POLITECNICO DI MILANO PhD... · 2.4 Docking 49 3. Method validation: fluorescein behavior in water solutions 52 3.1 Sodium fluorescein 53 ... A.2.3 Cross-linking polyacrylic acid

MOLECULAR MODELING OF TRANSPORT PHENOMENA IN

HYDROGELS

Tesi di Dottorato di TOMMASO CASALINI

Matricola 753868

Coordinatore: prof. Tiziano Faravelli

Tutore: prof. Maurizio Masi

Relatori: prof. Carlo Cavallotti

dr. Giuseppe Perale

POLITECNICO DI MILANO

DIPARTIMENTO

DI

CHIMICA,

MATERIALI

E

INGEGNERIA CHIMICA

Giulio Natta

Dottorato di Ricerca in

Chimica Industriale e

Ingegneria Chimica (CII)

XXV ciclo

2010 - 2012

ii

Entia non sunt moltiplicanda

praeter necessitatem.

Frustra fit per plura quod

fieri potest per pauciora.

iii

iv

Publications and Conferences

Publications on International ISI journals

1. F. Rossi, R. Ferrari, S. Papa, D. Moscatelli, T. Casalini, G. Forloni, G. Perale, P. Veglianese,

Tunable hydrogel nanoparticles release system for sustained combination therapies in the spinal

cord, Colloids and Surface B: Biointerfaces, accepted, in press.

2. T. Casalini1, F. Rossi1, E. Raffa. M. Masi, G. Perale, Bioresorbable polymer coated drug eluting

stent: a model study, Molecular Pharmaceutics, 2012, 9 (7), 1898 1910; 1equally contributed.

3. T. Casalini, M. Masi, G. Perale, Drug eluting sutures: a model for in vivo estimations, International

Journal of Pharmaceutics, 2012, 429, 148 157.

4. T. Casalini, M. Salvalaglio, G. Perale, M. Masi, C. Cavallotti, Diffusion and aggregation of sodium

fluorescein in aqueous solutions, Journal of Physical Chemistry B, 2011, 115, 12896 12904.

5. F. Rossi, T. Casalini, M. Santoro, A. Mele, G. Perale, Methylprednisolone diffusivit in Agar-

Carbomer based hydrogel: a promising tool for local drug delivery, 2011, Chemical Papers, 65 (6),

903 908.

6. F. Rossi, M. Santoro, T. Casalini, G. Perale, Characterization and degradation behavior of Agar

Carbomer hydrogels for drug delivery applications: solute effect, International Journal of Molecular

Science, 2011, 12 (6), 3857 3870.

7. T. Casalini, F. Rossi, M. Santoro, G. Perale, Structural characterization of PLLA and PGA

oligomers in water, International Journal of Molecular Science, 2011, 12 (6), 3394 3408.

8. F. Rossi, M. Santoro, T. Casalini, G. Perale, Synthesis and characterization of lanthanum bonded

Agar Carbomer hydrogel: a promising tool for biomedical research, Journal of Rare Earths, 2011,

29 (3), 259 264.

9. G. Perale, T. Casalini, V. Barri, M. Muller, S. Maccagnan, M. Masi, Lidocaine release from

polycaprolactone threads, Journal of Applied Polymer Science, 2010, 117 (6), 3610 3614.

Isi conference proceedings

10. T. Casalini, F. Rossi, P. Torriani, M. Masi, P. Veglianese, G. Perale, Mathematical modeling of in

vivo drug release from hydrogels for spinal cord injury treatment, European Cells and Materials,

2012, 23 suppl 5, 33.

v

Abstract or posters presented at international conferences

11. T. Casalini, G. Perale, M. Masi, C. Cavallotti, Molecular dynamics investigation of Agar

Carbomer hydrogels, poster presentation at eSSENCE workshop in Multiscale material modeling

2012, 7 8 june Uppsala (Sweden).

12. F. Rossi, T. Casalini, E. Raffa, M. Masi, G. Perale, Bioresorbable polymer coated drug eluting

stent: a model study, poster presentation to 9th World Biomaterials Congress 2012, 1 5 june,

Chengdu (China).

13. T. Casalini, M. Masi, G. Perale, Drug eluting sutures: a model for in vivo estimations, poster

presentation to 9th World Biomaterials Congress 2012, 1 5 june, Chengdu (China).

14. T. Casalini, F. Rossi, P. Torriani, M. Masi, P. Veglianese, G. Perale, Mathematical modeling of in

vivo drug release from hydrogels for spinal cord injuries treatment, poster presentation to 5th Annual

Meeting of the Scandinavian Society for Biomaterials 2012, 8 9 may, Uppsala (Sweden).

Conference proceedings

15. G. Perale, P. Veglianese, F. Rossi, M. Peviani, D. Llupi, T. Casalini, E. Micotti, M. Masi, A novel

hydrogel-based system as a flexible tool for spinal cord injury repair strategies, 23rd

European

Conference on Biomaterials 2010, 11 15 september, Tampere (Finland).

16. G. Perale, T. Casalini, V. Barri, M. Muller, S. Maccagnan, M. Masi, Lidocaine eluting sutures: from

device design to experimental production, AES-ATEMA 2009 Fourth International Conference on

Advances and Trends in Engineering Materials and their Applicatons, 1 4 september, Hamburg

(Germany).

Book chapters

17. T. Casalini, G. Perale, Types of biodegradable polymers, in M. Jenkins and L. Overend, Durability

and reliability of medical polymers, Cambridge, Woodhead, Chapter 1; ISBN 978 1 84569 929 1

(2012).

18. T. Casalini, G. Perale, Processing of bioresorbable polymers, in M. Jenkins and L. Overend,

Durability and reliability of medical polymers, Cambridge, Woodhead, Chapter 1; ISBN 978 1

84569 929 1 (2012).

