Biochemomechanical Hormone Delivery Oscillator: Theory · PDF fileBiochemomechanical Hormone Delivery Oscillator: Theory and Experiments ... Titration. 31 Fits to Henderson ... Close

1

Biochemomechanical Hormone Delivery Oscillator:

Theory and Experiments

Ronald A. SiegelDepts. Of Pharmaceutics and Biomedical

EngineeringUniversity of Minnesota

2

NO

T EN

OU

GH

(IN

EFFC

ITVE

)JU

ST

RIG

HT

TOO

M

UC

H

(TO

XIC

)“Traditional” Drug Delivery “Nontraditional” Drug Delivery

3

Stimuli-Sensitive, Reversible Swelling/Collapse Transition in Hydrogels

Temperature

pHGlucose

UreaAntigen/Antibody

DNA HybridElectricityMagnetism

Light

CollapsedImpermeable

SwollenPermeable

4

FREQUENCYHORMONE (pulses/day)__

Growth Hormone 9-16, 29Prolactin 4-9, 7-22Thyroid Stimulating Hormone 6-12, 13Adrenocorticotropic Hormone 15, 54

Gonadotropin Releasing Hormone (GnRH) 12-24Luteinizing Hormone (LH)Follicle Stimulating Hormone (FDH)

Estradiol 8-16, 8-19Progesterone 8-12, 6-16Testosterone1 3, 8-12

-Endorphin 13Melatonin 18-24, 12-20Vasopressin 12-18Renin 6, 8-12Parathyroid Hormone 24-139, 23Insulin 108-144, 120Pancreatic Polypeptide 96Somatostatin 72Glucagon 103, 144Aldosterone 6, 9-12Cortisol 15, 39

MANY HORMONES DISPLAY ULTRADIAN PULSATILITY

Source: G. Brabant et al., Trends in Endocrinology and Metabolism, 3, 183-190 (1992).

5

Pulsatile vs. Continuous

Responses of LH & FSH levels to the intermittent (hourly) or continuous infusion of GnRH

Belchetz, P. E.; T.M. Plant; E. J. Keogh; E. Knobil Science 1978 202 : 631-633

6

Hypogonadotropic HypogonadismLack of endogenous pulsatile Gonadotropin Releasing Hormone(GnRH) secretion

Clues From Nature, Artificial Chemical Oscillators• Glycolysis—NADH

• Intracellular Calcium Oscillations

• Cellular Circadian Oscillators

• Chemical and pH Oscillators (BZ, Landolt, Bromate, etc.); Teorell Membrane Oscillator and Related Systems

Away from Equilibrium, Strong Feedback Loops

Huang, N., et al., Science, 2012

8

Lessons from Chemical Oscillators: Design Heuristics

1. Away from equilibrium

2. Bistable element

3. Feedback element

Epstein, Orban, Boissonade, DeKepper

9

Hormone Delivery Oscillator--Design

10

Product Removal

Constant Source

Aw

ay fr

om E

quili

briu

m

Bistable

Feedback

Flory-Rehner-Donnan-Langmuir Theory

bindingionicelasticmixingTotal GGGGG

Crosslink Density: polymer‐solvent interaction parameter

Polymer‐solvent mixing Network elasticity

COO‐Na+

Na+

Na+

Na+

Ion Osmotic Pressure

%MAA incorporated, pH, salt concentration

Binding (ionization)

pH

Flory-HugginsFlory-Rehner

Flory-Rehner-DonnanFlory-Rehner-Donnan-Langmuir

|--COO- + H+

|--COOH

Ka

Flory-Rehner-Donnan-Langmuir (FRDL) Model (1D)

