Padideh Kamali-Zare
Modeling Biophysical Mechanisms underlying
Cellular Homeostasis
Padideh Kamali-Zare
Doctoral Thesis
Cell Physics, Department of Applied Physics
Royal Institute of Technology (KTH)
Stockholm, Sweden
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Modeling biophysical mechanisms underlying cellular homeostasis
TRITA-FYS 2010:01
ISSN 0280-316X
ISRN KTH/FYS/--10:01--SE
ISBN 978-91-7415-546-4
KTH AlbaNova University Center
Cell Physics, Department of Applied Physics
SE-106 91 Stockholm
SWEDEN
Thesis for the degree of Doctor of Technology in Biological Physics to be presented with due
permission for public examination in FA32, AlbaNova University Center, Roslagstullsbacken 21,
Royal Institute of Technology (KTH), Stockholm, on Friday February 4, 2010, at 13.00
© Padideh Kamali-Zare, 2010
The following papers are reprinted with permission:
Paper V: copyright © 2008, American Chemical Society
Paper VI: copyright © 2009, American Chemical Society
Tryck: Universitetsservice US AB
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Padideh Kamali-Zare
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Modeling biophysical mechanisms underlying cellular homeostasis
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Padideh Kamali-Zare
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Modeling biophysical mechanisms underlying cellular homeostasis
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Padideh Kamali-Zare
Abstract
Cellular homeostasis is the effort of all living cells to maintain their intracellular content when facing
physiological change(s) in the extracellular environment. To date, cellular homeostasis is known to be
regulated mainly by time-consuming active mechanisms and via multiple signaling pathways within
the cells. The aim of this thesis is to show that time-efficient passive (physical) mechanisms also,
under the control and regulation of bio-physical factors such as cell morphology and distribution and
co-localization of transport proteins in the cell membrane, can regulate cellular homeostasis. This
thesis has been developed in an interface between physics and biology and focuses on critical cases in
which cells face physiologically unstable environments at their steady state and therefore may need a
constituent effort to maintain their homeostasis. The main hypothesis here is that the cell geometry is
oriented in such a way that cellular homeostasis is preserved in a given environment. For exploring
these cases, comparative spatial models have been developed that combine transporting function of
membrane proteins with simple versus complex geometries of cells. Models confirm the hypothesis
and show that cell morphology, size of extracellular space and intercellular distances are important for
a dynamic regulation of water and ion homeostasis at steady state. The main clue is the existence of
diffusion limited space (DLS) in the bulk extracellular space (ECS). DLS can, despite being ECS,
maintain its ionic content and water balance due a controlled function of transport proteins in the
membrane facing part of DLS. This can significantly regulate cellular water and ion homeostasis and
play an important role in cell physiology. In paper I, the role of DLS is explored in the kidney
whereas paper II addresses the brain.
The response of cells to change in osmolarity is of critical importance for water homeostasis. Cells
primarily respond to osmotic challenge by transport of water via their membranes. As water moves
into or out of cells, the volumes of intra- and extracellular compartments consequently change. Water
transport across the cell membrane is enhanced by a family of water channel proteins (aquaporins)
which play important roles in regulation of both cell and the extracellular space dimensions. Paper III
explores a role for aquaporins in renal K+ transport. Experimentally this role is suggested to be
different from bulk water transport. In a geometrical model of a kidney principal cell with several
DLS in the basolateral membrane, a biophysical role for DLS-aquaporins is suggested that also
provides physiological relevance for this study. The biophysical function of water channels is then
extensively explored in paper IV where the main focus has been the dynamics of the brain
extracellular space following water transport. Both modeling and experimental data in this paper
confirmed the importance of aquaporin-4 expressed in astrocytes for potassium kinetics in the brain
extracellular space.
Finally, geometrically controlled transport mechanisms are studied on a molecular level, using silicon
particles as a simplified model system for cell studies (paper V and VI). In paper V the role of
electrostatic forces (around the nano-pores and in between the loaded material and the silicon surface)
is studied with regard to transport processes. In paper VI the roles of pore size and molecular weight
of loaded material are studied. All together this thesis presents various modeling approaches that
employ biophysical aspects of transport mechanisms combined with cell geometry to explain cell
homeostasis and address cell physiology-based questions.
/ Padideh Kamali-Zare
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Modeling biophysical mechanisms underlying cellular homeostasis
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Padideh Kamali-Zare
Contents
Abstract ................................................................................................................................ v
List of papers ........................................................................................................................ ix
Summary of papers and author contribution ......................................................................... x
Introduction and Background ................................................................................................ 1
1. Biological physics and cellular biophysics ....................................................................................... 1
2. Modeling ......................................................................................................................................... 1
Classical modeling ........................................................................................................................... 2
Non-classical modeling but very classical scientific strategy .......................................................... 2
3. Cellular homeostasis ....................................................................................................................... 3
Cell membrane, volume and electrical potential............................................................................ 3
Perturbation in the extracellular space .......................................................................................... 4
Biophysical homeostasis ................................................................................................................. 4
4. Morphology of cells and geometrical modeling ............................................................................. 4
5. Diffusion .......................................................................................................................................... 5
Diffusion Limited Space (DLS) ......................................................................................................... 5
6. Transport proteins .......................................................................................................................... 6
Water channels ............................................................................................................................... 6
K+ channels ...................................................................................................................................... 8
Cl- channels ..................................................................................................................................... 8
7. Kidney principal cells and DLS ......................................................................................................... 9
K+ excretion and K+ recycling........................................................................................................... 9
8. Astrocytes and DLS.......................................................................................................................... 9
K+ spatial buffering and K+ siphoning ............................................................................................ 10
Aquaporins and Kir channels in astrocyte membranes ................................................................ 10
Cortical spreading depression (CSD) ............................................................................................. 11
Importance of diffusion in the extracellular space ....................................................................... 11
9. Mesoporous silica particles and DLS ............................................................................................. 11
Methods and Results ........................................................................................................... 13
1. The Main Toolbox ......................................................................................................................... 13
I) Transport mechanisms .............................................................................................................. 14
II) Confocal microscopy ................................................................................................................. 16
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Modeling biophysical mechanisms underlying cellular homeostasis
III) Fluorescence intensity and molecular release ........................................................................ 17
IV) Virtual cell modeling software................................................................................................. 19
2. Methods and results of different papers ...................................................................................... 19
Paper I) .......................................................................................................................................... 19
Paper II) ......................................................................................................................................... 21
Paper III) ........................................................................................................................................ 22
Paper IV)........................................................................................................................................ 23
Paper V)......................................................................................................................................... 25
Paper VI)........................................................................................................................................ 27
3. A key result.................................................................................................................................... 28
Biophysical functional couplings between water and K+ channels in DLS .................................... 28
Discussion and Conclusion ................................................................................................... 31
From morphology to homeostasis .................................................................................................... 31
From homeostasis to morphology .................................................................................................... 32
Modeling perspectives ...................................................................................................................... 32
Example 1. DLS modeling .............................................................................................................. 33
Example 2: ECS modeling .............................................................................................................. 33
Conclusion ......................................................................................................................................... 34
Acknowledgements ............................................................................................................. 35
References .......................................................................................................................... 41
Appendix A.......................................................................................................................... 46
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Padideh Kamali-Zare
List of papers
This thesis is based on the following papers that will be referred to by their roman numbers:
I)
Role of Diffusion Limited Space on water and salt homeostasis. Submitted
Padideh Kamali-Zare, Jacob M. Kowalewski, Marina Zelenina, Anita Aperia, Björn
Önfelt and Hjalmar Brismar
II)
Diffusion limited space contributes to K+ siphoning by regulation of K+ and water
homeostasis in astrocytes. Manuscript
Padideh Kamali-Zare, Jacob M. Kowalewski, Björn Önfelt, Anita Aperia and Hjalmar
Brismar
III)
A role for AQP4 in renal K+ transport. Manuscript
Marina Zelenina, Yanhong Li, Padideh Kamali-Zare, Shigeki Sakuraba, Nanna
MacAulay, Sergey Zelenin, Alexander Bondar, Hjalmar Brismar and Anita Aperia
IV)
Extracellular Space dynamics contribute to Potassium kinetics during cortical
spreading depression in Aquaporin-4 Deficient Mice. Manuscript
Xiaoming Yao, Zsolt Zador, Padideh Kamali-Zare, Hjalmar Brismar, Donghong Yan,
Devin K. Binder, Alan.S. Verkman and Geoffrey T. Manley
V)
Release and molecular transport of cationic and anionic fluorescent molecules in
mesoporous silica spheres. Langmuir 2008; 24(19):11096-102.
Jovice B.S. Ng, Padideh Kamali-Zare, Hjalmar Brismar, Lennart Bergström
VI)
Intraparticle Transport and Release of Dextran in Silica Spheres with Cylindrical
Mesopores. Langmuir 2010; 26(1):466-70.
Jovice B.S. Ng, Padideh Kamali-Zare, Malin Sörensen, Peter Alberius, Hjalmar Brismar,
Niklas Hedin, and Lennart Bergström
Additional papers not included in the thesis
1.
Water transport mediated cell volume manipulation for cell separation. Manuscript
Padideh Kamali-Zare, Sahar Ardabili, Hjalmar Brismar and Aman Russom
2.
Mechanical properties of primary cilia regulate the response to fluid flow.
Submitted
Susanna Rydholm, Gordon Zwartz, Jacob M. Kowalewski, Padideh Kamali-Zare,
Thomas Frisk, and Hjalmar Brismar
3.