vi

vii

Table of Contents

Publications and conferences iv

Table of Contents vii

List of Figures xiii

List of Tables xxv

Abstract 1

1. Introduction 4

1.1 Hydrogels 6

1.2 Diffusion phenomena in hydrogels 8

1.3 Atomistic modeling of hydrogels 10

1.3.1 Three chains model 11

1.3.2 Multiple chains model 13

1.3.3 Amorphous cell model 13

1.3.4. Coarse-grain models 14

1.4 Agarose Carbomer hydrogels: a brief overview 15

1.4.1 Raw materials 16

1.4.2 Hydrogel synthesis 19

1.4.3 Hydrogel characterization 20

1.5 Sodium fluorescein: a peculiar behavior 21

1.5.1 Experimental methods 22

1.5.2 Diffusion data 22

1.6 Summary and aim of the work 23

2. Methods and theoretical background 26

2.1 Molecular dynamics simulations 26

2.1.1 Charge derivation: R.E.S.P. method 29

viii

2.1.2 Einstein equation 31

2.1.3 Radial distribution functions 33

2.2 Free energy calculations 35

2.2.1 Free energy perturbation methods 36

2.2.2 Thermodynamic integration methods 39

2.2.3 Potential of mean force 39

2.2.4 Umbrella Sampling 40

2.2.5 Weighted Histogram Analysis Method 41

2.2.6 Umbrella Integration 44

2.2.7 Error estimation: systematic and statistical errors 45

2.3 Poisson Boltzmann equation 47

2.4 Docking 49

3. Method validation: fluorescein behavior in water solutions 52

3.1 Sodium fluorescein 53

3.2 Simulation protocol 55

3.2.1 Sodium fluorescein: structure and atomic charges 55

3.2.2 Molecular dynamics simulations 56

3.2.3 Dimer structure 57

3.2.4 Dimerization potential of mean force 57

3.3 Diffusion phenomena 58

3.4 Radial distribution functions 61

3.5 Aggregation phenomena 64

4. Molecular modeling of Agarose Carbomer hydrogels 77

4.1 Hydrogel model 77

ix

4.1.1 Carbomer 974P 78

4.1.2 Agarose 79

4.1.3 Matrix model 79

4.1.4 Atomic charges 80

4.2 Simulation protocol 81


4.2.2 Numeral density maps 83

4.3 Diffusion coefficients 84

4.3.1 Sodium fluorescein 84

4.3.2 Sodium and chlorine ions 95

4.3.3 Water 97

4.4 Coordination numbers 99

4.5 Aggregation phenomena 102

4.6 A simple model for diffusion 103

4.7 Mesh size effect on fluorescein self-diffusion 110

5. Molecular modeling of hyaluronic acid hydrogels 113

5.1 Bone morphogenetic protein 2 114

5.2 Hyaluronic acid 116

5.3 Computational protocol 118

5.3.1 BMP2 structure and atomic charges 118

5.3.2 Hyaluronic acid structure and atomic charges 118

5.3.3 Docking 119


5.3.5 Potential of mean force 120

5.4 Results 121

5.4.1 Wrist complex 121

x

5.4.2 Knuckle complex 123

5.4.3 Upper complex 125

5.4.4 Lower complex 127

5.5 Summary 128

6. Conclusions 130

A.1 Appendix A 137

A.1 Fluorescein structure and charges 137

A.2 Polyacrylic acid structure and charges 140

A.2.1 Polyacrylic acid 140

A.2.2 Polyacrylic acid with propylene glycol 141

A.2.3 Cross-linking polyacrylic acid 143

A.3 Agarose structure and charges 145

A.3.1 Agarose 145

A.3.2 Cross-linking agarose 148

A.4 Protonated glucuronic acid structure and charges 150

A.2 Appendix B 152

B.1 7 nm hydrogels 152

B.1.1 Diffusion coefficients 152

B.1.2 Radial distribution functions 154

B.1.3 Coordination numbers 155

B.1.4 Monomer molar fraction 156



xi



















B.6 Water and PBS solutions 177




B.7 Diffusion model 184

B.7.1 Mass conservation 184

B.8. Bone morphogenetic protein 2 hyaluronic acid complexes 185

B.8.1 Binding areas 185

xii

A.3 Appendix C 187

C.1 Hydrogel synthesis 187

C.1.1 Materials 187

C.1.2 Synthesis 187

C.1.3 Drug loading 187

C.2 Hydrogel characterization 188

C.2.1 Swelling and degradation behavior 188

C.2.2 Mesh size: Flory Rehner theory 188

C.2.3 Drug release experiments 189

C.2.4 Self-diffusion measurements 190

C.2.5 FTIR analysis 191

C.3 In vivo experiments 192

Bibliography 196

xiii

List of Figures

Chapter 1 - Introduction

Figure 1.1. Schematic representation of hydrogel release (a) and in vivo qualitative bioluminescence of

antibody Alexa 647 delivery at four different time points (b).

Figure 1.2. Hydrogel schematic representation.

Figure 1.3. Agarose Carbomer hydrogel.

Figure 1.4. Chitosan-based gel swelling with respect to added NaOH.

Figure 1.5. Three chain model.

Figure 1.6. Polyvinyl alcohol chain cross-linked with polymethacrylic acid.

Figure 1.7. From atomistic to coarse-grain model.

Figure 1.8. Representative sagittal spin-echo images of a spinal cord injured mouse pre and post AC

hydrogel injection. Arrows show the presence of hydrogel.

Figure 1.9. Drug release from AC hydrogels.

Figure 1.10. a) Carbomer structure and b) cross-linkers (allyl sucrose and allyl pentaerythritol,

respectively).

Figure 1.11. Agarose monomeric unit.

Figure 1.12. Agarose gelation mechanism.

Figure 1.13. Cross-linking agents: glycerol (a) and propylene glycol (b).

Figure 1.14. AC hydrogels synthesis summary.

Figure 1.15. AC hydrogels swelling kinetics.

Figure 1.16 Sodium fluorescein.

Figure 1.17. Self-diffusion coefficient (a) and normalized self-diffusion coefficient (b) increase with respect

to mesh size and solute concentration.

Figure 1.18. Self-diffusion coefficient (a) and normalized self-diffusion coefficient (b) increase with respect

to mesh size at low solute concentration (25 mg mL-1

).

xiv

Chapter 2 - Methods and theoretical background

Figure 2.1. Example of Lennard Jones potential.

Figure 2.2. Radial distribution function determined from 100 ps molecular dynamics simulation of liquid

argon at 100 K with a density of 1.396 g cm-3

.

Chapter 3 - Method validation: fluorescein behavior in water solutions

Figure 3.1. NMR spectra of sodium fluorescein aqueous solutions at 50 mg mL-1

(E), 120 mg mL-1

(D), 200

mg mL-1

(C), 250 mg mL-1

(B) and 500 mg mL-1

(A).

Figure 3.2. Resonant structures of dianionic fluorescein (hydrogen atoms are omitted for the sake of clarity).

Figure 3.3. Time dependence of dianionic fluorescein mean square displacement (a) and relative

logarithmic plot (b) in water solution.

Figure 3.4. Time dependence of dianionic fluorescein mean square displacement (a) and relative

logarithmic plot (b) in PBS solution.

Figure 3.5. Dianionic fluorescein diffusion coefficient value with respect to chosen time origin from which it

is computed (a) and relative log(t) vs. log(MSD) slope (b) in water solution.

Figure 3.6. Dianionic fluorescein diffusion coefficient value with respect to chosen time origin from which it

is computed (a) and relative log(t) vs. log(MSD) slope (b) in PBS solution.

Figure 3.7. Radial distribution function and coordination number between fluorescein oxygen atoms and

Na+ ions in water (a) and PBS (b) solution.

Figure 3.8. Na+ diffusion coefficient value with respect to chosen time origin from which it is computed (a)

and relative log(t) vs. log(MSD) slope (b) in water.

Figure 3.9. Na+ self-diffusion coefficient value with respect to chosen time origin from which it is computed

(a) and relative log(t) vs. log(MSD) slope (b) in PBS.

Figure 3.10. Radial distribution function and coordination number between Na+ ions pairs in water (a) and

PBS (b).

Figure 3.11. Radial distribution function and coordination number between fluorescein dianion and water

oxygen atoms in water (a) and PBS (b).

Figure 3.12. Radial distribution function and coordination number between Na+ ions and water oxygen

atoms in water (a) and PBS (b).

Figure 3.13. Radial distribution function and coordination number between Cl- ions and water oxygen

atoms (a), Cl- self-diffusion coefficient value with respect to chosen time origin from which it is computed (b)

and relative log(t) vs. log(MSD) slope (c) in PBS.

Figure 3.14. Different approaches for molecular surfaces.

xv

Figure 3.15. Sodium fluorescein dimer structure determined at the M062X/6-311+G(2d,2p) level using the

SAS (a) and VDW (b) methodology to determine the solute-solvent boundary.

Figure 3.16. Potential of mean force between two fluorescein monomers calculated as a function of the

distance between the centers of mass at different pH in water solution a) and in PBS b).

Figure 3.17. Evolution of the distance between the centers of mass of two fluorescein molecules that

aggregate as a function of simulation time.

Figure 3.18. Specific and average dimers lifetimes in water solution (a) and PBS (b).

Figure 3.19. Specific and average distances between monomers centers of mass in water solution (a) and

PBS (b).

Figure 3.20. Specific and average distances between the centers of mass of the benzoic rings in water (a)

and PBS (b).

Figure 3.21. Mean centers of mass distances between the benzoic and phenoxy rings involved in edge to face

interactions in water (a,b) and PBS (c,d). The a c and b d plots correspond to the two distinct

bonds that can be formed for each molecule.

Figure 3.22. Examples of dimeric (a,b,c) and trimeric structures (d) of fluorescein complexes.

Figure 3.23. Specific and average dimer lifetime after trajectories imaging, in water (a) and PBS (b).

Figure 3.24. Average and specific monomers centers of mass distances and benzoic rings distances in water

(a,b) and PBS (c,d).

Figure 3.25. Instantaneous and average monomer molar fraction in water (a) and PBS (b).

Figure 3.26. Fluorescein minimum energy structures predicted by quantum chemistry (a) and force field

methods (b).

Figure 3.27. Minimum energy dimer structure predicted by quantum chemistry (a,b) and force field (c,d)

calculations.

Figure 3.28. Monomer structure in vacuo obtained through energy minimization with Amber ff03 force field.

Chapter 4 Molecular modeling of Agarose Carbomer hydrogels

Figure 4.1. Hydrogel molecular model.

Figure 4.2. Starting structures for electrostatic potential computation: pure polyacrylic acid (a), polyacrylic

acid with propylene glycol (b) and polyacrylic acid with methyl group (c).

Figure 4.3. Examples of Carbomer/fluorescein complex (a) and agarose/fluorescein complex (b). Water and

ions are omitted for the sake of clarity.