)2/1(2

)1ln(0

00

2sww

wFRDL cvv

vRT

0)/()/1()1( 00 fcs

]101/[1 ionizedfraction )( pKapHf

Swelling Stress

Electroneutrality Inside Hydrogel

Binding Isotherm

L/mol) (0.018 water of memolar volu

groupMAA ofacidity energy malmolar ther

mol/L) (0.15ion concentratsalt externalsynthesisat

groupsMPBA of hydrogel) of (mol/Lion concentratsynthesisat polymer offraction volume

synthesisat hydrogel) (mol/Ldensity chain active)(parameter n interactiolvent polymer/so

0

0

0

21

w

s

v

pKaRT

cpH externalpH

hydrogelin polymer offraction volume

medium externalin NaCl s hydrogelin Cl ,Na

)//( ratioDonnan ionizedfraction

-ClNa

ccccf

ss

Fixed Parameters

where

mixing elastic ion osmotic

pH

Swel

ling

pKa pH

Swel

ling

pKa

Depends on polymer hydrophobicity ) initial crosslink density 0initial concentration of ionizable (MAA) groups in hydrogel 0 external salt

concentration (Cs)

Monotonic or Bistable Equilibrium Behaviors

pHΔFpH

Swelling Ratio V/V0 = 0/

pHΔFpH

Common hydrogel with MAA

NIPA-MAA hydrogel

0FRDL

VF

Swelling Ratio V/V0 = 0/

14

pH Stat pHMeter

Magnetic stirrer

DonorCell ReceptorCellGlucoseOxidase

GlucoseSolution In

Donor CellSolution OutRecorderpH Stat pH Controller

Magnetic Stirrer

DonorCell

ReceptorCell

Radio-labeledGlucose in

Donor CellSolution Out

Recorder

4.9

5.2

DPM

Time, hr

Receptor

pHR

eceptor Cell pH

DPM DPM

Time, hr

5.2

5.1

5.0

4.9

5.0

5.1

5.2 Receptor C

ell pHHysteresis in Glucose Flux Across MAA-NIPA Hydrogel Membrane

Baker J. P. and Siegel R. A., Macromol. Rapid Comm. 16, 1996

15

Device pH: HighMembrane: open

0

1

2

3

4

4.5 4.75 5 5.25 5.5Device pH

Perm

eabi

lity

(Dim

ensi

onle

ss)

out

Glucose in

In-line pH MeasurementpH Stat pH Meter

pH=7.4T=370 C

MEMBRANE

Magnetic Stirrer

Glucose GlucoseGlucose oxidase (GOX)

Drug

T=370 CDonor Cell Receptor Cell

50 mM Saline 50 mM Saline

Marble

16

Device pH: DecreasingMembrane: Open

0

1

2

3

4

4.5 4.75 5 5.25 5.5Device pH

Perm

eabi

lity

(Dim

ensi

onle

ss)

out

Glucose in


pH=7.4T=370 C

MEMBRANE

Magnetic Stirrer

Glucose Glucose

H+

Glucose oxidase (GOX)

Drug



Marble

17

0

1

2

3

4

4.5 4.75 5 5.25 5.5Device pH

Perm

eabi

lity

(Dim

ensi

onle

ss)Device pH: Low

Membrane: Closed

out

Glucose in


pH=7.4T=370 C

MEMBRANE

Magnetic Stirrer

Glucose Glucose

H+

Glucose oxidase (GOX)

Drug



Marble

X

X

18

Device pH: IncreasingMembrane: Closed

0

1

2

3

4

4.5 4.75 5 5.25 5.5Device pH

Perm

eabi

lity

(Dim

ensi

onle

ss)

out

Glucose in


pH=7.4T=370 C

MEMBRANE

Magnetic Stirrer


Drug



Marble

X

X

H+

19

Device pH: HighMembrane: open

0

1

2

3

4

4.5 4.75 5 5.25 5.5Device pH

Perm

eabi

lity

(Dim

ensi

onle

ss)

out

Glucose in


pH=7.4T=370 C

MEMBRANE

Magnetic Stirrer


Drug



Marble

20

Preliminary Oscillation Attempt

Leroux J. C. and Siegel R. A., Chaos, 9(2): 1999

Glucose Concentration

21

Hypothesis Regarding Lack of Sustained Oscillations and How to Fix

Closed

Steady State

Open

pH

Perm

eabi

lity

Slow Dynamics

Closed

Fast Dynamics

22

Acceleration of Oscillations using Marble

d[H+]/dt ~ [Glucose influx] – [H+ ion elimination]

MarbleCaCO3

H+ Ca2+ + HCO3-

kmarble+

Rábai, Gy.; Hanazaki, I. J. Phys. Chem. 1996, 100, 10615.