Developmental programming prevented by ouabain. Submitted
Juan Li, Georgiy R. Khodus, Markus Kruusmägi, Padideh Kamali-Zare, Xiao-Li Liu,
Ann-Christine Eklöf, Sergey Zelenin, Hjalmar Brismar and Anita Aperia
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Modeling biophysical mechanisms underlying cellular homeostasis
Summary of papers and author contribution
Paper I explores the role of DLS formed by membrane invaginations and intercellular
distances on the cell water and K+ homeostasis. Geometrical models tested and clarified this
role and explained the systematic function of all implemented components (i.e. aquaporins
and Kir channels) in maintaining the extracellular K+ level. The candidate developed the
models, and analyzed the results. The candidate also wrote the manuscript together with
Björn Önfelt, Jacob Kowalewski and Hjalmar Brismar.
Paper II explores the role of DLS and DLS-aquaporins on K+ and water transport in the
brain. These models consider two different DLS compartments in two different sides of the
extracellular space (opposite of each other), and investigate the role of polarized expression
of aquaporins on cell volume regulation. The candidate developed models of DLS in the
brain. A full geometrical model of astrocytes was also built up separately by the candidate
together with Jacob Kowalewski. The candidate also wrote the manuscript.
Paper III investigates a novel role for aquaporin-4 (a family member of aquaporins
expressed in kidney principal cells) in renal K+ transport. This role is shown to be different
from bulk water transport. The geometrical model of a kidney principal cell with membrane
invaginations forming DLS suggests a role for DLS-aquaporins in regulation of net K+
transport through the cell. This role is due to dynamic clearance of DLS region from the
excess K+, generation of net K+ efflux in DLS and conservation of the cell membrane
potential. The model therefore provides a physiological relevance for aquaporin-4 expressed
in the basal membrane invaginations. The candidate developed the models of this study and
contributed to the manuscript writing (modeling sessions).
Paper IV explores the role of dynamic changing of the brain ECS on K+ kinetics in cortical
spreading depression (CSD). The dynamic changing is modeled by considering water
transport following K+ transport into the cells and the recovery by normal diffusion within the
ECS. The models explore whether there is a requirement for a physical association between
the two proteins aquaporin-4 and Kir4.1 channel in the astrocytes, in order to perform an
efficient clearance of K+ from the ECS. Candidate did the modeling part of this study (models
of K+ release into- and clearance from the ECS in CSD with and without aquaporins). The
candidate also contributed to the manuscript writing (modeling sessions).
Paper V studies the internal distribution and release of cationic and anionic dye molecules
from mesoporous silica spheres. Confocal microscopy is used for long time-imaging
experiments on an optical section within the particles. The release profiles were then fitted
with different models of normal diffusion from the particles (in case of Oregon Green 488containing spheres) and normal diffusion affected by electrostatic forces around the surface
of the spheres (in case of Rhodamin 6G-containing spheres). The candidate did the image and
data analysis of this study and tested different models to clarify which one best explains the
observed behaviors. The candidate also contributed to writing the model-related parts of the
manuscript.
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Padideh Kamali-Zare
Paper VI presents experimental data and a model to investigate the dependence of release
rates to the molecular weight of the loaded material in silica particles. This study shows that
when the size of the guest molecules in sphere is about the same as the pore size (6.5 nm), the
release profile is controlled by the molecular interaction between the guest molecules and the
pore walls of the channel. Candidate did the image and data analysis of this study and helped
with describing the molecular models for this transport.
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Modeling biophysical mechanisms underlying cellular homeostasis
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Padideh Kamali-Zare
“The key to every biological problem must finally be
sought in the cell, for every living organism is, or
some time has been, a cell.”
--E. B. Wilson, cell biologist, 1925
Introduction and Background
This thesis describes a modeling toolbox that is applicable for studying cellular homeostasis
and its underlying mechanisms. The important factors of all models are cell's geometry, colocalization of transport proteins in the cell membrane and physiologically uneven
extracellular ionic concentrations that are gathered to address homeostatic mechanisms in
cellular biophysics.
1. Biological physics and cellular biophysics
Biological physics is the research area that links physics and biology. In my view, the goal of
physics is mainly to understand, quantify and define the relationship between physical
parameters of a system [1]. How spatial or temporal dimensions of a system change
depending on the magnitude and form of physical forces, or how a thermodynamic statement
of a system develops depending on the chemical energy, are examples of problems in
physics. However in biology it is not possible to define all parameters affecting a living
system such as a cell, instead the goal can be said to build models and hypothesis for the
function, and through experiments collect supporting data [2]. This thesis deals with
questions that are about how a cell as a living system maintains its balance and communicates
with the physical world. This is defined in the domain of cellular biophysics as a branch of
biological physics [3, 4] that focuses on single cells. A single cell is different from a typical
physical system in the sense that if a physical system is exposed to a change (a force field), it
develops in time according to that force field. However a cell although primarily behaves
similarly and adopts some changes, has the tendency or driving force of getting back to its
initial statement. This will act by mechanisms called homeostatic mechanisms. In the
following section “modeling” in a general term is explained and some examples of classical
and non-classical modeling are mentioned. The section will then be followed by other
important concepts of the thesis.
2. Modeling
There are different interpretations for the term “modeling” depending on the desired level of
complexity in the problem and the questions addressed. In the most general terms, modeling a
system is to translate what we see in our daily observations, experiments and life experiences
to a bunch of words, expressions, equations, and constant relationships. This is to provide a
better understanding of the events around and to formulate things in a way that is more
familiar. We do this modeling also to scientifically communicate our findings with the rest of
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Modeling biophysical mechanisms underlying cellular homeostasis
the world. No matter which language the model is presented in, it can help to figure out the
universal rules of a system and predict what may come next. As the knowledge obtained by
such models relies on the observations, the more extensive the initial data, the more realistic
and reliable the resulting models can be.
Classical modeling
Often in biology, the definition of “modeling” more or less equals to “simulating” the
observed reality. And, to simulate a system, any individual parameter has to be taken into
account as equally important as any other parameter in the system. So the complexity in such
models is an advantage that can make the model closer to the real world situation. In cases
where there is a lack of knowledge of the system, as is often the case in biology, the models
with reduced number of parameters are less informative than experiments in which all the
parameters are present. Modeling a system can be also “to gather” and “put together” all the
known existing relationships between the individual parameters of a system in order to obtain
a new set of interactions and relationships. The knowledge obtained by such models relies on
the combined effects of individuals. This is applicable for modeling biological networks as is
done in systems biology [5-9].
Modeling a system can also describe mechanisms underlying processes and characterize the
state of the system in an exact manner. Mathematical modeling is an example of this type of
modeling. Mathematical models mostly combine different reactions to explain the system
function. They can highlight the most informative experiments and provide useful testable
predictions for future experiments [10, 11]. They can also bring new insights and solid
concepts into biology. Although mathematical models are expressed in mathematical terms
and equations, they are highly formed and shaped by well-known mechanisms, values and
parameters derived from experiments [12, 13]. So, if there is little known about a biological
system because of lack of available experimental data, mathematical modeling cannot be the
best method to explore the system. Often, even if there are some experimental data available,
it is hard to know exactly under which conditions those data have been obtained and whether
the model satisfies those conditions. In contrast, in cases where enough experimental data is
available, mathematical models can fit well with the real world and provide solid information
about the system and its function.
Non-classical modeling but very classical scientific strategy
In a non-classical way, modeling a system can hypothesize the reason for any observation to
take place and then try to test and clarify that hypothesis based on simulations and
experiments. This hypothesis-driven modeling can provide results that can be applied to
experiments. These models can provide logical statements and theories for the modeled
system that can be applied to similar cases. This although, for modeling, is called nonclassical, is a very classical scientific strategy. Simplicity in such models is an advantage and
complexity is a disadvantage. This is because the focus and central part of the model is to
find the absolute reason for any incident and not the side effects of having multiple
interactions together. Furthermore, the goal of this modeling is to identify the cause of any
particular effect, therefore the system’s parameters do not equally contribute to that particular
cause-effect interaction and instead get classified into key/ important versus unimportant
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Padideh Kamali-Zare
factors. In this thesis key factors are those which actively respond to a perturbation (normally
a physical force field) in the system and then help the system to regain its initial statement.
The knowledge provided by such models relies on the response of the system involving key
factors to a particular perturbation. This type of modeling can be called hypothesis-driven
modeling and is a central part of this thesis. This modeling method is then used to investigate
mechanisms underlying cellular homeostasis. Below is a schematic figure showing how a
cell in an interaction with a physical force field can get classified into key versus non-key
components.
B
A
A physical force field
Physics
Interaction with key components
Cell
Key components
Model
Biology
Cell
Figure 1. A. Schematic view of a cell with all its individual components that form a very crowded
compartment. B. A cell in interaction with a physical force field (perturbation) is classified into key
versus non-key components. Key components interact with the physical force field, and this
interaction can be modeled in an interface between physics and biology.
3. Cellular homeostasis
Cells could be called as the smallest "living" units in nature; and the main characteristic of a
living unit, in my opinion, is that it performs active functions in order to survive in this world.
From a physical point of view, if cells stop those active mechanisms, they rapidly disintegrate
and merge into the extracellular fluid. This is because the natural tendency of physical
gradients (i.e concentration, electrochemical, osmotic) is to balance things towards the least
energy and highest entropy. These laws are certainly not valid when it comes to cells as they
contain a soup of intracellular fluid with drastically different concentrations of ions and other
material than the extracellular medium. Cellular homeostasis is the effort of all living cells to
maintain their intracellular content when facing physiological change(s) in the extracellular
environment [4, 14]. Regulation of cellular homeostasis has several physiological aspects and
is important for tissue function [15, 16].