Figure 4.4. Sodium and fluorescein concentration profile with respect to the distance from Carbomer

monomers center of mass (from 7nm simulation).

xvi

Figure 4.5. Fluorescein center of mass numeral density isosurface at 2.152 10-3

molecules -3

(from 7 nm

simulation).

Figure 4.6. Mean square displacement vs. simulation time representative of hindered diffusion regimen

(from 14 nm simulation).

Figure 4.7. Mean square displacement vs. simulation time representative of hopping diffusion regimen

(from 7 nm simulation).

Figure 4.8. Mean square displacement vs. simulation time representative of free diffusion regimen (from 12

nm simulation).

Figure 4.9. Average mean square displacement evolution with respect to simulation time and relative

logarithmic plot (from 9 nm simulation).

Figure 4.10. Fluorescein Diffusion coefficients histograms in PBS (a), 7nm gel (b), 14 nm gel (c) and 19 nm

gel (d).

Figure 4.11. Theoretical and experimental normalized diffusion coefficients with respect to mesh size at low

solute concentration (25 30 mg mL-1

).

Figure 4.12. Log(t) vs. log(MSD) plot slope values with respect to mesh size.



).

Figure 4.14. Log(t) vs. log(MSD) plot slope values with respect to mesh size.

Figure 4.15. System density values with respect to mesh size.

Figure 4.16. Sodium ion diffusion coefficient with respect to mesh size (a) and relative slope of log(t) vs.

log(MSD) plot (b).

Figure 4.17. Sodium diffusion coefficients histograms in PBS (a), 7nm gel (b), 14 nm gel (c) and 19 nm gel

(d).

Figure 4.18. Chlorine ion diffusion coefficient with respect to mesh size (a) and relative slope of log(t) vs.

log(MSD) plot (b).

Figure 4.19. Chlorine diffusion coefficients histograms in PBS (a), 7nm gel (b), 14 nm gel (c) and 19 nm gel

(d).

Figure 4.20. Comparison between computed and experimental water normalized self-diffusion coefficient

with respect to mesh size.

Figure 4.21. Logarithmic plot slope with respect to mesh size.

Figure 4.22. Fluorescein oxygen atoms sodium (a) and sodium fluorescein oxygen atoms (b)coordination

numbers with respect to mesh size.

Figure 4.23. Carboxylic groups oxygen atoms sodium (a) and sodium carboxylic groups oxygen atoms

(b) coordination numbers with respect to mesh size.

xvii

Figure 4.24. Fluorescein water (a) and chlorine water (b) coordination numbers with respect to mesh

size.

Figure 4.25. Sodium water coordination numbers with respect to mesh size.

Figure 4.26. Sodium sodium coordination numbers with respect to mesh size.

Figure 4.27. Time averaged fluorescein monomer molar fraction with respect to mesh size.

Figure 4.28. Dimer average lifetime with respect to mesh size.

Figure 4.29. Mean monomers centers of mass distances (a) and mean benzoic rings centers of mass

distances (b) with respect to mesh size.

Figure 4.30. Second order statistical moment of concentration distribution vs. simulated time (a) and

relative logarithmic plot (b) for diffusion equation analytical solution.


relative logarithmic plot (b) for diffusion equation numerical solution.


relative logarithmic plot (b) for diffusion equation numerical solution in sub-diffusion regimen.

Figure 4.33. Second order statistical moment of fluorescein concentration distribution vs. simulated time (a)

and relative logarithmic plot (b) for diffusion equation numerical solution in super-diffusion regimen.

Figure 4.34. Second order statistical moment of sodium concentration distribution vs. simulated time (a) and

relative logarithmic plot (b) for diffusion equation numerical solution in super-diffusion regimen.

Chapter 5 Molecular modeling of hyaluronic acid hydrogels

Figure 5.1. In vitro BMP2 release from hyaluronic acid-based hydrogels.

Figure 5.2. Active BMP2 structure and detail of disulfide bond.

Figure 5.3. BMP2 wrist knuckle structure.

Figure 5.4. Hyaluronic acid.

Figure 5.5. Hyaluronic acid-based hydrogel synthesis summary.

Figure 5.6. -D-glucuronic acid (protonated state).

Figure 5.7. BMP2 considered binding areas: wrist (blue), upper (green), lower (red) (a) and knuckle

(black) (b).

Figure 5.8. Potential of mean force for wrist complex at pH 4.5 (a) and pH 7 (b).

Figure 5.9. Example of stable geometry of wrist complex at pH 4.5: front (a) and lateral view (b).

Figure 5.10. Residues involved in H-bond attainment for wrist complex.

Figure 5.11. Potential of mean force for knuckle complex at pH 4.5 (a) and pH 7 (b).

xviii

Figure 5.12. Front and lateral view of knuckle complexes at pH 4.5 (a,b) and pH 7 (c,d).

Figure 5.13. Residues involved in H-bond attainment for knuckle complex at pH 4.5 (a) and pH 7 (b).

Figure 5.14. Potential of mean force for upper complex at pH 4.5 (a) and pH 7 (b).

Figure 5.15. Front and lateral view of upper complexes at pH 4.5 (a,b) and pH 7 (c,d).

Figure 5.16. Residues involved in H-bond attainment for upper complex at pH 4.5 (a) and pH 7 (b).

Figure 5.17. Potential of mean force for lower complex at pH 4.5 (a) and pH 7 (b).

Figure 5.18. Front (a) and lateral (b) view of lower complex at pH 4.5.

Figure 5.19. Residues involved in H-bond attainment for lower complex at pH 4.5.

Chapter 6 Conclusions

Figure 6.1. Hydrogel molecular model.

Figure 6.2. Examples of Carbomer/fluorescein complex (a) and agarose/fluorescein complex (b). Water and

ions are omitted for the sake of clarity.



). Normalization is made with respect to self-diffusion coefficient in

PBS.

Figure 6.4. Log(t) vs. log(MSD) plot slope values with respect to mesh size for fluorescein.

Figure 6.5. Potential of mean force for wrist complex at pH 4.5 (a) and pH 7 (b).

Appendix A

Figure A.1. Fluorescein minimum energy structure.

Figure A.2. Polyacrylic acid dimer structure.

Figure A.3. Polyacrylic acid dimer with propylene glycole structure.

Figure A.4. Cross-linking polyacrylic acid dimer structure.

Figure A.5. Agarose structure.

Figure A.6. Cross-linking agarose structure.

Figure A.7. Protonated glucuronic acid structure.

xix

Appendix B

Figure B.1. Fluorescein mean square displacement with respect to simulation time (a) and relative

logarithmic plot (b).

Figure B.2. Free fluorescein molecules mean square displacement with respect to simulation time (a) and

relative logarithmic plot (b).

Figure B.3. Sodium ions mean square displacement with respect to simulation time (a) and relative


Figure B.4. Chlorine ions mean square displacement with respect to simulation time (a) and relative


Figure B.5. Water mean square displacement with respect to simulation time (a) and relative logarithmic

plot (b).

Figure B.6. Normalized diffusion coefficient histograms for fluorescein (a), sodium (b) and chlorine (c).

Figure B.7. Fluorescein oxygen atoms sodium ions (a) and sodium ions fluorescein oxygen atoms (b)

radial distribution function and coordination numbers.

Figure B.8. Carboxylic moieties oxygen atoms sodium ions (a) and sodium ions carboxylic moieties

oxygen atoms (b) radial distribution function and coordination numbers.

Figure B.9. Sodium ions sodium ions (a) and sodium ions water oxygen atoms (b) radial distribution

function and coordination numbers.

Figure B.10. Chlorine ions water oxygen atoms (a) and fluorescein center of mass water oxygen atoms

(b) radial distribution function and coordination numbers.


coordination numbers.


oxygen atoms (b) coordination numbers.

Figure B.13. Sodium ions sodium ions (a) coordination numbers.

Figure B.14. Fluorescein monomer molar fraction with respect to simulation time.









xx


plot (b).

























plot (b).








xxi


















plot (b).


















xxii








plot (b).
















Figure B.71. Fluorescein mean square displacement evolution with respect to simulation time (a) and

relative logarithmic plot (b) in water environment. Time origin corresponds to attainment of Brownian

regimen.