230 24 48 72 96 120 144 168

-1

0

1

2

3

4

5

6

7

Rate

of D

eliv

ery

(nm

ol/m

in)

Time (Hours)

1000 1100 1200 1300 1400 1500-1012345678

Rat

e of

Del

iver

y (n

mol

/min

)

Time (minutes)

Proof of Concept-Rhythmic, Pulsed Delivery of GnRH

G. P. Misra and R. A. Siegel, J. Controlled Release 81-1:2, 1 (2002)

50 mM glucose

0 2 4 6 84 .4

4 .6

4 .8

5 .0

Rec

epto

r cel

l pH

T im e (D ays)

24

Questions

• When can oscillations occur?

• Why do pH oscillations stop?

• Can we model this system?

25

Parameters We Can Control

• Upstream glucose concentration• Membrane composition (MAA content)• Marble surface area (Heterogeneous reactivity)

• Enzyme concentration (Presently in excess)• Volume of reaction chamber (Fixed for now)

26

5 mol % MAA

0

2

4

6

8

10

12

14

0 20 40 60 80

Glucose Concentration (mM)

OscillationNo Oscillation

Surf

ace

area

exp

osed

mar

ble

(sq.

cm

)

Range Phase Diagram

4.7

4.9

5.1

5.3

5.5

5.7

0 20 40 60 80

pH

18 23 30 13 mM

Time (Hours)

Upstream Glucose Concentration (mM)

Surface area marble exposed = 10 sq. cm

27

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

0 20 40 60 80 100 120 140Time (Hours)

pH

33 40 55 70 103 110

0

2

4

6

8

10

12

14

0 50 100 150Glucose Concentration (mM)

Surf

ace

area

exp

osed

mar

ble

(sq.

cm

)

10 mol % MAA

Range Phase Diagram

OscillationNo Oscillation

Upstream Glucose Concentration (mM)

Surface area marble exposed = 10 sq. cm

28

Shift In Glucose RangeE

xpos

ed M

arbe

Sur

face

Are

a

Glucose Concentration (mM)

29

Questions




30

Evolution of Periodicity, Contents of Reaction Volume

4.55

4.65

4.75

4.85

4.95

5.05

5.15

0 1 2 3 4 5 6 7Time (Days)

pH 0

50

100

150

200

250

1 11 21 31 41 51 61 71

Pulse (#)

Puls

e D

urat

ion

(Min

utes

)

2.3

2.8

3.3

3.8

4.3

4.8

5.3

0 0.2 0.4 0.6 0.8 1 1.2 1.4Volume of 0.009 N HCl added (mL)

pH

0.178 0.704 1.146 1.77

2.158 2.77 3.69 4.04

4.68 5.69 6.77 0

Days

pH Oscillation Period Increase

Titration

31

Fits to Henderson-Hasselbalch Equation

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6 7

Time (Days)

Buf

fer C

once

ntra

tion

(mM

)

3.2

3.25

3.3

3.35

3.4

3.45

3.5

3.55

3.6

pKa

Buffer ConcentrationCell I Glucose ConcentrationpKaAverage pKa

Close to pKa of gluconic acid (~3.6)

32

Further Evidence Implicating Gluconate

20 40 60 80 100

4.50

4.75

5.00

5.25

5.50

pH

Time (hrs)