Cell membrane, volume and electrical potential
Cell membranes are the stable walls of cells separating them from the rest of the world. All
the needed material for the cells interior passes through the membrane barriers either by
diffusion or by means of transport proteins. Such proteins sit in the cell membranes and
selectively transport sugar, water and important ions for the maintenance of cell’s electrical
properties. Cells have a certain volume that is crucial for their physiological functions. Bulk
water transport into/out of cells can change cell's volume, thereby giving importance to water
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Modeling biophysical mechanisms underlying cellular homeostasis
transport regulation mechanisms for cell volume homeostasis. Cells also have certain
membrane potential that varies by ion transport. Equally important as the cell volume is the
maintenance of the cell membrane potential which contributes to passive ion transport via
electrochemical gradients. A counteraction between active and passive processes drives the
system towards equilibrium [4, 17, 18].
Perturbation in the extracellular space
Any change in the extracellular space can be considered a perturbation that pushes the cell
towards an imbalanced condition (perturbed state). This change can be either a change in
ionic concentration that can risk the balance of the membrane potential or a change in
osmolarity that can lead to water transport or volume change. The primary response of cells
to these perturbations usually follows physical laws and principles, whereas the secondary
response is mainly biologically oriented and is considered a fundamental characteristic of a
living cell.
Biophysical homeostasis
This thesis investigates that cells, via biophysical mechanisms, can regulate their homeostasis
and that these mechanisms are under control of cell morphology. This is done through
biophysical aspects of cellular components that, in contrast to biochemical active processes,
take a shorter time to run. A short-time passive homeostasis is the basic foundation of the
models described in this thesis that lead to identifying the role of unknown parameters. In
such way, without taking into account the biochemical aspects of transport proteins, these
biophysical roles could be explored. Below is a schematic figure of passive and active
homeostasis.
Perturbed state
Initial state
Long time
Short time
Active
Passive
Figure 2. Schematic figure of cellular homeostasis using passive and active mechanisms. A
perturbation shifts the cell from the initial state to the perturbed state. The cellular homeostasis is then
the effort of cellular components to regain the initial state. This can either happen biophysically, in
short time, or biologically, in long time.
4. Morphology of cells and geometrical modeling
Morphology of a cell is the shape and orientation of the cell membrane in the extracellular
space. In compartmental modeling the geometry of cells is implemented and can regulate the
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Padideh Kamali-Zare
distribution of concentrations in space. Below is a figure showing few examples of cells
geometry that have been implemented in the models described in this thesis.
A
ECS
E
B
ECS
Cell
Cell
C
ECS
ECS
Cell
ECS
F Cell
D
Cell
Cell
ECS
ECS
Figure 3. Examples of cells geometry (A-F). Cell compartments can be either simple (only
rectangles), or complex expressing invaginations or morphological features. Extracellular space
(ECS) can be large, narrow or a compartment having both large and narrow regions.
5. Diffusion
Diffusion is the random motion of molecules or small particles caused by thermal energy
[19]. Diffusion is an important mechanism underlying transport of ions and materials across
the cell membrane as well as within the intra- and extracellular compartments. Diffusion
across the cell membrane is controlled by the function of transport proteins as they passively
pass material down the chemical or electrochemical gradients. Diffusion in intra- or
extracellular spaces, however, is controlled by the geometry of cells that can provide
conditions of free versus limited diffusion. Diffusion in the brain extracellular space has been
extensively studied by diffusion analysis, measured by microelectrodes [20, 21] and
quantified using large flexible random coils of dextrans [22].
Diffusion Limited Space (DLS)
Taking into account the geometry of cells and the extracellular space, a domain can be
formed in complex geometries with membrane invaginations that generate DLS. DLS is the
region where diffusion of substrates is limited by the geometry of the cell. This region can be
either filled with extracellular fluid such as basal invaginations in epithelial cells or with the
cytoplasm such as in ciliary bodies. Also, when two cells are placed in close proximity of
each other, i.e. astrocytes and neurons, DLS can be formed. Intercellular or lateral spaces in
the kidney are other examples of DLS (Figure 4). This thesis explores whether DLS can be
considered as a part of a microdomain linking the function of co-localized proteins to each
other. This is, of course, different from physical associations between the co-localized
proteins, hence the classification as a non-classical association. This can occur via a local
ionic concentration in the extracellular environment, regulated by the restricted substrate
diffusion in a DLS. A possible physiological role for structures like DLS has been suggested
before [23, 24]. However, their microdomain role and regulating function is unknown.
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Modeling biophysical mechanisms underlying cellular homeostasis
Diffusion Limited Space (DLS)
B
A
Intercellular space
Lateral space
Basal infolding
Figure 4. Examples of DLS in the body [25] (printed with permission). A. Schematic picture of
principal cell in the kidney collecting duct. B. Electron microscopy picture of a lateral space in the
kidney.
6. Transport proteins
Transport proteins in the cell plasma membrane are important for regulation of e.g. water and
ion balance in cells. These proteins play significant roles for conservation of the cell volume
and membrane potential. By maintaining the biochemical content of the cell, transport
proteins enable cells to survive, function and proliferate in various extracellular milieus. This
makes them important as a link between the intra- and extracellular compartments.
Water channels
Water channels (aquaporins or AQPs) are integral membrane proteins that facilitate transport
of water across the cell membrane. There are 13 family members of water channels so far
discovered in mammalians [26, 27]. Water channels are important for regulation of water and
volume homeostasis in the body [28].
History of water channels
The problem of membrane water permeability has for many years been addressed by
biophysicists and physiologists. In the 1920s, when the lipid bilayer was discovered, people
were speculating that water passes through the plasma membrane by simple diffusion.
However, it was shown that in certain tissues such as red blood cells, renal tubule and
secretory glands, very high water permeability cannot be explained by simple diffusion
through the plasma membrane. This ultimately led to the prediction that there must be
specific proteins assigned to water transport across the cell membrane. In 1970 an important
observation was made that proved this prediction. While it was known that simple diffusion
cannot be inhibited by any reagent, one observed that mercuric chloride can reduce water
permeability of red blood cells. This provided strong evidence that water channels must exist
and they do have a specific pore characteristic that make them sensitive to mercury [27]. In
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Padideh Kamali-Zare
1992 expression of aquaporins was established in Xenopus oocytes expressing red cell
CHIP28 protein in the laboratory of Peter Agre [29], and in 2003 the Nobel Prize in
chemistry was awarded to Agre for the discovery of aquaporins.
High permeability and selectivity of aquaporins
Comparing two mechanisms of water transport across the cell membrane, with and without
aquaporins, indicates that water channels have a high capacity for water transport. Some
water channels can, in addition to water, transport glycerol and small gas molecules.
Aquaporins have a narrow pathway with 2.8 Å pore diameter. An arginine side of the pore
produces a repulsive force for cations, e.g. protons that can explain the mechanism for water
selectivity of aquaporins [27].
Physiological importance of water channels
Aquaporins are especially important for the kidney function since this is where bulk water
filtration and re-absorption take place in the body [30]. About 180 L of water needs to be
passed through kidney cells everyday for normal regulation. The lack of proper re-absorption
of that water would result in death from dehydration and, if aquaporins could transport both
water and protons, the body would become acidotic [27, 31, 32].
In the brain, aquaporins are important for water and ion homeostasis. Formation of brain
edema is believed to be somehow connected to aquaporins . However, the mechanism is not
known. Two types of brain edema exist: cytotoxic brain edema that occurs when the brain
glial cells swell, and vasogenic, when the extracellular space swells [33]. Both types of brain
edema are caused by water imbalance in the brain and must therefore be linked to the
function of aquaporins in cells. Thus a proper regulation of water channels in the brain can be
critical for preventing brain edema [34, 35].
In the lung, fluid movement between the air space and vascular compartments happens
through highly water-permeable airway epithelia. This is important for water re-absorption of
alveolar fluid in the neonatal period that prepares the alveolar for respiration [36].
Osmosis as mechanism for water movement via aquaporins
The mechanism behind water movement through aquaporins is osmosis. Osmotic gradients
are created when the osmolarity of intra- and extracellular environments differ. This drives
water into or out of cells depending on the direction of the gradient (inward when the
extracellular solution is hypotonic and outward when it is hypertonic) [4]. The water
movement then leads to cell swelling or shrinkage. If aquaporins are abundantly expressed in
a cell, even a small difference between intra- and extracellular osmolarity can lead to a bulk
water transport and a serious change in the cell volume. That is why red blood cells rapidly
shrink in seawater (a hypertonic solution) and swell in fresh water (a hypotonic solution)
[27].
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Modeling biophysical mechanisms underlying
u
cellular homeostasis
Figure 5. A model of aquapo
porin-mediated water transport in a local envir
vironment around the
aquaporin. Color gradient shows
ws water concentration ([H2O]) around the channel.
el. As water leaves the
cell to the extracellular space, water
wa concentration inside the cell decreases and increases
inc
in the ECS.
K+ channels
There are two different types
es of potassium channels: voltage gated potass
assium (Kv) channel
[37, 38] and potassium inward
ard rectifying (Kir) channels [39, 40]. Kv cha
hannels are gated by
voltage and no gating is known
wn for Kir channels. They are open when thee membrane
m
potential
is highly negative and pass potassium
pot
down the electrochemical gradient. The
T conductance of
+
the channel is high for inward
rd K current [41, 42].