Figure B.72. Fluorescein self-diffusion coefficient with respect to time origin from which it is computed (a)

and relative slope of MSD vs. t logarithmic plot (b) in water environment.

Figure B.73. Sodium ions mean square displacement evolution with respect to simulation time (a) and

relative logarithmic plot (b) in water environment. Time origin corresponds to attainment of fluorescein

Brownian regimen.

Figure B.74. Sodium ions self-diffusion coefficient with respect to time origin from which it is computed (a)

and relative slope of MSD vs. t logarithmic plot (b) in water environment.

xxiii

Figure B.75. Fluorescein mean square displacement evolution with respect to simulation time (a) and

relative logarithmic plot (b) in PBS environment. Time origin corresponds to attainment of Brownian

regimen.

Figure B.76. Fluorescein self-diffusion coefficient with respect to time origin from which it is computed (a)

and relative slope of MSD vs. t logarithmic plot (b) in PBS environment.

Figure B.77. Sodium ions mean square displacement evolution with respect to simulation time (a) and

relative logarithmic plot (b) in PBS environment. Time origin corresponds to attainment of fluorescein

Brownian regimen.

Figure B.78. Sodium ions self-diffusion coefficient with respect to time origin from which it is computed (a)


Figure B.79. Chlorine ions mean square displacement evolution with respect to simulation time (a) and

relative logarithmic plot (b) in PBS environment. Time origin corresponds to attainment of fluorescein

Brownian regimen.

Figure B.80. Chlorine ions self-diffusion coefficient with respect to time origin from which it is computed (a)


Figure B.81. Normalized diffusion coefficient histograms for fluorescein (a), sodium (b) and chlorine (c) in

PBS environment.


radial distribution function and coordination numbers in water environment.


function and coordination numbers in water environment.

Figure B.84. Fluorescein center of mass water oxygen atoms radial distribution function and coordination

numbers in water environment.


radial distribution function and coordination numbers in PBS environment.


function and coordination numbers in PBS environment.


(b) radial distribution function and coordination numbers in PBS environment.

Figure B.88. Monomer molar fraction in water (a) and PBS (b) environment.

Figure B.89. Mass conservation for diffusion analytical model (a) diffusion numerical model (b) and

hindered diffusion model (c).

Figure B.90. Mass conservation for fluorescein (a) and sodium (b) for diffusion under electrostatic potential

gradient.

Figure B.91. Wrist binding area.

Figure B.92. Knuckle binding area.

Figure B.93. Upper binding area.

xxiv

Figure B.94. Lower binding area.

Appendix C

Figure C.1. a) 1H NMR spectra of sodium fluorescein aqueous solutions at 50 mg mL

-1 (A), 120 mg mL

-1 (B),

200 mg mL-1

(C), 250 mg mL-1

(D) and 500 mg mL-1

(E) and b) 1H sodium fluorescein NMR spectra in AC6

hydrogels at different solute concentration: 50 mg mL-1

(A), 100 mg mL-1

(B), 250 mg mL-1

(C).

Figure C.2. Fourier transform infrared spectra of AC1 (blue line) and AC6 (red line) hydrogels, Carbomer

974P and agarose (black lines).

Figure C.3. a) AC1 hydrogel FTIR spectra before and after 2 and 4 weeks of degradation in PBS (black

lines) and b) dried AC1 hydrogel FTIR spectra before and after 2 and 4 weeks of degradation in PBS (black

lines).

Figure C.4. Coronal spin echo images of a mouse spinal cord (AC) compared to sham operated mouse

(CTR). White arrows show hydrogel presence.

Figure C.5. a-f) bright field; b-g) spinal cord, low magnification, 10x, scale bar 250 m; c-e) and h-j) spinal

cord, high magnification, 20x, scale 70 m.

xxv

List of Tables

Chapter 1 - Introduction

Table 1.1. Atomistic hydrogels models.

Table 1.2. AC hydrogels formulations and compositions.

Table 1.3. PBS composition.

Table 1.4. AC hydrogels structural parameters.

Chapter 3 - Method validation: fluorescein behavior in water solutions

Table 3.1. Simulated PBS composition.

Table 3.2. Sodium fluorescein dimer interaction energies (kcal/mol) calculated on structures optimized at

the M062X/6-311+G(2d,2p) level using the SAS and VDW methodologies to compute the solvent-solute

surfaces and several density functionals.

Chapter 4 Molecular modeling of Agarose Carbomer hydrogels

Table 4.1. Number of molecules in the system with respect to mesh size; the number of sodium ions contains

both counter-ions and PBS salt contributions.

Chapter 5 Molecular modeling of hyaluronic acid hydrogels

Table 5.1. Electrostatic and Van der Waals contribution for wrist complex from MD simulations and US

trajectories; energies are in kcal mol-1

.

Table 5.2. Electrostatic and Van der Waals contribution for knuckle complex from MD simulations and US


.

Table 5.3. Electrostatic and Van der Waals contribution for upper complex from MD simulations and US


.

Table 5.4. Electrostatic and Van der Waals contribution for lower complex from MD simulations and US


.

Table 5.5. BMP2/HA complexes interaction free energies with respect to pH value.

xxvi

Chapter 6 Conclusions

Table 6.1. Comparison between calculated and experimental self-diffusion coefficients and interaction

energies.

Table 6.2. BMP2/HA interaction free energies (in kcal mol-1

) with respect to pH.

Appendix A

Table A.1. Atomic coordinates and atom types of dianionic fluorescein optimized structure.

Table A.2. Dianionic fluorescein atomic charges.

Table A.3. Atomic coordinates and atom types of polyacrylic acid dimer optimized structure.

Table A.4. Polyacrylic acid dimer atomic charges.

Table A.5. Atomic coordinates and atom types of polyacrylic acid dimer with propylene glycol optimized

structure.

Table A.6. Polyacrylic acid dimer with propylene glycol atomic charges.

Table A.7. Atomic coordinates and atom types of cross-linking polyacrylic acid dimer optimized structure.

Table A.8. Cross-linking polyacrylic acid dimer atomic charges.

Table A.9. Atomic coordinates and atom types of agarose optimized structure.

Table A.10. Agarose atomic charges.

Table A.11. Atomic coordinates and atom types of cross-linking agarose optimized structure.

Table A.12. Cross-linking agarose atomic charges.

Table A.13. Atomic coordinates and atom types of protonated glucuronic acid optimized structure.

Table A.14. Protonated glucuronic acid atomic charges.

Appendix C

Table C.1. Agarose Carbomer hydrogels formulations and compositions.

1

Abstract

This PhD thesis is focused on the molecular modeling of hydrogels, i.e. hydrophilic amorphous

polymeric matrices, able to retain up to 99% of water in weight terms, nowadays widely used for drug

delivery applications, in tissue engineering and regenerative medicine fields. In particular, this work is aimed

at understanding and describing the mechanisms, which govern transport phenomena that take place in the

bulk phase of these materials. The here presented investigation is performed using molecular models, which

contain an atomistic detail: this is fundamental in order to achieve an exhaustive overview and a full

understanding of solute diffusion in such systems.

Many macroscale mathematical models have been developed in order to compute a solute diffusion

coefficient, each based on different theories (free volume, obstruction effects, and so on) aimed at properly

describing hydrogel environment. However, the assumptions behind every approach constitute a limit for

these models, thus reducing the field where they can be employed. A molecular modeling approach

introduces the necessity of an atomistic model of the polymeric matrix, but removes all the assumptions that

limit the application of a specific model, hence leading to a wider-ranging approach.

Moreover, being molecular diffusion the main mechanism governing the drug release rate both in in

vitro and in vivo environments, it is immediate to discern that such approach can become a complementary

and synergic tool for experimental activity. The atomistic detail offered by molecular modeling can help

researchers to understand the peculiarities of various materials formulations, thus leading to a smart material

design according to various exigencies and also allowing a better and more efficient experimental design.

This thesis work considered the diffusion of sodium fluorescein in hydrogels made of Carbomer

974P (an high molecular weight cross-linked polyacrylic acid) and agarose (a polysaccharide), cross-linked

with glycerol and propylene glycol through esterification, performed by means of microwave assisted

synthesis approach. Sodium fluorescein was chosen since it possesses a steric hindrance and a molecular

weight (376.28 g mol-1

) similar to commonly used drugs. The formulation and the synthesis of agarose

Carbomer hydrogels have been conceived and developed by our research group specifically for tissue

engineering purpose. This material was widely characterized in terms of chemical/physical properties and

mechanical and release behavior. Moreover, agarose Carbomer hydrogels have been fruitfully tested in in

vivo conditions proving to be suitable and promising drug delivery devices for spinal cord injury treatments.