Receptor cellsolution replaced

0 5 10 15 20 254.4

4.5

4.6

4.7

4.8

4.9

5.0

5.1 Gluconic Acid

pH

Time (hours)

33

Ramped pH–Glucose Permeability Profile

0.0 2.5 5.0 7.5 10.0 12.5 15.010.0

10.5

11.0

11.5

12.0

12.5

IVIIIII

Time (hrs)

Vol 0

.1N

NaO

H

I

4.3

4.4

4.5

4.6

4.7

4.8

pH

Hypothesis: Membrane phase separates due to stress, causing increased glucose flux at III

Glu

cose

Per

mea

ted

pH=4.3Glucose

pH 7.0III

Swollen membrane

pH=4.8Glucose

pH 7.0

I,

IV

pH=4.3Glucose

pH 7.0Swollen

layer

Collapsed layer

IIStress

34

NaOH

HCl

Microscope Video Monitor

Cell B

Cell A

Water B

ath

Light

Drain

Metal Ring

Membrane

Saline Reservoir

Saline Reservoir

Setup to Study Membrane Morphology

35

1 mm

P Q

R1 R2

S pH=3

Optical Video Micrographs of Membrane Under Stress

0 2 4 6 8 10 12 144.00

4.25

4.50

4.75

5.00

5.25

5.50

S

Q R

P

pH

Time (hrs)

pH profile in Cell BCell A = 7.4

2 cm

R1 R2

36

Rapid pH swings promote sustained oscillations

• Using marble and ↑ glucose ↑ frequency

• Gluconate buildup ↓ frequency

• Slow oscillations might lead to intermediate permeability due to stress-induced membrane heterogeneity. Intermediate permeability permits system to find a stationary attractor.

Summary of Mechanistic Analysis

37

Questions




38

Simple Biphasic Pseudolinear “Toy” Model

Steady state “attractors”

h

HgLss

h

HgHss P

GPh

PGP

h ,,

,, ;

x

x

x

xG = external glucose concentrationh = H+ concentration in chamberV = volume of chamberA = area of membranePgH = “high” glucose permeabillityPgL = “low” glucose permeabillityPh = H+ permeability

39

Characteristics of Biophasic Pseudolinear “Toy” Model

Criterion for Oscillations

TH

TLPgL

PgH

HLLss

LHLss

hL

LHHss

HLHss

hH

hhhh

APVT

hhhh

APVT

,

,

,

,

ln

ln

Time Parameters

LHGL

hHL

GH

h hPPGh

PP

40

Effects of Marble

h

marmar

APA

Reaction of H+ with marble renormalizes Ph by factor (1+) where

LHh

Lg hPP , G

Osc

(m

arb

le )

HLh

Hg hP

P ,

LHGL

hHL

GH

h hPPGh

PP )1()1(

HLLss

LHLss

hL

LHHss

HLHss

hH

hhhh

APVT

hhhh

APVT

,

,

,

,

ln1

ln1

Effect of MAA “Doping”

41

LHGL

hHL

GH

h hPPGh

PP

Criterion for Oscillations

conc. ionizable groups fraction ionized: Mass action (Langmuir) binding

Algebra:

G

Osc

(mar

ble

)Osc

Conclusion: Increase in MAA doping leads to increased glucose concentrations needed for oscillations

42

“Toy” Lumped Mathematical Model

21 2

1(1 )[ln(1 ) ( ) ( ) ( 2)]2

ow ww o s

o

dL k v v Cdt

Swelling:

Acid in Membrane: [ ( )] (1 )( 2 )COOH

M M II MH H H H

d L C C k C Cdt

Acid in Device:0

(1 )( )]II II M IIG HH G H H marble H

Ak Akd C e C C C k Cdt V V

0 0(1 )( 1/ ) ( / ) 0sC f Membrane Electroneutrality:

1 1(1 / ) (1 10 )a IIpK pHMH af C K Fraction Ionized:

Conservation of Polymer: L =L0

Mixing Elastic Ion Osmotic

Bound Free

Acid DiffusionGlucose Inflow

“Free Volume Theory”

Marble Shunt

43

0.0 0.1 0.2 0.3 0.4

1

2

3

4Sw

ellin

g R

atio

L/L

0

f0 (M)4.0 4.5 5.0 5.5 6.00

1

2

3

4

0=0.28M

0=0.20M

0=0.15M

0=0.38M

Swel

ling

Rat

io L

/L0

pHII

0 20 40 60 800.1

0.2

0.3

0.4

0.5

U

S

S

0 (M

)

CG (mM)

OSC

DecayDecayDecay

Decay

Bifurcation Diagram

0 25 50 75 100 1254.25

4.50

4.75

5.00

5.25

5.50

5.75

6.00

0=0.20 M, CG = 10mM

0=0.26 M, CG = 22mM

0=0.38 M, CG = 50mM

pH

Time (hrs)0 20 40 60

4.2

4.4

4.6

4.8

5.0

5.2

5.4

5.6

5.8

2% MAA, 15mM Glucose



pH

Time (hours)

“Theory” Experiment

44

Distributed, PDE Modeling (Jon Urban)• Assume Now That Concentrations, Stress

Field, Electrical Potential, etc., Vary Across Thickness of Membrane, 0<x<L(t).

• Laboratory, Reference Frames

• A Sampler of PDE’s Used:x

x0

dx0

dx=α(x0)dx0

0

0

L0

L(t)

oxtxf

ERTFCz

xC

Dxt

CCKKC

CC

HiHHo

H

Ha

aoAT

HH

o2

ClClNaNa

ClCl

NaNa

CDCDx

CDx

CD

FRTE

xK

KC

xC

Dxt

CC

C S

S

SSSo

So

SS

o

1

Hydrogel Deformation

Hydrogen Ion Transport, Binding

Electric Field

Glucose Transport

dx

45

Some Predictions

00.5

11.5

22.5

33.5

4

0 200 400 600Membrane Coordinate (µm)

α

0 hrs

0.5 hrs1 hr

1.5 hrs

0.E+001.E-052.E-053.E-054.E-055.E-056.E-057.E-05


CH (M

)

0.5 hrs

0 hrs

1 hr1.5 hrs

020406080

100120140160


CA- (

mM

)

0.5 hrs

0 hrs

1 hr1.5 hrs

-1000-500

0500

1000150020002500300035004000

0 200 400 600

Membrane Coordinate (µm)

E (V

/m)

0.5 hrs

0 hrs

1 hr

1.5 hrs

46

Summary• Autonomous, rhythmic, pulsed delivery of GnRH, fueled by

constant level of glucose, has been achieved from hydrogel/enzyme oscillator based on NIPA/MAA membrane and glucose oxidase. Oscillations are due to membrane bistabilityand feedback from enzyme reaction.

• Marble required for sustained oscillations, due to speeding up of membrane transitions.

• Models presented: Simple Relay, ODE, 1-D PDE• Not covered: Oscillations persist for several days but slow down

due to accumulation of gluconic acid, a buffering species, in the chamber. Oscillations halt when oscillations become too slow.

• Halting of oscillations correlated with phase separation (nucleation/growth or spinodal decomposition) in membrane, leading to intermediate permeability to glucose and establishment of stable steady state.

47

Acknowledgments

• NIH, NSF, Amgen• Profs. Carme Calderer, Yoichiro Mori• Postdocs: John Baker, Jean-Christophe Leroux, Gauri Misra, Eric

Nuxoll, Bingtuan Li, Xiaoqin Zou, Lingxing Yao• Graduate Students: Anish Dhanarajan, Siddharthya Mujumdar,

Amardeep Bhalla

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Biochemomechanical Hormone Delivery Oscillator: Theory · PDF fileBiochemomechanical Hormone Delivery Oscillator: Theory and Experiments ... Titration. 31 Fits to Henderson ... Close