An electrochemical gradient
nt is built up by the difference between thee resting membrane
potential and the Nernst pot
otential for K+. Nernst potential is a concen
centration dependent
+
potential that varies when the extracellular K level changes. When thee extracellular K+ is
increased the direction of electrochemical
ele
gradient is inward. This iss against
a
the normal
+
concentration (chemical) grad
adient which is from the high K concentration
tion inside the cell to
the low concentration outside
de the cell [4, 39, 40, 43]. Kir channels are invo
nvolved in regulation
of many important cellular processes
pro
such as renal potassium transport [44]
[44 and maintenance
+
of the membrane resting poten
tential as well as intra- and extra-cellular K levels
lev [45].
Cl- channels
Chloride is the most abunda
dant physiological anion. In the kidney chlo
hloride channels are
present in both the plasma membrane
me
and intracellular organelles and hav
ave a very important
role for the salt filtration in the
th kidney [46, 47]. In the brain chloride chan
hannels participate in
the neuronal astrocytic inter
teraction, playing a role in the control of extracellular K+
homeostasis [42, 48], bufferin
ring extracellular pH [49] and regulating nerve
rve activity [50]. The
transport of Cl ions throughh ion
i channels in both excitable and non-excita
itable cells underlies
important physiological functi
ctions such as stabilizing the cell resting membr
brane potential [51].
Since the concept of K+ excret
retion after the uptake by the cell relies on thee transient
t
change in
the cell membrane potentiall (depolarization
(d
of the membrane potential), Cl transport into the
cell may diminish this transie
sient change and therefore compete with the K+ spatial buffering
by the cell [51-53].
8
Padideh Kamali-Zare
7. Kidney principal cells and DLS
Principal cells in the kidney collecting duct are an example of cell types in the body that
express complex geometrical features. They are located in the cortical collecting tubule
(CCT) and are polarized cells having basolateral (basal) and apical membranes. In their
basolateral side they face the blood and the apical side, face the urine. The physiological role
of principal cells is to regulate K+ levels in the blood [25, 54].
K+ excretion and K+ recycling
When K+ levels are elevated, kidney principal cells take up K+ from the blood side and
secrete it into the urine. This mechanism is known as K+ excretion that is highly dependent
on the function of active transporter, Na+,K+-ATPase, in the basolateral membrane. A small
portion of K+ is released back to the basal extracellular space. This K+ recycling is believed to
play an important role in maintenance of the cell membrane potential, but the underlying
mechanism is not known [55-57].
Principal cells of the collecting duct have deep membrane invaginations in their basal
membrane. These form DLS within the extracellular space. In a geometrical model, described
in paper I of the thesis, the possible role of membrane invaginations, has been tested on K+
excretion and K+ recycling. Kidney principal cells express low conductance Kir channels,
Kir7.1 [58]. They also express three different AQP-isoforms; AQP4 and AQP3 are present in
the basolateral membrane and AQP2 are in the apical membrane [59]. In paper I,
mathematical models tested the possible role of Kir channels and aquaporins in facilitating
the net K+ transport by these cells at steady state, and in paper III, a novel role for AQP4 in
renal K+ transport is investigated.
8. Astrocytes and DLS
The brain consists of two different cell types: neurons and glial cells. Neurons are electrically
excitable cells that, by transferring signals along the nerve fibers, communicate with the rest
of the body and produce our senses. Astrocytes are star-shaped glial cells in the brain that
tightly surround neurons and support them in many ways, including transporting glucose for
their energy supply and clearing the ECS from the excess ions released during the neuronal
activities [60, 61]. Astrocytes occupy the space between neurons and blood capillaries and
are in contact with the blood brain barrier [62]. Although astrocytes are more abundant in the
brain than neurons, much less attention has been given to them and little is known about their
function. One of the basic differences between neurons and astrocytes is that expression of
aquaporins is found in astrocyte membranes but not in neurons. This enables astrocytes to be
the main response to water imbalance in the brain, thereby protecting neurons against severe
volume changes.
The extracellular ECS between the neurons and astrocytes is fairly tight in contrast to the big
volume of astrocytes. This makes the exchange of water between the astrocytes and the
extracellular space a sensitive mechanism that can unfavorably lead to a prolonged shrinkage
of the ECS. This thesis focuses on mechanisms by which water transport from the ECS to the
astrocytes can occur in a well controlled manner that protects the ECS from serious
shrinkage.
9
Modeling biophysical mechanisms underlying cellular homeostasis
One of the well known functions of astrocytes is the clearance of K+ from the ECS during
neuronal activities [13, 63, 64]. When neurons are fired by action potential, they release K+
into the ECS. Fast removal of the excess K+ from the ECS is important for the continual
neuronal function. Diffusion within the ECS as the mechanisms for this removal is rather
slow due to the narrow ECS. Astrocytes do speed up this clearance by taking up K+ from the
ECS. This is believed to occur via the active transporter Na+,K+-ATPase and the NaKCl
transporter [13, 65].
K+ spatial buffering and K+ siphoning
Transport of extracellular K+ from regions of high concentration to regions in the
extracellular space with lower concentration is called K+ spatial buffering. This K+ spatial
buffering is done through the cells. K+ flows into the cell in the form of inward current and
leaves the cell in the form of outward current [24]. Astrocytes and retina glial cells are
believed to spatially buffer the excess K+ from the synapse where high neuronal activity
occurs. This was first proposed by Orkand et al. in 1966 [66]. Below is a schematic figure of
K+ spatial buffering that results in balanced ECS K+ levels.
K+
K+
Kir Pump
channel
K+- channel
Cell
K+
Figure 6. Schematic figure of K+ spatial buffering. The excess K+ from the regions of high ECS K+ is
taken up by the cell and is released to the other parts with low ECS K+.
If the excess K+ get released from the cell and directed towards blood capillaries or other
fluid volumes, K+ spatial buffering is called K+ siphoning [67-69].
Aquaporins and Kir channels in astrocyte membranes
Astrocytes express high conductance K+ inward rectifying (Kir) channels in their membranes
[43]. It has also shown that endfoot membranes of astocytes and retina glial cells have higher
K+ conductance than the rest of the membrane [43, 70]. This thesis investigates the
circumstances under which the passive K+ transporters such as Kir channels can facilitate the
K+ uptake. Recent studies have shown that astrocytes also express aquaporin-4 in their
membranes [52, 69, 71]. Aquaporin-4 density is shown to be higher in the endfoot membrane
of astrocytes, proposing a physiological role for their polarized expression [52].
Recent studies suggest a role for aquaporin-4 expressed in astrocyte membrane in facilitating
K+ siphoning [34, 69, 71-75]. However the underlying mechanism is not known. In a
10
Padideh Kamali-Zare
geometrical model described in paper II, a possible role for aquaporins and Kir channels in
astrocytes K+ siphoning is investigated.
Cortical spreading depression (CSD)
Distribution of K+ and water channels in the brain and retina is involved in edema and
seizures [52, 71, 76, 77]. Phenotype studies on AQP4 deficient mice, have revealed that
AQP4 is involved in neuroexcitation. Seizure threshold is increased and its duration
prolonged in AQP4 deficient mice [78]. Lack of AQP4 also affect other neuronal electrical
responses involved in auditory [79], olfactory [80] and retinal systems [81]. A lack of AQP4
also impaired K+ uptake from the extracellular space in models of neuroexcitation [82]. All
these suggest a role for AQP4 in neuroexcitation. However, the underlying mechanism is not
known. One model for investigating this role is cortical spreading depression (CSD) which is
a self-propagating wave of neuronal depolarization with increased extracellular K+ level [83].
In paper IV of this thesis, a biophysical involvement of AQP4 in K+ kinetics in CSD is
explored.
Importance of diffusion in the extracellular space
Understanding the function of neuronal-released substrates in the extracellular space (ECS),
need characterization of their distribution in the tissue. For this reason morphology of ECS is
of critical importance. There have been several experimental techniques to estimate the size
of ECS and the effective diffusion coefficients for the released or injected substrates [21, 22,
84-88]. It is shown that slow diffusion of drugs in the brain can be a major contributing factor
to the limited penetration of them, therefore suggesting ways to modify the chemical’s design
for overcoming this diffusion limitation. [89]. In this thesis, a proper connection between the
tissue morphology, a diffusion limited space and a microdomain is proposed.
9. Mesoporous silica particles and DLS
Mesoporous silica particles (Figure 7) are used in this thesis as models for transport studies
on a molecular scale. They could resemble the real cell’s transporting processes since the
pores on the silica surface function in a way similar to passive transporters, i.e. ion or water
channels. Mesoporous particles are used for biomedical applications, especially in drug
delivery, for their capacity to host macromolecules and drug molecules for extended time
periods. Release of host molecules from the particles is hindered as the diffusion of
macromolecules within the confined space inside the particles is limited (a DLS) [90-92].
This increases the release time compared to release of macromolecules loaded in a diffusion
free space particle. In paper V of this thesis, diffusion of loaded material from the particles to
the bulk solution is studied in two different conditions: one where there was no electrostatic
force between the host molecules and the silica surface (using negatively charged Oregon
green as loaded material), and one where there was an attractive electrostatic force between
the surface of the silica (with negative charge) and the loaded material (Rhodamin 6G with a
positive charge). Transport of host molecules could then be studied by the models for
concentration/chemical gradient mediated release versus electrochemical gradient mediated
release. In paper VI, the role of host molecule molecular weight was studied in either Dextran
11
Modeling biophysical mechanisms underlying cellular homeostasis
3k or Dextran 10k. In conclusion, both studies could give insights into transport studies in
vitro.