The scientific issue behind this work lies in a counter-intuitive behavior observed during

experimental activity, i.e. fluorescein self-diffusion enhancement with respect to water solutions, found

through HRMAS DOSY technique. This self-diffusion coefficient increase has, moreover, a dependence on

both hydrogel mesh size and solute concentration.

2

The methods offered by computational chemistry, and in particular by molecular dynamics

simulations, allow to compute a theoretical self-diffusion coefficient, which implicitly contains all the effects

due to the peculiarity of the environment and the interactions with the other species present in the system,

such as water, ions, and other solute molecules. The obtained values can be directly compared with

experimental data, while the detail offered by the simulations can suggest some insights about this particular

behavior.

The first part of the thesis is aimed at verifying if the chosen computational approach (that is, the

adopted force field along with charge derivation protocol) is suitable to describe fluorescein diffusion; this is

assessed analyzing fluorescein behavior in water solutions, thus in systems that can be easily described and

that are not dramatically different from hydrogel environment, because of their very high water content. In

particular, fluorescein behavior is investigated by computing a self-diffusion coefficient and a dimerization

free energy, quantities that can be directly compared with experimental data obtained by our research group

or available in literature.

The core of this thesis is constituted by the development of a representative molecular model of

agarose Carbomer hydrogel and the study of sodium fluorescein transport phenomena inside the bulk phase

of these materials. Molecular trajectories obtained through molecular dynamics simulations do not only

allow computing a theoretical self-diffusion coefficient, but also highlighting the main mechanism behind

the curious self-diffusion enhancement.

The last part of the work is dedicated to the investigation of transport phenomena involving bone

morphogenetic protein 2 in hyaluronic acid-based hydrogels, made in collaboration with prof. J. Hilborn

research group of polymer chemistry Department at the Uppsala University. Experimental activity showed

that protein release rate does not exhibit an expected initial burst release phase, but a continuous and slow

delivery from the polymeric matrix. This behavior has been explained by means of electrostatic interactions

between the solute and the matrix, but this hypothesis has never been verified, up to authors best

knowledge. Molecular modeling is employed to study the attainment of complexes between bone

morphogenetic protein 2 and hyaluronic acid chains, in order to check the proposed explanation and thus to

understand the experimental evidence.

Despite this topic might sound completely different from the main subject of this thesis, it is not the

case: this part of the work was, indeed, aimed at investigating diffusion phenomena in hydrogels, where the

atomic detail becomes an essential feature in order to understand the macroscopic system behavior.

The final purpose of the entire work is, indeed, to propose and validate a wide-ranging modeling

approach, underlining how molecular modeling can become an useful tool when coupled with experimental

activity in the sparkling and ever-growing drug delivery field.

At a final glance, this thesis represents the modeling part of a wider project which involved, as said,

the application of the agar-carbomer hydrogels as promising devices for controlled drug delivery in spinal

3

cord injury repair strategies: the insights emerged from this thesis will be directly used for a smart material

design for future experimental campaigns.

4

1. Introduction

Si autem acceperis septimam decimam

partem ignis, et commiscueris cum hiis

[terra, aqua et aere] destillando et

imbibendo sicut dictum est, evenit tibi

lapis rubeus, clarus, simplex, non

adurens, de quo si parum super multum

Mercurii proieceris, convertet eum in

purissimum solem obrisium. Iste ergo

est modus de lapide minerali, ut dictum

est

Tractatus D. Thomae De Aquino,

ordinis praedicatorum de lapide

philosophico

Se ora prenderai una diciassettesima

parte del fuoco e la mescolerai coi tre

sopraddetti elementi [terra, acqua,

aria],distillandoli e inzuppandoli come si

detto, otterrai una pietra rossa,chiara,

semplice, non bruciante, della quale, se tu

getterai una piccola parte su molto

mercurio, questo sar convertito in Oro

obrizzo purissimo. Questo il metodo per

conseguire la Pietra Minerale.

Tommaso dAquino, Trattato della Pietra

Filosofale

This work is focused on the investigation at molecular level of transport phenomena inside

hydrogels; these are hydrophilic polymeric matrices, able to retain up to 99% of water (in weight terms) and

commonly used as drug release devices or scaffolds for tissue engineering1. In particular, this thesis deals

with the diffusion of sodium fluorescein in hydrogels made of agarose (a polysaccharide) and Carbomer (a

polyacrylic acid). The methods offered by computational chemistry, in primis molecular dynamics

simulations, have been applied to analyze a counter-intuitive behavior observed during experimental activity:

a solute diffusion enhancement in hydrogels with respect to water solutions. Diffusivity increase has also a

dependence on hydrogel structure and solute concentration.

A proper understanding of diffusion enhancement mechanisms is not a merely scientific curiosity,

but it also has practical consequences. Once the fundamentals phenomena involved are highlighted, material

design can be modified in order to tune properly drug diffusion in the matrix, and thus drug release rate in in

vivo environments; this allows to perform an optimal device design, according to various exigencies.

Such a modeling approach can also be used as a tool for optimizing the experimental activity, since

the effects of alterations in materials formulation can be previously qualitatively estimated through

simulations. Last but not least, a detailed atomistic approach aimed at transport phenomena can support the

efforts of diffusion mathematical modeling in hydrogels. In this framework, several theories have been

developed starting from different theoretical backgrounds (statistical mechanics, free volume theory, and so

on), but every mathematical description has its own limitations which reduce the application field.

Introduction

5

This work constitutes the modeling part of a wider project, which involves the hydrogel synthesis

and its application as potential drug delivery device for spinal cord injury repair strategies. Amongst the

secondary events that occur after a trauma in the spinal cord area, the formation of a connective scar tissue is

particularly critical because it prevents axons reconnections and hence leads to the chronic neurological

outcomes typical of this pathology. The use of an hydrogel as an in situ drug delivery device would allow to

limit this secondary effect, by releasing active compounds that can limit the formation of a scar connective

tissue (methyprednisolone), or remove it (Chondroitinase ABC) or even inhibit it2,3

.

Hydrogel release and mechanical behaviors have been extensively characterized2-11

and material

biocompatibility has already been extensively assessed through in vivo experiments using mice models2,12

. In

particular, a drug-loaded gelling solution is injected into the damaged spinal cord area, leading to the

formation of an hydrogel in the injured zone, thanks to in situ gelation process, which allows to avoid

invasive surgical implantations12

. Moreover, the hydrogel spontaneously degrades both in vivo and in vitro

because of hydrolysis process, eliminating the exigency of an invasive device removal. In vivo and ex vivo

imaging showed the drug release in the damaged area, confirming that agarose Carbomer hydrogels are

promising tools for spinal cord injury treatment2.

a)

b)

Figure 1.1. Schematic representation of hydrogel release (a) and in vivo qualitative bioluminescence

of antibody Alexa 647 delivery at four different time points (b) (from Perale et al.2).

Chapter 1

6

This satisfactory experimental campaign strengthens the need of a deep investigation of transport

phenomena inside agarose Carbomer hydrogels, in order to highlight the main factors, which influence

diffusion, and hence optimize material formulation. The presence of strong interactions between a drug and

the matrix, e.g., could imply a delayed delivery of the active compound, a condition that must be avoided

when a fast release is needed. The assessment of such interactions can be qualitatively and quantitatively

determined through an atomistic model, which provides details that are not experimentally accessible.

1.1 Hydrogels

Hydrogels are three dimensional water-swollen network constituted by cross-linked structures of

hydrophilic homopolymers and copolymers1. Generally speaking, cross-links can be formed by strong

chemical linkages (covalent or ionic bonds), permanent or temporary physical entanglements,

microcrystallite formation or weak interactions, like van der Waals forces or hydrogen bonds. The main

hydrogels peculiarity is the ability to absorb and retain a large amount of water (up to 99% in weight terms)

thanks to their hydrophilic behavior, that leads to the swelling of the entire network. Polymeric structure

remains insoluble in water because of chemical and/or physical cross-links between macromolecular chains.

Swelling can be explained in thermodynamics terms, according to Flory Rehner theory13

.

