Research Projects - UPR-RPepsilon.uprrp.edu/aniel/Lectures/ResearchPresentation.pdf · Outline...

Research Projects

Aniel Nieves-Gonzalez

December 2016

Aniel Nieves-Gonzalez () Research projects December 2016 1 / 19

Outline

Projects on kidney physiology.I A mathematical model of a thick ascending limb (TAL) cell.I A mathematical model of the TAL segment.

Project on polyp population dynamics.I A mathematical model of A. cervicornis and its environment.I Asymtotic and structural stability analysis.

Funding oportunities


Kidney physiology

Nephrons are:the functional unit of thekidneys. ∼106 nephrons perkidney.embedded into the interstitialfluid, whose osmolarityincreases w/ depth.aren’t very accessible. Thisimplies having a rather sparsedata set to describe theircharacteristics.


Mathematical Model: Kidney physiology

The model of the TAL segment consists of an ensemble of model TALcells (70) that make up the tubule walls. We assume:

Mass and water conservationin luminal and cytosoliccompartments.Water impermeable TAL.A well-stirred & dilute cytosol.Solute electrodiffusion followsGHK constant-field fluxequation.



The conservation laws for the luminal compartment are

∂

∂xFw(x, t) = 0

∂

∂tCa

i (x, t) +Fw(t)A

∂

∂xCa

i (x, t) =2πrA

(Ja

i (Cc,Ca)− Jpi (Ca,Cb)

)




∂

∂xFw(x, t) = 0

∂

∂tCa

i (x, t) +Fw(t)A

∂

∂xCa

i (x, t) =2πrA

(Ja


)whereas in the cytosolic compartment of each cell we have

d

dtCc

ij(t) =(Cc

ij(t)AbJb

w(Cc,Cb)−(AaJa

i (Cc,Ca) +AbJbi (Cc,Cb)

)) 1Vj(t)

d

dtVj(t) = −AbJb

w(Cc,Cb)




∂

∂xFw(x, t) = 0

∂

∂tCa

i (x, t) +Fw(t)A

∂

∂xCa

i (x, t) =2πrA

(Ja


)whereas in the cytosolic compartment of each cell we have

d

dtCc

ij(t) =(Cc

ij(t)AbJb

w(Cc,Cb)−(AaJa

i (Cc,Ca) +AbJbi (Cc,Cb)

)) 1Vj(t)

d

dtVj(t) = −AbJb

w(Cc,Cb)

The model tracks the time evolution and the space dependency alongthe TAL segment of the cell volume, cytosolic, and luminalconcentrations of Na+, K+, Cl−, H+, NH+

4 , NH3 and others.



This is solved with a parallelized simple-splitting method with aflux-limiter for the homogeneous part, and a BDF for thenonhomogeneous part.



The model represent the solute fluxes as having two components:An electrodiffusive component.A carrier mediated component.

Additional equations are solved to ensure:Electroneutrality of the compartmentspH homeostasis.Cell volume regulation.


Conclusions and future goals: Kidney physiology

We learned about:The efficiency of the workload distribution among the TAL cellsthat comprise the TAL segment.The importance of NH+

4 cycling in TAL cells for Na+ along theTAL segment.

We will proceed:Incorporate Mg2+ transport along the segment.Study how the filtering properties of the segment arise from thedynamical properties of the cells.


Polyp population dynamics

ENVIRONMENTAL MONITORING Sea Surface Temperature (SST)

Solar Radiation (SR) Shallow—Deep

Spring—Summer—Fall—Winter

GENE EXPRESSION analysis of stress-

related genes

Biochemical analyses of photo-protective

proteins

SOMATIC GROWTH

REPRODUCTIVE EFFORT

TRANSPLANT EXPERIMENTS Deep !" Shallow

Of individual corals to determine coral resiliency in a changing environment.

Authentic Research Experience for

Undergraduates Incorporate research

into laboratory courses and gauge impact on student learning and retention in STEM

Community Outreach & Impact

Policymakers Website to share results through

interactive maps and science articles for

general public.

PREDICTIVE MATHEMATICAL MODEL Interactions between Acropora cervicornis and its environment

Coral Resiliency



1 The objective of this project is to study the dynamics of theinteraction of the polyps of the A. cervicornis coral with twosubstances that are part of their inmune system: melanine andflourecent protein (FP).

2 The synthesis and the breakdown of those substances affect thepolyp growth.

3 Environmental factors like solar radiation and water temperatureaffect the synthesis and breakdown of melanine and FP.



The model represents three variables:

1 Polyps of A. cervicornis per unit area (cm2).2 Concentration of melanin (in µM).3 Concentration of fluorescent protein (FP) (in µM).



The model represents three variables:1 Polyps of A. cervicornis per unit area (cm2).

2 Concentration of melanin (in µM).3 Concentration of fluorescent protein (FP) (in µM).



The model represents three variables:1 Polyps of A. cervicornis per unit area (cm2).2 Concentration of melanin (in µM).

3 Concentration of fluorescent protein (FP) (in µM).



The model represents three variables:1 Polyps of A. cervicornis per unit area (cm2).2 Concentration of melanin (in µM).3 Concentration of fluorescent protein (FP) (in µM).



The model represents three variables:1 Polyps of A. cervicornis per unit area (cm2).2 Concentration of melanin (in µM).3 Concentration of fluorescent protein (FP) (in µM).

Melanin and FP are proteins produced by the coral immune system tocombat different pathogens.