10μm
Figure 7. Scanning electron microscopy image of mesoporous spheres.
12
Padideh Kamali-Zare
Cell and tissue, shell and bone, leaf and flower, are so
many portions of matter, and it is to the obedience to the
laws of physics that their particles have been moved,
moulded, and conformed.
--D’Arcy Thompson, On growth and form, 1917
Methods and Results
1. The Main Toolbox
The methods used in this thesis are all taken from an interface between physics and biology
where concepts, principles and a physics toolbox are applied to understand and explain
biology. The tools offered by physics in the context of biology can then address questions
that are medically relevant. This is a part of biological physics that links different scientific
areas in order to understand living systems in a logical way. Mathematical equations and
computer simulations are a part of the toolbox used to formulate and quantify mechanisms
underlying biological phenomena. In this thesis biology as the context has been roughly
viewed as a field with structural versus functional (physiological) parts. The structural part
has been put outside of the toolbox whereas the functional (physiological) part has been the
major focus. Below is a schematic diagram summarizing biological physics as a toolbox used
in this thesis.
Physics
Math
Computer Biological
Physics
Biology
(Structure) (Function)
Medicine
Figure 8. Biological physics, as an interface between physics and biology, applies tools of physics
into the context of biology and explains medically relevant questions. All together, this scheme
presents a toolbox that links different scientific areas in order to understand living phenomena.
13
Modeling biophysical mechanisms underlying cellular homeostasis
I) Transport mechanisms
Diffusion
Diffusion of any substance (s) in the space, as well as across the cell membrane, can be
described by Fick’s second law:
∂ 2Cs
∂Cs
= Ds 2 ,
∂x
∂t
(1)
Where Ds is the diffusion coefficient of the substance in units of m2/s and Cs is the
concentration of substance S. This relates the temporal and spatial distribution of the
substance concentration (Cs) to each other [4, 93].
Electrodiffusion
Electrodiffusion of ions occurs when the driving force for the movement of ions is electrical.
Across the cell membrane electrodiffusion takes place when there is a difference between the
resting membrane potential of the cell and the local Nernst potential for any specific ion [4].
Nernst potential for ion ‘n’ is described as:
⎛ Cno
RT
Vn =
ln ⎜
z n F ⎜⎝ C n i
⎞
⎟
⎟
⎠
(2)
where R is the molar gas constant (8.31 J.K-1.mol-1), F is Faraday’s constant ( 9.65 × 10 4
C/mol), Zn is the valence of the ion ‘n’, and T is the absolute temperature. At (tc=24o C) the
Nernst potential is calculated as:
Vn =
⎛ Cno
59
log 10 ⎜⎜ i
zn
⎝ Cn
⎞
⎟
⎟
⎠
(3)
mV
i
o
where C n is the intracellular concentration of ion ‘n’ and Cn is the extracellular
concentration.
If the Nernst potential for ion ‘n’ differs from the membrane potential, a current will be
generated as:
In = γ n (Vm −Vn )
Where
(4)
γ is the conductance of the membrane to the ion ‘n’ and Vm is the membrane potential
[4].
Potassium inward rectifying (Kir) channels
Kir channels can allow a passive transport of K+ driven by electrodiffuion. When the
extracellular concentration of K+ locally changes, the Nernst potential for K+ increases and
creates an inward current of K+. The conductance of Kir channels for K+ varies among the
family members. Kir4.1 channel expressed in astrocytes has a relatively high K+
conductance, about 27pS, whereas Kir7.1 expressed in kidney principal cells has a low K+
14
Padideh Kamali-Zare
conductivity, about 50 fS [94]. The current density of K+ across the membrane enriched by
Kir channels is described by:
jK = NKirG(Vm −Vn )
(5)
Where NKir is the number of Kir channels per µm2 of the membrane and G is the conductance
of each single channel to K+.
Molecular efficiency of Kir channels
As 1 K+ ion has a charge of 1.6 × 10 −19 coulomb and the conductance of a Kir7.1 channel to
Kir7.1
= 50fS [94], knowing that 1S=1 Ampere = 1 Coulomb/ Second , at a voltage of 100
K+ is: G
Volt
Volt
Q × ( N / s)
. Therefore
mV, the conductance can be calculated as: G Kir 7.1 = 50 fS = 5 × 10−14 S =
V
the total number of ions passing through a channel per second will be:
G Kir 7 .1 × V 5 × 10−14 × 0.1
N/s =
≈ 3 × 104 ions/ sec . This value for Kir4.1 channel can be
=
−19
Q
1.6 × 10
calculated as:
G
×V 27 ×10−12 × 0.1
Q × ( N / s)
⇒ N/s =
=
≈ 1.68×107 ions/ sec
−19
V
Q
1.6 ×10
Water transport and volume change
Kir 7.1
G Kir 4.1 = 27 pS = 27 ×10−12 S =
Cellular membranes are highly permeable to water. The primary response of cells to the
change in osmotic pressure is water transport. Mechanisms for water transport via the
membranes are through the porous membrane and via water channels.
The Van’t Hoff law states that the osmotic pressure at equilibrium caused by different ion
concentrations in and out of the cell can be calculated as:
πi −πo = RT(Cni − Cno )
(6)
The osmotic-driven water flux through the water permeable membrane is calculated as:
φW = pW (Cno − Cni ) = Lv RT(Cno − Cni )
where
(7)
pW = LvRTis the osmotic permeability and Lv is hydraulic conductivity [4].
The cell volume changes following water transport across cell membranes. Assuming that the
volume of the transferred water is V i , dynamics of V i can be calculated as:
φW =
−1 dV i
o
i
= Lv RT(Cn (t ) − Cn (t ))
A(t ) dt
(8)
Flux of water through the pores of molecular dimension
15
Modeling biophysical mechanisms underlying cellular homeostasis
The osmotic permeability of a pore of molecular dimension is calculated as:
a
RTπr 4
pW = N
G( W )
r
8υ W ηd
(9)
where N is the number of water molecules that can fit in the pore of the channel,
radius of the pore,
υW
is partial molar volume of water (volume of a mole of water),
viscosity, d is the length of the pore,
η is
aW is the radius of a single water molecule and
G(
aW
) is the hindrance factor calculated as:
r
G(
aW
a
a
a
) = 1 − 0.8341( W ) 2 + 0.8977 ( W ) 3 − 1.0586 ( W ) 4
r
r
r
r
In the case that aw is very small compared to r, the ratio (
factor tends to 1: ( G(
r is the
(10)
aW
) tends to zero and the hindrance
r
aW
) →1 ). This indicates that the pore is large enough to avoid any
r
resistance to water movement. The flux of water through the pore will then be:
φW = N
RTπr 4 aW
o
i
G ( ).(Cn − Cn )
8υWηd
r
(11)
Flux of water through water channels (aquaporins)
Physiologically aquaporins act as water pores with a small pore diameter [4, 32]. Water
transport through the aquaporins is osmotically driven. Aquaporins have narrow pathways
with diameter of about 2.8 Å in the narrowest part of the pore that is big enough for a single
water molecule. Although the pathway for water transport is narrow, the channel provides no
resistance for water molecules to pass so the hindrance factor can be considered as 1. This
occurs because no hydrogen bonding can occur between water molecules that are spaced
within the pore at intervals [27, 32]. For this reason, water channels have high single channel
permeability.
Molecular efficiency of aquaporins
3
Assuming that water permeability is 15×10−14 cm (for aquaporin-4) [95], a single channel can
s
9
pass 5×10 water molecules per second.
II) Confocal microscopy
With confocal microscopy it is possible to optically section a sample [96]. In this thesis
confocal microscopy was used for cell volume measurements [97] as well as molecular
16
Padideh Kamali-Zare
release studies from silica particles [98, 99]. In those studies it was crucial to record
fluorescence intensity from thin and well-defined optical sections, why the pinhole was kept
as small as possible. This made it possible to directly interpret a change in fluorescence
intensity as a change in dye concentration. The centro-symmetric geometry of mesoporous
spheres allowed the use of a 2D diffusion model for the analysis of fluorescence intensity and
dye release.
III) Fluorescence intensity and molecular release
Rate of molecular release
Silica particles have a negatively charged surface. This property provides an electrostatic
attraction to positively charged dye molecules such as Rhodamin 6G loaded in the particles.