Figure 1.2. Hydrogel schematic representation (TEM image of a freeze-dried mannose-based

hydrogel, from Y. E. Shapiro14

).

Introduction

7

When a polymeric network is swollen, polymeric chains tend to assume a stretched configuration,

although this conformational arrangement is hindered by an opposite elastic force. This process is also

subjected to an entropic effect, due to the entropy increase caused by polymer solvent mixing, which favors

hydrogel swelling towards lower free energy configurations. The final swelling behavior derives from the

equilibrium between these opposite effects.

Figure 1.3. Agarose Carbomer hydrogel.

Hydrogels can be classified in several ways, according to morphology, constituents or network

physical properties. For example, hydrogels can be neutral, anionic, cationic, ampholytic or zwitterionic

depending on side chain groups in the structure backbone. According to composition, it is possible to

distinguish homopolymer hydrogels, copolymer hydrogels or interpenetrating polymeric hydrogels. In the

second case, two or more monomers are subjected to copolymerization; at least one monomer must be

hydrophilic. Interpenetrating polymeric network can be obtained by polymerizing and cross-linking a

monomer in a presence of an already cross-linked structure, or by polymerizing different monomers with

substantially different processes. Hydrogels can be prepared starting from synthetic polymers, such as

polyethylene glycol (PEG), polyvinyl alcohol (PVA) and polyacrylates, or using naturally-occurring

macromolecules like agarose, alginate, chitosan, and so on. Synthetic polymers can be tailored to obtain

certain desired properties (degradation rate, mechanical strength, et cetera), while natural polymers shows an

inherent biocompatibility and a reduced risk of inflammatory response for in vivo applications.

Focusing on morphology, hydrogels can be amorphous, semicrystalline, H-bonded, supramolecular,

or constituted by colloidal aggregates. In amorphous hydrogels chains are randomly arranged, while in

semicrystalline ones there are dense regions of ordered polymeric chains. In H-bonded and supramolecular

hydrogels network formation is due to long range interactions (hydrogen bonding, interactions, van der

Waals forces).

Hydrogels structure is quantitatively described by several molecular parameters1 but an exhaustive

overview is given by few quantities like the average molecular weight between two following cross-links Mc

Chapter 1

8

(in g mol-1

), cross-link density c (defined as the ratio between the number of moles between two cross-links

and the volume of dry polymer, in mol cm-3

) and mesh size (defined as the distance between two following

cross-link, in nm). These parameters are usually determined through equilibrium swelling theory (originally

developed by Flory and Rehner, and modified by Peppas and coworkers1) and rubber elasticity theory

15.

Nowadays, hydrogels have gained a key role in drug delivery16,17

, tissue engineering18

and

regenerative medicine field19

. This ever growing use is the result of several key properties of hydrogel. In

primis their good biocompatibility, which is due to the ability to simulate natural tissue environment, thanks

to the high water content. This leads to a wide range of applications, which comprehends the delivery of

drugs16,20-25

, proteins26-29

, genes30,31

and growth factors32-34

. In tissue engineering field, hydrogels can be used

as scaffolds for cellular growth35,36

and more in general for cell delivery purposes in regenerative

medicine37,38

. Moreover, hydrogel properties can be tailored during synthesis, according to specific

applications. Hydrophilicity, e.g., can be modified by tuning composition or adding hydrophilic/hydrophobic

monomers in the copolymer. Mechanical strength can be improved by increasing cross-link density (thus

reducing mesh size and increasing network rigidity), by formation of interpenetrating polymer networks or

by crystallization, which induces crystallite formation and structure reinforcement. Formulations can be also

tuned in order to modify delivery properties, such that drug release is performed only where, or when, it is

required. For example, in pH-sensitive hydrogels the protonation of ionic moieties in the backbone

determines the swelling of the matrix according to the electrostatic repulsion between charged groups;

swelling enhancement increases the volume occupied by the solvent and reduces polymer volume fraction,

leading to the creation of wider diffusive paths and thus to drug release. This feature is particularly useful

when drug release should be performed during inflammation, where pH value locally decreases because of

macrophages action.

Figure 1.4. Chitosan-based gel swelling with respect to added NaOH (from Bhattarai et al.39

).

1.2 Diffusion phenomena in hydrogels

While the estimation of diffusion coefficients in gas and liquid phases is consolidated40

, a general

and reliable theory for solid matrices still lacks. Diffusion in hydrogels is not an exception, and nowadays

Introduction

9

transport phenomena in such systems are a discussed topic. Solute diffusion strongly depends on polymer

concentration and degree of swelling in a complex way, feature that makes diffusion estimation a

challenging problem, yet very interesting41

: a reliable description of transport phenomena acquires a greater

importance in biomedical applications, being molecular diffusion the key mechanism which governs the

release of active compounds. The simplest way to deal with diffusion is Ficks first law, which relates

diffusive flux to concentration gradient:

cDJ (1.1)

Where J is the diffusive flux, D is the diffusion coefficient and c is solute concentration. Ficks first

law is the starting point of numerous models of diffusion in polymeric systems. Diffusion behavior in

polymers, however, can have different behaviors according to network physical properties and solvent

(solute) interactions with the network. A first evaluation can be made by distinguishing diffusion

mechanism: Fickian (Case I) and non-Fickian (Case II and anomalous diffusion). Fickian diffusion

mechanism is usually observed when polymer system is above glass transition temperature Tg; polymer

chains possess an high mobility that allows a fast solvent penetration, and solvent diffusion rate is slower

that polymer relaxation rate. In anomalous diffusion regime they have the same order of magnitude, while in

Case II diffusion polymer relaxation is slower than solvent diffusion rate.

Korsmeyer Peppas equation42-44

is a semi-empirical formula which allows to discriminate between

diffusion mechanisms:

nt ktM

M

(1.2)

Where Mt is mass released at time t, M is mass release when time tends to infinite, k is a

proportionality constant and n is the diffusional exponent which discriminates between diffusion

mechanisms. When n is equal to 0.5, diffusion mechanism is Fickian, while for n equal to 1 Case II

mechanism takes place. Anomalous diffusion occurs for n values between 0.5 and 1.

Backgrounds of existing theories developed to estimate diffusion in hydrogels can be summarized as

follows45

:

Obstruction theories: polymer chains are considered as motionless if compared to

solute molecules (solvent and/or solute). This approximation is supported by the evidence that

polymer self-diffusion coefficient is much smaller than solvent and solute ones. The presence of

polymeric chains, regarded as fixed and impenetrable rods in solution, increases the length of the

mean path that molecules experience to reach a certain point of the system;

Chapter 1

10

Hydrodynamic theories: this description takes into account hydrodynamics

interactions, like the frictional ones that interest all system (solute/solvent, polymer/solvent,

solute/polymer, polymer/polymer). This approach is suitable to describe concentrate systems, where

polymeric chains start to overlap (and thus obstruction theories fail);

Free volume theories: free volume (a well known concept in polymer chemistry) is

defined as the volume not occupied by matter or, more generally, as the volume of system at a

certain temperature minus the volume of the same system at 0 K. Free volume rearrangements create

pores and cavities where diffusing species are able to pass through; thus, free volume is considered

to be the main factor which determines molecular diffusion.

Efforts have been made also in polyelectrolyte gels field46

, where a proper inclusion of electrostatic

effects is essential to describe the diffusion of charged solutes. Amsden et al.47

developed a model based on

obstruction theories which takes into account polymer ionization degree. Fatin-Rouge et al.48

studied the

motion of ions (Na+, Li

+, F

-, Cl

-, calcein

3-) in agarose hydrogels, taking into account, along with steric

effects, electrostatic interactions through Donnan potential. Results suggest that local electrostatic potential

has a not negligible influence on ion motion. Vega et al.49

also investigated ion diffusion in agarose

hydrogels, and proposed a model that explicitly underlines the effect of gel porosity (i.e., mesh size) and ions

size in solute diffusion. Hyk et al.50

modeled ion diffusion in polyacrylate gels by means of Manning theory

of polyelectrolytes51

.

Generally speaking, literature offers several examples of theories about diffusion in hydrogels and

their applications52

. Nevertheless, every approach has its own limitations, which reduce the field where it can

be fruitfully applied. Many mathematical models, moreover, are strictly system dependent since they contain

one or more parameters that must be obtained through fitting of experimental data.

