(1 − 2n)(n!)2 (vTsαm/h)n [vTs + αm/h + αk]2n−1 .

• A. Packer

Cell Quota Based Models + Applications : — 61 / 72

Prostate Cancer and Androgen Deprivation Therapy Uptake

Closer look at uptake

The error for T(n) is 0 if Ts = 0 and is an increasing function of Ts. Error is approximated by T − T(n) ≈ (vTs + αm/h + αk) 2 (2n + 2)! (1 + 2n)((n + 1)!)2 (vTsαm/h)n+1 [vTs + αm/h + αk]2n+1 ≤ 2vTsαm/h vTs + αm/h + αk

• 4vTsαm/h

(vTs + αm/h + αk)2 n < 2vTsαm/h vTs + αm/h + αk 4−n < max {αm/h, vTs} αk 4−n. (6) Therefore T(1) and T(2) are good approximations for T if αm/h is sufficiently small or αk is sufficiently large. Since v < 1 and Ts < αk for applications to rat and human, the condition αm/h is more pertinent.

• A. Packer

Cell Quota Based Models + Applications : — 62 / 72

Prostate Cancer and Androgen Deprivation Therapy Uptake

Closer look at uptake

Approximations: T(1) = vTs D(1)

I

= αm βD vTs vTs + αk T(2) = vTs vTs + αk vTs + αk + αm/h D(2)

I

= αm βD vTs vTs + αk (vTs + αk)2 (vTs + αk)2 + αkαm/h . Uptake: V(Ts) = (Rt − Q) kT

a T(Ts) + kD a D(Ts)

V(1)(Ts) = vTs

• kT

a + kD a

αm βD 1 vTs + αk

• V(2)(Ts) = vTs

vTs + αk vTs + αk + αm/h

• kT

a + kD a

αm βD vTs + αk + αm/h (vTs + αk)2 + αkαm/h

• A. Packer

Cell Quota Based Models + Applications : — 63 / 72

Prostate Cancer and Androgen Deprivation Therapy Single population model

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 64 / 72

Prostate Cancer and Androgen Deprivation Therapy Single population model

Simplified Model

We now have a model with the same structure of Portz et al. (2012): X′ = µX − δm (qδ)n Qn + (qδ)n X − δ0X, Q′ = (Rt − Q)(kT

a T + kD a D) −

• (1 − fD)kT

d − fDkD d

• Q − µQ,

P′ = σ Qp Qp + (qσ)p X + σ0X − βPP, where T = vTs vTs + αk vTs + αk + αm/h, D = αm βD vTs vTs + αk (vTs + αk)2 (vTs + αk)2 + αkαm/h . µ(Q) = µm Qm Qm + (qµ)m .

• A. Packer

Cell Quota Based Models + Applications : — 65 / 72

Prostate Cancer and Androgen Deprivation Therapy Results

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 66 / 72

Prostate Cancer and Androgen Deprivation Therapy Results

Two Subpopulations: Cases 1 to 4

200 400 600 800 1000 1200 1400 2 4 6 8 10 12 14 16 18 20 days Serum PSA (ng/mL) and Cells (millions) PSA data PSA model CS cells CR cells 200 400 600 800 10 20 30 40 50 60 days Serum PSA (ng/mL) and Cells (millions) 200 400 600 800 1000 10 20 30 40 50 60 70 days Serum PSA (ng/mL) and Cells (millions) 200 400 600 800 1000 5 10 15 20 25 30 35 days Serum PSA (ng/mL) and Cells (millions)

Model with two cell subpopulations fitted to patient data from Akakura et al. (1993).

• A. Packer

Cell Quota Based Models + Applications : — 67 / 72

Prostate Cancer and Androgen Deprivation Therapy Results

Two Subpopulations: Cases 5 to 7

100 200 300 400 500 600 20 40 60 80 100 120 days Serum PSA (ng/mL) and Cells (millions) 100 200 300 400 500 600 10 20 30 40 50 60 days Serum PSA (ng/mL) and Cells (millions) 200 400 600 800 1000 2 4 6 8 10 12 14 16 days Serum PSA (ng/mL) and Cells (millions)

Model with two cell subpopulations fitted to patient data from Akakura et al. (1993).

• A. Packer

Cell Quota Based Models + Applications : — 68 / 72

Prostate Cancer and Androgen Deprivation Therapy Results

One Population: Cases 1, 3, 4

200 400 600 800 1000 1200 1400 20 40 60 80 days Serum PSA (ng/mL) and Cells (millions) PSA data PSA model cells 200 400 600 800 1000 10 20 30 40 50 60 70 days Serum PSA (ng/mL) and Cells (millions) 200 400 600 800 1000 10 20 30 40 days Serum PSA (ng/mL) and Cells (millions)

Model with one cell population fitted to patient data from Akakura et al. (1993).

• A. Packer

Cell Quota Based Models + Applications : — 69 / 72

Prostate Cancer and Androgen Deprivation Therapy Results

One Populations: Cases 5 to 7

200 400 600 20 40 60 80 100 120 days Serum PSA (ng/mL) and Cells (millions) 200 400 600 10 20 30 40 50 60 days Serum PSA (ng/mL) and Cells (millions) 200 400 600 800 1000 5 10 15 20 25 days Serum PSA (ng/mL) and Cells (millions)

Model with one cell population fitted to patient data from Akakura et al. (1993).

• A. Packer

Cell Quota Based Models + Applications : — 70 / 72

Prostate Cancer and Androgen Deprivation Therapy Future Work

Outline

1

Introduction

2

Neutral Lipid Synthesis in Green Microalgae Introduction Model Results Conclusion

3

Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction Model Equilibria Linear Functional Response Holling Type II Functional Response

4

Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation Uptake Single population model Results Future Work

• A. Packer

Cell Quota Based Models + Applications : — 71 / 72

Prostate Cancer and Androgen Deprivation Therapy Future Work

Future Work

Predictions

Current models are poor at making predictions. Can we formulate a mechanistic model that accurately predicts treatment

• utcomes?

Stochastic methods

Bayesian inference. Forecasting.

• A. Packer

Cell Quota Based Models + Applications : — 72 / 72