This electrostatic attraction restricts free diffusion of dye molecules from the mesoporous
spheres. Therefore Fick’s second law of diffusion (equation-1) as the simple mechanism for
transport of dye molecules from the silica spheres cannot be applied. For negatively charged
Oregon Green 488-containing spheres the release is controlled by normal diffusion described
by Fick’s second law. Although the solution to equation-1 varies by the initial concentration,
we can still assume that the release of dye (m) through the pores follows steady state
diffusion:
φm = Pm(ΔCm)
where
(12)
Pm is the rate of the dye molecule transport through the pores, and ΔCm = Cmi − Cmo is
inside
the concentration gradient between inside ( Cm
outside
) and outside of the particle ( Cm
). So
the differential equation-1 will change to:
∂(Cminside − Cmoutside)
∂C( x, t )
= Pm
∂t
∂x
(13)
Assuming that steady state applies at each instant in time in the pore, the differential
equation-16 will reduce to:
∂C ( t )
= k ( C mi − C mf )
∂t
(14)
Equation-14 is a first-order ordinary linear differential equation with rate constant coefficient
k . The solution to equation-14 is an exponential function varying in time between initial (at
time=0) concentration, C m and final (at time= ∞) concentration, C m :
i
f
C (t ) = ⎡⎣ (Cmi − Cmf ) exp( − kt ) ⎤⎦ + Cmf
(15)
The decay of the fluorescence intensity can then be represented by a simple rate equation
∂ I (t )
= k ( I mi − I mf ) resulting in an exponential decay:
∂t
17
Modeling biophysical mechanisms underlying cellular homeostasis
I (t ) = ⎡⎣( I mi − I mf ) exp(−kt ) ⎤⎦ + I mf
(16)
i
where I (t ) is the fluorescence intensity of dye d, I m is fluorescence intensity of dye at time
f
zero, Im is fluorescence intensity at time
∞, and k
is the rate of decay. Assuming nominal
bleaching of the dye, it can be concluded that the decrease in the fluorescence intensity of the
dye in the spheres during the study represents the release of the dye molecules. Accordingly,
we can conclude that molecular transport studies can be performed using fluorescence
imaging and a 2D diffusion model for dye molecules.
Effective diffusion coefficients for dye molecules
The molecular transport and release of material from silica particles can follow different
models depending on the charge and molecular weight of the loaded material. In the simplest
case (if there is no extra interaction between the loaded material and the pore or the surface of
the particles), a spherical diffusion model can be used as a model for release. This can be
described with a simple Fickian model (equation-1) and means that the effective diffusion
coefficient can be derived from the rate of release. If the loaded material is a fluorescent dye,
the change in intensity of the particle in time can be related to the rate of the release. Using
the theory for sorbate uptake in isothermal spherical particles [93], an effective diffusion
coefficient Deff could then be calculated by:
It
6
= 2
I0 π
− n 2π 2 Dt
1
exp(
)
∑1 n 2
R2
∞
(17)
where It is the intensity in time ‘t’, I0 is the intensity at time zero and R is the radius of the
spherical particle. The expression can be simplified by analyzing the asymptotes at short or
long times only. The long time asymptote estimation is calculated as:
It
6
− π 2 Dt
exp(
)
=
I0 π 2
R2
(18)
For more complicated cases such as release of dextran molecules, the release behavior is
neither exponential nor biexponential in nature. If the release kinetics conforms to a
logarithmic time dependency, such as for chemisorptions and formation of, e.g., oxide films
on metals [100], the model for release can be described by:
dq
= CasN
dt
(19)
where q is the number of molecules sticking to the surface, s is the site density, a is the
effective contact area, C is a coefficient constant, a is the approximate molecular cross
section, N is the number of impacts made by the molecules with the surface per unit area per
unit time. In summary time dependent concentration can be monitored by:
q* = q0 −
1 ⎛ t
ln ⎜
b ⎜⎝ t ∞
⎞
⎟⎟
⎠
(20)
18
Padideh Kamali-Zare
where b is the effective surface area and q* can be expressed as the time-dependent
concentration. This equation can be deduced from the Temkin model [101, 102]. Johnson and
Arnold have used similar approaches to quantify reversible protein adsorption onto a nonuniform surface [103]. They showed that an increase in the binding sites between the protein
and the surface increases the apparent binding affinities and maximum capacities, i.e.
increasing binding strength.
IV) Virtual cell modeling software
Mathematically, a simplified biological system can be described by ordinary differential
equations (ODEs) including time derivatives that track changes in time. Often ODEs do not
have simple and exact solutions. However, in simplified cases, with the use of numerical
methods, the system can be described by partial differential equations (PDEs) in which both
spatial and temporal derivatives are present. PDEs can then have simple solutions by which
describe several phenomena such as transport of material within the intra- and extracellular
spaces as well as across the cell membrane. Fick’s laws of diffusion are also described in
PDEs [4, 18]. Virtual Cell (NRCAM, NRCC, NIH; www.vcell.org) is the software used for
geometrical models in this thesis. It uses the finite element method (FEM) that finds the
approximate solutions for PDEs, and numerically integrates the solution over a certain
distance [104]. The full description of models: DLS in kidney principal cells, DLS in
astrocytes, K+ accumulation in the brain and K+ clearance in CSD involving all the values,
parameters and equations are publicly available at www.vcell.org/ bio-models, under the user
padideh. A full description of the model of DLS in kidney can be found in Appendix A.
2. Methods and results of different papers
Paper I)
This paper investigates the role of DLS formed in epithelial cells, on water and ion
homeostasis. K+ levels in different parts of the extracellular space are assumed to reflect this
role.
Method
A series of spatial models are developed to study the dynamic distribution of K+ in the
diffusion limited space (DLS) formed in kidney principal cells. Models are constructed in
three modes: with AQPs, without AQPs, and with polarized expression of AQPs. The
geometry used in the models (shown in Figure 9A) resembles the kidney cell morphology
with several DLS (Figure 4). Parameters that are modeled are [K+], [Cl-], [H2O] and
membrane potential of the cell. K+ and water channels are located in the cell membrane. Cell
membranes are assumed to be permeable to Cl-. Water transport by the cell is modeled as an
osmotic-driven flux, where [K+] and molecular [H2O] are the factors contributing to the
osmotic gradient. Permeability of the cell membrane without AQPs is set to 17 µm/s. In
membranes with AQPs, the permeability increased 7-fold to 119 µm/s. The mechanism of K+
and Cl- transport is electrodiffusion. The initial perturbation used is an elevation of
19
Modeling biophysical mechanisms underlying cellular homeostasis
extracellular K+ from 4 mM to 6 mM. The response of the cell to this elevation is monitored
in time. Two data sets for potassium concentration [K+] are compared, one in the bulk
extracellular space (ECS) and one in the DLS region. A full description of the Virtual cell
modeling, specific settings and all the parameters are described in Appendix A.
Results
The models show that following both K+ transport within the ECS and a passive inward
current from the basal side into the cell, an outward K+ current is generated in DLS. This
recycling can transiently maintain the cell membrane potential. However as the elevated K+
diffuses into DLS from the bulk ECS, the outward current will be inhibited at a steady state.
Our models show that if water channels are localized in DLS together with Kir channels, K+
recycling can run at steady state. This highlights a role for DLS and aquaporins that can be
physiologically important for the continual recycling at steady state and the maintenance of
the cell membrane potential.
Models show that aquaporins can efficiently maintain the K+ level at the DLS region by local
dilution of the excess K+ diffused into the region from the bulk extracellular space. Kir
channels in DLS membrane can then generate outward K+ currents since membrane potential
is already depolarized following the bulk K+ uptake, but the Nernst potential is maintained.
K+ recycling through Kir channels in DLS can re-polarize the cell membrane potential
(Figure 9).
The maintenance of the cell membrane potential then leads to a continual passive uptake of
K+ through the rest of the basal membrane. This is important for the passive K+ excretion into
the urine. Aquaporins in DLS, by maintaining K+ levels, can also inhibit the high activity of
the energy consuming K+ transporter, Na+,K+-ATPase, under conditions of high K+ in the
bulk extracellular space.
A
B
ECS (without AQPs)
DLS (without AQPs)
5.5
T=1s
[K ], mM
5
+
T=0
D
6
ECS (with AQPs)
4.5
Membrane lined
with Kir channels
(uniform) and
AQPs (polarized)
K+
DLS (with AQPs)
4
0
0.2
0.4
0.6
0.8
1
Time, s
Diffusion limited Space
(DLS)
C
-82
Local Nernst potential
+
for K , mV
Increased K+ level (6mM)
K+
ECS (without AQPs)
DLS (without AQPs)
-84
water
-86
K+
ECS (with AQPs)
-88
Diffusion free space (ECS)
-90
DLS (with AQPs)
-92
-94
0
0.2
0.4
0.6
Time, s
20
0.8
1
Padideh Kamali-Zare
Figure 9. Model of DLS in a kidney principal cell. A) The geometry used in the model. B) Dynamics
of the extracellular [K+] is shown in the bulk extracellular region and in DLS (part of the extracellular
space surrounded by membrane infoldings). Initially, [K+] in both regions of extracellular space
increase (following the elevation in K+ at the lower edge of the geometry). However, if AQPs are
present, [K+] is regulated by water transport and tends to its normal level. The regulation with AQPs
is enhanced in DLS, and DLS [K+] quickly gets back to its normal value. C) The Nernst potential for
K+ in four different models is shown. The membrane potential in this case follows the Nernst potential
at steady state. D) Direction of K+ and water flux across the membrane.
Paper II)
This paper investigates the role of DLS in the brain.
Method
Geometry used in this model is shown in Figure 10A. Three different compartments are
implemented in the model, representing a neuron, an astrocyte and a wall of endothelial cells.
Two DLS exist in the model: one is in the synapse and one between the astrocyte and the
endothelial cells. Two different modes of synapse geometry are implemented: one when the
synapse is a DLS, and one when it is a diffusion free space (DFS). [K+], [Cl-], [H2O] and
membrane potential are the major parameters that are modeled. Initially the extracellular K+
level in the synapse is elevated to about 15 mM. The [K+] in different parts of the
extracellular space are tracked in time. Also the role of polarized expression of AQPs (10fold increased in the endfoot region) is studied. A full description of the model can be found
in the VCell database under the user padideh.