The use of atomistic models introduces the necessity of a representation at molecular level of the

polymeric matrix, but removes all the assumptions that limit the application of a specific model and widens

the applicability and the suitability of the approach. Moreover, the detail provided by these methods allows

to verify the phenomena which give the most important contribution to diffusion (steric hindrance,

hydrodynamics interactions, and so on) as well as specific interactions with the matrix, that also influence

the diffusive behavior of the solute and which are not described in a complete way by mathematical models.

1.3 Atomistic modeling of hydrogels

In the last two decades thanks to a growing diffusion of hydrogels applications, a deeper

understanding of their peculiarities and an increasing availability of computational resources, molecular

modeling of hydrogels is becoming a discussed topic in literature. Polymer networks are usually investigated

by means of molecular dynamics or Monte Carlo simulations, with atomistic detail or with coarse-grain

models. Computations are aimed at characterizing mechanical behavior, diffusion phenomena, network

Introduction

11

structure or solvation effect on polymer matrices. The examples, that literature can offer, are hereby

classified according to the approach adopted to describe the matrix. Models types and their main features are

summarized in table 1.1.

Model Description Application References

Three chains model A basic structure of three

cross-linked

perpendicular polymeric

chains forms a cubic

lattice by means of

periodic boundary

conditions;

Transport phenomena;

mechanical properties;

water/polymer

interactions;

54 - 62

Multiple chains model One or more cross-

linked long polymeric

chains are arranged in a

lattice that forms an

infinite polymeric

network through

periodic boundary

conditions;

Water/polymer

interactions;

63 - 69

Amorphous cell model Polymeric chains are

arranged according to

energy minimization

criteria in order to

reproduce a disordered

bulk phase;


thermodynamics;

70 - 73

Coarse-grain model Groups of atoms are

embedded in beads

with proper charge and

hindrance. Loss of

atomistic details.


swelling;

ion/polyelectrolyte gels

interactions;

74 - 86

Table 1.1. Atomistic hydrogels models.

1.3.1 Three chains model

This approach represents hydrogel structure with a basic element formed by three cross-linked

polymeric chains perpendicular each other. By means of periodic boundary conditions, this element is

repeated in space and forms a cubic lattice that approximates the polymeric network, as represented in figure

1.5.

Chapter 1

12

Figure 1.5. Three chain model (from Jang et al.53

).

This is a statistical representation, since hydrogel random network is reduced to a cubic lattice

having mean structural properties such mesh size or average molecular weight between adjacent cross-links.

The structure, moreover, is very ideal, since it does not possess free dangling chains or self-looping. This

approach is usually coupled with molecular dynamics simulations carried out in NPT ensemble with explicit

solvent molecules; water is usually modeled through SPC/E or TIP3P formalisms.

Such approach was firstly introduced by Chiessi et al.54,55

who studied dynamical behavior of water

in matrices made of polyvinyl alcohol, obtaining a good agreement with Quasi Elastic Neutron Scattering

(QENS) experimental data and providing a first validation of this approach.

Lee et al. extensively investigated copolymer hydrogels made of N-vinyl-2-pyrrolidone (VP) and 2-

hydroxyethyl methacrylate (HEMA), analyzing the effect of monomeric sequence and hydration on

mechanical56,57

and transport properties58

. In particular, they studied the diffusion of ascorbic acid and D-

glucose at different hydration levels (from 20% to 80% in weight terms) and chain compositions; they found

that, at low hydration level, solute experiences anomalous diffusion since it is confined into the matrix by

virtue of enhanced interactions with the polymer, while at high water content molecules are more likely to be

solvated. Jang et al.53

investigated mechanical and transport properties of interpenetrating networks of

polyacrylic acid (PAA) and polyethylene oxide (PEO); like Lee and coworkers, they studied diffusion of D-

glucose and ascorbic acid both in single PAA and PEO networks and in interpenetrating networks. Wu et

al.59

studied rhodamine diffusion in polyethylene glycol-based hydrogels, focusing on the effect of cross-link

density on transport phenomena. They compared the results with Amsden theory, which combines

obstruction and hydrodynamics theories, obtaining a good agreement. In these works, an experimental value

to be compared with simulation results is not given, but the computed diffusion coefficient are in good

agreement with experimental data related to similar systems and solutes with comparable hydrodynamic

radii.

He et al.60,61

studied the effect of water content on zwitterionic hydrogels of carboxybetaine acrylate,

focusing on different swelling equilibrium states. Zhao et al.62

proposed a variation of this approach (called

dummy cube model) in order to evaluate polymer/water interactions in a polymethacrylic acid network.

Introduction

13

1.3.2 Multiple chains model

Multiple chains model represents hydrogels with one or more long cross-linked polymeric chains. In

some cases, chains are arranged in a lattice, which constitutes an infinite network through periodic boundary

conditions; in other works, the system is constituted by one or more finite cross-linked chains. This is

consistent with the definition of hydrogel structure, since in real systems these cross-linked chains swell in

presence of water and are able to form networks thanks to physical entanglements and long range

interactions. Simulation protocol is usually analogous to three chains model (NPT ensemble and explicit

solvent).

Figure 1.6. Polyvinyl alcohol chain cross-linked with polymethacrylic acid (from Paradossi et al.63

).

Netz et al.64

characterized the interactions between water and polyacrylamide hydrogels simulating

different systems, changing degree of polymerization and cross-link density of the chain. Walter et al.

studied hydrogels made of poly(N-isopropylacrylamide) (PNIPAM) both in water65

and in water/methanol

mixture66

, highlighting conformational behavior and solvent interactions, while Deshmukh et al.67

instead

focused their attention on cross-link peculiarity (flexibility and chain length) and temperature on PNIPAM

hydrogels behavior in water.

Paradossi et al.63

built a network of one polyvinyl alcohol chain formed by 800 monomers cross-

linked with a short chain of polymethacrylate, characterizing swelling behavior and solvent distribution at

different temperatures. Chiessi et al.68

modeled thermoresponsive behavior of hydrogels made of polyvinyl

alcohol and poly(methacrylate-co-N-isopropyl acrylamide), while Tnsing et al.69

focused on PNIPAM gels.

1.3.3 Amorphous cell model

Amorphous cell models generate initial configurations by adding polymeric chains (with a certain

degree of polymerization) in a tetragonal or orthorhombic cell according to minimum energy criteria through

a Metropolis-like algorithm. This approach is aimed at obtaining an equilibrated configuration of a

disordered bulk phase, which can also reproduce experimental density.

Chapter 1

14

Bermejo et al.70-72

studied the cross-link effect on glass transition temperature and mechanical

properties of swollen matrices of polyvinyl alcohol, with a good agreement with experimental data, while

Zhang et al.73

characterized water and ethanol diffusion.

1.3.4 Coarse-grain models

Coarse-grain models are intended to give a physical meaningful description of the system by

reducing the detail and thus the computational cost of the simulation. Groups of atoms are included into

single beads thus losing the atomistic detail. A methyl group, e.g., can be represented by a single bead with

proper charge and steric hindrance.

Figure 1.7. From atomistic to coarse-grain model (from Gautieri et al.74

).

Gautieri et al.74

used a coarse-grain model to describe the diffusion of benzene in polyvinyl alcohol

matrices, obtaining a good agreement with experimental data. Pei et al.75

modeled an agarose gel as a cubic

lattice of cylindrical fibers, and studied diffusion of model chains which mimics DNA through Brownian

dynamics. Miyata et al.76

computed the self-diffusion coefficient of a generic charged particle in a swollen

counter-charged hydrogel through Brownian dynamics, underlining the effect of electrostatic interactions

between the solute and the matrix.

Claudio et al.77

simulated a polyelectrolyte microgel, comparing the results of a coarse - grain

charged bead spring model with a Poisson Boltzmann cell model, where the gel is modeled as single

charged colloid in a spherical cell. Results showed that ion distribution is in good agreement with the

different approaches. Yin et al.78,79

and Yan et al.80

characterized swelling of polyelectrolyte gels through

coarse-grain molecular dynamics simulations, with explicit divalent and monovalent counter-ions in implicit

solvent.

Edgecombe et al. extensively investigated polyelectrolyte hydrogels through Monte Carlo

simulations in implicit solvent. In their works, they analyzed swelling behavior and structure peculiarity

varying cross-link density, charge density and chain stiffness81

, in the presence of macroions82

or of a

reservoir of salt solutions83

; they also extended the analysis to interpenetrating networks84

.