The main assumptions are:1 Logistic growth and linear “natural” decay for the polyp colony.

2 Polyp colony also decays as a linear function of melanin and FPconcentration.

3 The following statement of conservation of “energy” holds:

I At greater depth (where solar radiation and temperature are lower)the polyps grow stronger and the protein synthesis is weaker. Atlesser depth (where solar radiation and temperature are higher) thepolyps grow diminishes and the protein synthesis is stronger.



The main assumptions are:1 Logistic growth and linear “natural” decay for the polyp colony.2 Polyp colony also decays as a linear function of melanin and FP

concentration.

3 The following statement of conservation of “energy” holds:





concentration.3 The following statement of conservation of “energy” holds:





concentration.3 The following statement of conservation of “energy” holds:



Mathematical model: Polyp population dynamics

The model equations are a dynamical system:

x1(t) = F1(d)βax1

(1− x1

κa

)− δax1 − (δafx2 + δamx3)x1

x2(t) = F2(d)βfx2x1 − (δfx2x1 + δfmx3x2)

x3(t) = F3(d)βmx3x1 − (δmx3x1 + δmfx2x3)

and the environmental function is

F1(d) = e6d

F2(d) = e−100d = F3(d)

The numerical solution of the model show how the polyp and theprotein concentrations evolve in time. This allows us to makepredictions about the system.


Asymtotic stability analysis: Polyp pop. dynamics

Important definitions:Equilibrium point.Lyapunov stability of an equilibrium point.Asymptotic stability of an equilibrium point.An equilibrium point can be hyperbolic or nonhyperbolic.Hyperbolic equilibria can be further classified into

I SourceI SinkI saddle



Important mathematical “tools” for the analysis.

Theorem (Hartman-Grobman)

Let x∗ be a hyperbolic equilibrium point of a C1 vector field f(x) withflow φt(x). Then there is a neighborhood N of x∗ such that φ isequivalent (topologically conjugate) to its linearization on N .




Definition (Lyapunov function)

A continuous function L : Rn → R is a strong Lyapunov function for anequilibrium point x∗ of a flow φt(x) on Rn if there is a neighborhood Uof x∗ such that L(x∗) = 0, L(x) > 0 ∀x 6= x∗ and L(φt(x)) < L(x) forall x ∈ U {x∗} and ∀t > 0. A weak Lyapunov function satisfiesL(φt(x)) ≤ L(x) rather than the strict inequality.




Theorem

Let x∗ be an equilibrium point of a flow φt(x). If L is a weak Lyapunovfunction in some neighborhood U of x∗, then x∗ is stable (Lyapunovstable). If L is a strong Lyapunov function in some neighborhood U ofx∗, then x∗ is asymptotically stable.



The objective is to characterize the stability of the equilibria andinterpret the results in biological terms.



The analysis yield results like...

Table: Stability conditions for equilibria1

EPs Stability condition Unstability condition Physical feasibility

x∗1 p1 < 0, p2 < 0, p3 < 0 N/A unconditionalx∗2 p1 > 0, p2 < 0, p3 < 0 p1 < 0, p2 < 0, p3 < 0 p1 ≥ 0x∗3 saddle N/A p1 ≥ 0, p2 ≥ 0, p3 ≥ 0x∗4 p1 < 0, p2 < 0, p3 < 0 N/A always unphysicalx∗5 p1 < 0, p2 < 0, p3 < 0 N/A always unphysical1 p1 = F1βa − δa, p1 = F2βf − δf , and p1 = F3βm − δm.


Structural stability analysis: Bifurcations

Informally, a bifurcation is a qualitative change in the flowassociated to the solution of an ODE system that is produced by achange in a parameter of such system.

For example:

I Saddle-node bifurcation.I Andronov-Hopf bifurcation.

Bifurcations can be local or global.Structurally stable equilibria means that they can’t be removed bysmall changes in the equations (parameters).The existence of bifurcations in a model might have a relevantinterpretation for the physical system that the model represent.Numerical continuation methods are used to find and studybifurcations.



Informally, a bifurcation is a qualitative change in the flowassociated to the solution of an ODE system that is produced by achange in a parameter of such system.For example:






I Saddle-node bifurcation.

I Andronov-Hopf bifurcation.











Bifurcations can be local or global.

Structurally stable equilibria means that they can’t be removed bysmall changes in the equations (parameters).The existence of bifurcations in a model might have a relevantinterpretation for the physical system that the model represent.Numerical continuation methods are used to find and studybifurcations.





Bifurcations can be local or global.Structurally stable equilibria means that they can’t be removed bysmall changes in the equations (parameters).

The existence of bifurcations in a model might have a relevantinterpretation for the physical system that the model represent.Numerical continuation methods are used to find and studybifurcations.





Bifurcations can be local or global.Structurally stable equilibria means that they can’t be removed bysmall changes in the equations (parameters).The existence of bifurcations in a model might have a relevantinterpretation for the physical system that the model represent.

Numerical continuation methods are used to find and studybifurcations.







Conclusions and further work

The mathematical model can be used as a tool for coralconservation, by allowing us to address questions like:

I Given the changes in solar radiation and ocean temperature due toglobal warming, in the long run, what is the polyp growth?

I How does the polyp population evolves when they are “planted” ata particular depth? Is there an optimal depth for planting thecorals?


Funding oportunities

NIHNSFNOAA


Questions. . .


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