Results
Simulations in geometrical models of a synapse region show that, following a step increase in
synaptic [K+], K+ is taken up by the astrocyte. This enhances the clearance of K+ from the
synaptic region (Figure 10B). The model also shows that AQPs play a role in K+ clearance in
DLS but not in DFS. AQPs showed no effect on chloride transport in the synapse. The
geometry of the extracellular space both in the synapse region and in the endfoot is shown to
be an essential component for the local changes in the cell and ECS volume (Figure 10C and
paper II). Furthermore, simulations show that polarized expression of AQPs in the astrocyte
endfoot strengthened the functional coupling between K+ and water channels. This results in
an efficient removal of K+ from the tight region between the astrocytes and the endothelial
cells. Therefore, DLS formed between astrocyte endfoot and blood capillaries, although
limiting the free diffusion of the excess K+ from the region, facilitates K+ siphoning by the
help of AQPs (Figure 10D). This is important for the maintenance of membrane potential and
the net K+ spatial buffering capacity in astrocytes.
21
A
B
Neuron
ECS1 (synapse) DLS
Astrocyte
endfoot
ECS2 DLS
Capillary wall (endothelial cells)
[K+] in the synapse, mM
Modeling biophysical mechanisms underlying cellular homeostasis
16
14
12
10
6
4
0
Local Cell Swelling
108
106
102
20
40
60
Time, ms
80
astrocyte
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
Time, s
D7
104
100
0
0.001
6
+
110
ECS=DFS
When
ECS= DFS
ECS=DFS, without
AQPs
ECS=DLS,
without
AQPs
ECS=DFS,
with
uniform
AQPs
ECS=DLS,
with
uniforma
AQPs
ECS=DFS, with
AQPs
polarized
to
ECS=DLS,
with
polarized
AQPs
the endfoot
ECS2 [K ], mM
Cell Volume (%)
C 112
Diffusion in the ECS
Diffusion +uptake by the astrocyte
8
100
without AQPs
without AQPs
with uniform AQPs
with uniform AQPs
with AQPs polarized to
with polarized AQPs
the endfoot
5
4
0
10
20
Time, ms
30
40
Figure 10. A) Geometry used for the model of astrocytes facing two DLS in the extracellular space.
B) [K+] in the synapse (ECS1) is tracked in time, comparing with and without astrocyte K+ uptake. C)
Volume of the cell when ECS is a free diffusion space compared to when it is a DLS. Three different
modes of AQPs expression are also compared. D) [K+] in ECS2 is tracked in time in three different
modes of AQPs expression.
Paper III)
This paper investigates a novel role for AQP4 in renal K+ transport. Experiments of this study
showed that water permeability of AQP4, but not AQP3, is modulated by extracellular
potassium concentration. This suggested a specific role for AQP4 in K+ transport.
Experiments also showed that AQP4, but not AQP3, assembles with Kir7.1 and Na+,K+ATPase in rat renal medullary tissue.
Modeling Method
To see whether water flow via AQP4 coupled to K+ transport may play a physiological role in
the steady state transport of K+ in kidney principal cells, a geometrical model is developed.
This model describes the dynamic distribution of K+ in the DLS region (Figure 11A) for
cases of increased and decreased ECS [K+]. The presence of a low conductance K+ channel
(Kir7.1) is also of importance for this model. Simulations of the dynamic distribution of K+ in
the DLS were performed under three conditions: 1) no AQPs are present in the plasma
membrane; 2) AQPs are present, but are not regulated; 3) AQPs are present and regulated by
changes in K+ concentration in the DLS ([K+]DLS).
Modeling Results
Simulations show that if AQPs are not present in the model, the membrane water
permeability is low, showing no significant influence on [K+]DLS. This results in rapid
changes in [K+]DLS following variations in ECS [K+] that lead to similar values for resting
[K+] in both DLS and ECS. However, if AQPs are present in the model and expressed in DLS
membrane, a high water permeability allows an outflux (in case of increased ECS [K+]) or
22
Padideh Kamali-Zare
influx of water (in case of decreased ECS [K+]). This leads to a counteraction between K+ and
water that balances the K+ level while diffusing from ECS to DLS (in case of increased ECS
[K+]) or from DLS to ECS (in case of decreased ECS [K+]). The resulting [K+]DLS at steady
state is close to the normal value. In a model where AQPs in the plasma membrane are
regulated by extracellular K+, the regulation of [K+]DLS is faster (Figure 11 B). The regulation
of AQPs in the model is through an osmotic effect and not due to a physical association with
K+ transporters.
Figure 11. Model of a kidney principal cell (A) Geometry used in the model modified from [105] (B)
The time course for [K+]DLS under three conditions: without AQPs (red traces), with AQPs (green
traces), and with regulated AQPs (dark blue traces). The upper traces are from a model of increased
ECS [K+] and the lower traces from a model of decreased ECS [K+]. (C) Images taken from the
models reflecting the values of [K+] in colors. The color gradient is from dark blue (low [K+]) to red
(high [K+]). The first series of images are from a model of “without AQPs”, the second series are from
a model of “with AQPs” and the last series, from a model of “with regulated AQPs”.
Paper IV)
This paper investigates a biophysical role for AQP4 in K+ kinetics in CSD. The hypothesis is
that AQPs in the brain extracellular space cause dynamic changing of ECS, and subsequently
regulate the ECS [K+]. Supporting the hypothesis means that AQP4 can play a role in K+
clearance without a need for a physical association with a co-localized K+ channel (Kir4.1) in
the astrocyte membrane. Thus the geometry of ECS and its dynamic changes are of critical
importance to a proper biophysical coupling between AQPs and Kir channels.
Modeling Method
Geometrical models are developed that describe a neuron releasing K+ into the dynamically
changing extracellular space in synapse The models take into account the astrocyte K+ uptake
and calculate how much the rate of K+ uptake would change with expression of AQPs. The
geometry implemented in this model is shown in Figure 12B that describes a system
consisting of two cells (one as a neuron and one an astrocyte) sharing an extracellular space
23
Modeling biophysical mechanisms underlying cellular homeostasis
with width ‘w’. Important concepts and parameters in the model are the initial size of the
extracellular space, dynamic changes of the extracellular space following water transport,
water permeability and K+ conductance of the cell membrane. The role and function of water
and K+ channels are defined by the homeostatic rules and equations.
In the models of K+ accumulation, the ECS K+ level is controlled by two mechanisms; the
release from the neuron and the free diffusion within the ECS. The mechanism for K+ release
from the neuron into the ECS was defined as either diffusion down the chemical gradient or
via voltage gated potassium (Kv) channels. In addition, a small electrodiffusion of K+ from
the cell into the extracellular space was assumed to occur following anion transport by the
neuron. The initial level of [K+] in the neuron was set to 130 mM then decreased down to 83
mM by the release of K+. A boundary condition maintaining the neuronal K+ level was set at
the upper edge of the neuron (the opposite side of the synapse) in order to balance the [K+] in
the neuron while releasing K+ into the ECS. Two modes, representing WT and AQP4 KO,
were defined by their difference in the initial ECS width (70nm for WT and 90nm for KO).
In the models of K+ clearance, the ECS K+ level ([K+]) after release from the neuron is
controlled by three mechanisms: passive diffusion within the ECS, electro-diffusion via Kir
channels and counteraction by water (dynamic changing in ECS). The K+ diffusion
coefficient in ECS was set to 100 µm2/s and K+ conductance of the Kir channels to 27pS
(Kir4.1) [106]. Initial values of intra- and extracellular K+ were set to 150mM and 35mM,
respectively, and the base level of ECS K+ was set to 2.5 mM. Boundary conditions were set
so that the vertical edges of the geometry were assumed to be in contact with the rest of the
extracellular space, including a fixed level of [K+]. This value was varied depending on the
initial size of the extracellular space.
ECS volume is controlled by water transport across the cell membrane and free diffusion
within the ECS (recovery of the volume change). The models of K+ clearance were performed
under two different modes, one with AQPs (WT) and one without AQPs (KO). Following the
recent study on AQP4 deficient mice and the increased ECS [107], the initial size of the ECS
width was set to 28% increased in the mode without AQPs, 90 nm for KO versus 70 nm for
WT. In addition to this experimentally known role of AQPs, two more possible aspects of
AQPs were tested in the models:
1. AQPs as dynamic water transporters. The water permeability of the WT membrane
was set to 7-fold increased over the AQP4 KO membrane.
2. AQPs as components that can be associated with K+ channels in the cell membrane
and increase the conductance of the K+ channels.
Modeling Results
The models show that the ECS width plays an important role in accumulation of K+ in the
synapse. Decreased ECS results in a faster K+ accumulation since smaller region is to be
occupied by the excess K+. The models show that K+ uptake by the cell is the main
mechanism for K+ clearance. Diffusion within the extracellular space is the second important
mechanism. ECS width was shown to be important for the mechanism of K+ clearance.
24
Padideh Kamali-Zare
Decreased ECS causes a higher electrochemical gradient for K+ to pass across the cell
membrane and therefore results in a faster K+ uptake by the cell. This result could partly
explain the experimental data showing that K+ clearance takes place faster in WT and
furthermore could suggest a biophysical role for AQP4 on K+ clearance via decreased ECS
and higher K+ uptake by the cell. This effect of decreased ECS on K+ clearance was shown to
be enhanced by the dynamic contraction of ECS following water transport by the cell. This
indicates that AQP4, by allowing a fast water transport across the cell membrane and a fast
ECS counteraction, may facilitate K+ transport by the cell (Figure 12).