Introduction

15

Mann et al.85,86

used a bead spring model to characterize the swelling of polyelectrolyte hydrogel

both in a good and in a poor solvent, studying structure conformations by changing charge density along the

chain.

1.4 Agarose Carbomer hydrogels: a brief overview

Amongst all hydrogel formulations that have been proposed in literature, this work is focused on

hydrogels made of Carbomer 974P (a poly-acrylic acid) and agarose (a polysaccharide), whose synthesis has

been conceived and developed by our research group9. During hydrogel synthesis, a chemical cross-linking is

performed with polyols, i.e. glycerol and propylene glycol, in order to create ester bonds thanks to

polycondensation of Carbomer carboxylic groups and hydroxyl functional groups of cross-linking agents.

Agarose Carbomer (AC) hydrogels have been investigated as potential drug delivery devices in the

spinal cord injury repair strategies, particularly for the treatment of acute cases. Indeed, one of the secondary

effects, which are consequence of a trauma in the spinal cord area, is the formation of a connective glial scar

tissue, which prevents the reconnection of damaged axons and in the end contributes to the chronic

neurological outcomes. The use of a biomedical device, like this hydrogel, in the injured portion could allow

the delivery of active compounds that can interfere with scar tissue formation and eventually inhibit the

formation of this kind of barrier.

AC hydrogels have been completely characterized in terms of physical-chemical and rheological

behavior4,6,9,10,12

and they have been also widely investigated for what regards release and mechanical

behavior2,3,5,7

. Experimental activity also proved that the addition of lantanium salts8 during synthesis phase

does not alter material properties, allowing exploiting rare earths luminescence properties for matrix analysis

at the very early stage. Lantanium is and uniformly distributed and bonded to the polymeric backbone, which

does not interfere with La3+

emission spectra. Hydrogels biocompatibility has been assessed by means of in

vivo experiments in mice models. In particular, AC hydrogels proved to perform in situ gelation12

; this means

that it is possible to inject in the damaged area a drug-loaded gelling solution, which will spontaneously form

the hydrogel in tissue environment, thus avoiding invasive (and hence risky) surgical placement of the

device.

Figure 1.8. Representative sagittal spin-echo images of a spinal cord injured mouse pre and post AC

hydrogel injection. Arrows show the presence of hydrogel (from Perale et al.2).

Chapter 1

16

Moreover, AC hydrogels are subjected to controlled in vivo degradation, thus eliminating the need of

an invasive surgical removal of the device after drug release. Degradation was, hence, investigated both in

vitro and in vivo and was confirmed to occur because of hydrolysis mechanism: water molecules break ester

bonds which are formed during cross-linking phase and act as bridges between macromolecules. In vivo and

ex vivo imaging show hydrogel formation and drug release in the injured volume, confirming the suitability

of AC hydrogels as drug delivery device in the spinal cord injury field. Moreover, in vivo experiments allow

first discovering and highlighting the importance of cerebrospinal fluid (CSF), which flows from rostral to

caudal direction and directs drug release towards caudal direction with respect to hydrogel placing point.

This effect has been recently understood and modeled87

.

Figure 1.9. Drug release from AC hydrogels (from Perale et al.2).

In this context, indeed, this work represents a complementary and synergic part of experimental

activity. Being, as said, molecular diffusion the main mechanism which rules drug delivery from hydrogels

(and, in general, from biomedical devices), a detailed study at atomistic level can give useful insights about

formulation peculiarities that can be applied in the experimental framework. As said, the presence of strong

drug/matrix interactions is not desirable when a fast release is needed, but it is useful when the context

requires a slow prolonged release; atomistic investigation can quantify these interactions, suggesting a

material formulation tuning according to specific exigencies.

1.4.1 Raw materials

Generally speaking, Carbomers are synthetic high-molecular weight polymers of acrylic acid, cross-

linked with either allyl sucrose or allyl pentaerythritol, with a percentage of carboxylic groups comprised

from 52% to 68% on dry basis. Formulations differ by degree of polymerization, solvent used for synthesis,

and cross-link density. Molecular weight values have been theoretically estimated, and are comprised in a

range between 7 105 to 4 10

9 g mol

-1 88

according to the specific formulation. It should be noted that, since

Introduction

17

Carbomers are cross-linked macromolecules, molecular weight is infinite by definition89

and reported values

can be representative of degree of polymerization.

a) b)

Figure 1.10. a) Carbomer structure and b) cross-linkers (allyl sucrose and allyl pentaerythritol,

respectively).

Carbomers are used in pharmaceutical formulations as rheology modifiers, emulsifying or gelling

agents, and in drug delivery systems thanks to their bioadhesive properties. In water environment, at neutral

pH, Carbomers form physical gels; this is due to the swelling behavior of the macromolecules, caused by the

repulsion of the ionized carboxylic moieties in the backbone; Carbomer pKa is equal to 6.0 0.5 88

, and most

of carboxylic groups are in fact ionized at neutral pH values. Elasticity theory of swollen gels have been

applied in order to compute the average molecular weight between crosslinks; values comprised between

about 100,000 and 240,000 g mol-1

have been reported88

.

Despite the increasing use of this compound, an exhaustive Carbomer overview still lacks in

literature, which offers fragmentary and brief insights about macromolecular structure.

United Stated Pharmacopeia (USP) firstly distinguishes Carbomer formulations according to

benzene use as polymerization solvent. If this compound is not employed in the synthesis, then classification

is made according to viscosity measurements (type A, B and C as viscosity range increases). In particular,

Carbomer 974P is considered as a medium cross-linked polymer, with a viscosity range between 25,000 and

45,000 mPa s and classified as Carbopol homopolymer type B, according to USP standards. The letter P

in the formulation number indicates that it is a pharmaceutical grade polymer.

Agarose is a linear polysaccharide, constituted by an alternating sequence of (1-4)-linked D-

galactose and (1-3)-linked 3,6 anhydro-L-galactose. The main monomeric component is thus a disaccharide

called agarobiose. Along with agaropectin (a sulphated ionic polysaccharide), agarose is one of the main

component of Agar, a polysaccharide complex which is extracted from the agarocytes of Rhodophyceae

algae; it is responsible of the thermoreversible gelling behavior of Agar water solution.

Agar is soluble in water at high temperatures; when the solution is cooled to about 30 40 C, gel

formation can be observed.

Chapter 1

18

Figure 1.11. Agarose monomeric unit.

At high temperatures, Agarose chains are in the random coil state and, in the cooling phase, they

form left handed dual helix thanks to hydrogen bonding and van der Waals interactions. A further

aggregation between helices, again due to long range interactions, leads to network stability and rigidity and

consequently to hydrogel formation.

Figure 1.12. Agarose gelation mechanism (from Hsu et al.90

).

Moreover, gel melting point is about 85C; this high difference between gelling and melting points is

called gelation hysteresis. Nowadays, Agarose gels are used as supports for affinity chromatography and

DNA electrophoresis. Generally speaking, instead, Agar is employed in food application as a stabilizing or

gelling agent, and in pharmaceutical field in drug release devices88

.

Propylene glycol and glycerol are polyols, here employed as cross-link agents because of their

functionalities, equal to 2 and 3 respectively. They are commonly used in pharmaceutical formulations as

solvents, emulsifiers, and plasticizers88

. All hydrogels constituents are approved by American Food and Drug

Administration (US-FDA) and by the European MEdicines Agency (EMEA).

a) b)

Figure 1.13. Cross-linking agents: glycerol (a) and propylene glycol (b).

Introduction

19

1.4.2 Hydrogel synthesis

AC hydrogels represent a library of materials, since various formulations have been developed by

varying the ratio between carboxylic groups and hydroxyl groups added in the gelling solution. By

increasing the amount of cross-linking agents, and thus the number of hydroxyl functional groups, hydrogel

mesh size decreases and network stiffness increases, leading to better mechanical properties10

.

An example of different formulations is given in table 1.2.

AC1 AC3 AC4 AC5 AC6

PBS [mL] 49.75 49.1 48.75 47.75 34.25

Carbomer 974P [g] 0.25 0.25 0.25 0.25 0.25

Agarose [g] 0.25 0.25 0.25 0.25 0.25

Propylene

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