Moreover, the models show that the biophysical roles of AQP4 highly depend on the ECS
and disappear in large ECSs where transport of water cannot efficiently decrease the ECS
width. The models predict that in order to get the similar result (faster K+ clearance in WT) in
large ECSs (bigger than 500 nm width), the AQPs should have a positive regulatory effect on
the K+ permeability of the membrane.
A
Phase 1
B
Phase 3
Phase 2
Cell1
Cell1
Neuron
ECS
Neuron
Neuron
Excess K+
Water
K+
w: width of the ECS
width
AQPs and Kir channels
Cell2
Membrane lined with AQPs and Kir channels
Water
Astrocyte
Astrocyte
Cell2
D 40
110
AQP4 KO (Modeling)
AQP4 KO (Experimental)
WT (Modeling)
WT (Experimental)
100
without AQP4-mediated
ECS dynamics
with AQP4-mediated
ECS dynamics
90
80
70
60
50
ECS [K+], mM
Rate of K+ uptake (%)
C
Astrocyte
ECS
ECS: Extracellular Space
40
30
30
20
10
20
10
0
10 303950 70 90 110 130
160
190
230
0
500
ECS width (nm)
50
100
150
200
Time, s
250
300
Figure 12. A) Different phases for K+ and water transport in the brain. B) Geometry used in the
models of K+ kinetics in CSD. C) Rate of K+ uptake depending on the static/initial ECS width (green
bars) and the dynamics of ECS initiating from the water transport via AQP4. D) ECS K+ kinetics in
WT (70 nm ECS width) versus AQP4 KO (90 nm ECS width).
Paper V)
This paper studies molecular transport of cationic and anionic dye molecules from the silica
particles.
Method
Dye loading: Two types of fluorescent dyes were used in this study. The positively charged
Rhodamin 6G and the negatively charged Oregon green 488. Experiments were performed in
PBS (phosphate buffered saline solution). The mesoporous particles were immobilized in
25
Modeling biophysical mechanisms underlying cellular homeostasis
gelatin (2x10-6 mM) on glass bottom microwell dishes. The immobilized particles were then
loaded with dyes by incubation in PBS with respective dye concentration of 0.025 mM.
Imaging: A region of interest with a population of at least 15 isolated spheres was selected.
An inverted confocal microscope with a 40 x /1.3 NA oil objective was used for imaging. A
488 nm Argon laser was used for Oregon green excitation, and a 543 nm HeNe laser was
used for Rhodamin 6G. The z-sectioning (with 1µm thickness) was performed so that each
particle was optically cut into 5-10 sections. Images were taken once in an hour and
continued for 7 hours.
Results
Below is a schematic figure showing the difference between Oregon Green release as an
anionic dye (with no electrostatic attraction to the silica surface) and Rhodamin 6G as a
cationic dye. The behavior of Oregon Green is as expected; decrease in fluorescence intensity
in time resulted from the release of dye molecules from the particle (region 1) to the bulk
solution (region 2). This behavior is different for Rhodamin 6G-containing spheres.
Following normal diffusion from the core of the particle to the surface, the core looses the
intensity in time. However due to the electrostatic attraction around the surface, the edge of
the particle remains highly fluorescent. This phenomenon creates four different regions of
intensities within the particle.
After a few hours (when the saturation on the surface occurs), the release profile from the
surface to the bulk solution will follow a slow exponential decay.
t (zero)
t (end)
Oregon Green
2
2
1
t (end)
4
Rhodamin
3
2
1
1
Figure 13. Schematic comparison of two different type of release in between Oregon Green- and
Rhodamin 6G-containing spheres. As no electrostatic attraction exists for Oregon Green dye
molecules, the release is normal diffusion driven and results in an even distribution of the dye
molecules at t(end). For Rhodamin, the electrostatic attraction with the surface of the particle, results
in an uneven distribution of dye molecules at t(end).
For Oregon Green the spatial distribution of the fluorescence intensity as a function of time
fitted with a Gaussian function, C ( x, t ) =
b
π
exp(− x2 ) , described in Figure 14. This
suggested that the mechanism underlying release of Oregon Green from silica spheres is a
26
Padideh Kamali-Zare
normal diffusion from the interior of the particle to the bulk solution. Detailed analysis of the
transport mechanisms is explained in paper V and previously in this section.
0.8
70 min
100 min
0.6
150 min
0.4
220 min
290 min
0.2
0
-8
b)
-6
-4
-2
0
2
4
6
F
30 m in
7 0 m in
Arbitary intensity
Fluorescence
Intensity
0.8
0.6
0.4
0.2
0
0
8
Distance,µm
µm
Distance,
E
1
Fluorescence intensity
a)
D
C
3 min
1
Fluorescence intensity
B
FluorescenceIntensity
Intensity
Fluorescence
A
12 0 m in
21 0 m in
30 0 m in
40 0 m in
1
1
2
Period 1
3
4
5
Period 2
0.8
0.6
Surface
0.4
Average
0.2
Core
Near surface
b)
0
5
10
Distance,
µm )
S ize (m icron
15
6
0
0
Core
1
2
3
τ, h
4
5
6
7
Figure 14. Release and molecular transport of anionic (Oregon Green) and cationic (Rhodamin 6G)
from silica spheres. A) Confocal imaging of a single particle while releasing Oregon green. B)
Spatial distribution of fluorescence intensity (Oregon Green) and C) Temporal distribution
following an exponential decay with a rate constant of 0.33 h-1. D) Confocal imaging of a sphere
when releasing Rhodamin 6G. E) Spatial distribution of fluorescence intensity (Rhodamin 6G) and
F) Temporal distribution of Rhodamin having two periods; Period two follows a very slow
exponential decay with a rate constant of 0.10 h-1.
Paper VI)
This paper studies the role of molecular interaction between the pore of silica particles and
big dye molecules (approximately the same size as the pore) on molecular release.
Method
A similar study to paper V, was performed to compare the release profile of the loaded
material depending on their molecular weight and the molecular interaction with the pore of
the spheres (paper VI). A different type of silica particle was used for this study, made
through emulsion and solvent evaporation (ESE) method. ESE particles are mesostructured
silica spheres with large pores that have been produced with the recently developed method
[108]. Dex3k and Dex10k were used as the model dye molecules for this study since they
have molecular weights in a range that gives hydrodynamic radii similar to the diameter of
the cylindrical pores (6.5 nm).
The spatio-temporal dependencies for molecular transport of macromolecules inside particles
of mesoporous silica were explored using a different analysis of the observed release from the
previous analysis descried in paper V.
Results
Comparing the rate of Dextran3K transport to Dextran10k shows that the smaller the loaded
material (Dex 3k), the faster the release will be. This suggests a dynamic interaction with the
27
Modeling biophysical mechanisms underlying cellular homeostasis
pore of the sphere. Also, concentration profiles inside the ESE spheres showed no variation
with the distance to the external surface. This strongly indicated that the release of dextran
from the ESE spheres cannot be well described by a Fickian-type model (characterized by
Gaussian-shaped intraparticle concentration profiles as shown in paper V). Instead, the flat
profile suggests that the release is controlled by the pores close to the external surface of the
spheres. There are studies showing that many pore channels bend back into the particle, and
only a few channels having pore openings facing the external solution. According to this
structural feature the flat concentration profile inside the particles can fit well with Temkin
model instead of Fickian model.
A
B
50 μm
Fluorescence Intensity
Fluorescence
Intensity
C
1
Dex 10k
I=Exp(-022*t-0.09)
0.9
0.8
0.7
0.6
Dex 3k
I=Exp(-071*t-0.16)
0.5
0.4
0.3
0.2
0.1
0
2
4
6
8
10
12
14
16
18
Time, h
Time,
h
Figure 15. A) Electron microscopy images of the calcined spherical ESE particles. B) Confocal
image of the spheres loaded with Dex3k. C) Exponential comparing between the release profile of
Dex 3k and Dex10k.
3. A key result
Biophysical functional couplings between water and K+ channels in DLS
A possible functional coupling between water and potassium channels localized in a DLS is
studied by spatial models starting with different initial conditions. Water and potassium
permeability of the membrane is constant in all models and only the localization of the
change in the extracellular K+ varies. Direction and rapidity of fluxes via K+ and water
channels in response to a certain change in the extracellular K+ level could be studied in two
separate models: one describing the condition where the edge of the ECS is clamped to a
changed K+ level while the rest of the ECS is at a normal level (Figure 16A and 16B), and the
other where DLS is clamped to a changed K+ level and the rest of ECS is at a normal level
(Figure 16C and 16D). Two modes for each model has been tested: one for the case of
increased K+ (A and C) and one for decreased K+ (B and D). The mechanism for K+ transport
28
Padideh Kamali-Zare
via Kir channels is electrodiffusion, and that for water transport is osmotic driven flux. K+
and water channels are located only in DLS. Simulations of K+ and water transport across the
cell membrane show that the response of the cell to the local K+ change in the extracellular
space is step-wise; either K+ or water channels initially respond to the change, resulting in
regulation of the function of the other channel type. In the case of changed K+ levels at the
edge of the ECS (Figure 16A), water transport is faster and will be followed by K+ transport.
This is because osmotic-driven water flux caused by the exchange between ECS and DLS K+
is more efficient than the electrodiffusion of K+ that requires a severe change in the local
Nernst potential for K+ in DLS. When the extracellular K+ changes in DLS, the response
order will be the opposite. In such a case, K+ channels respond to the change first and are
followed by water channels. This is because if [K+] is highly changed around the Kir
channels, the electrodiffusion of K+ is faster than the osmotic driven water flux. These data
suggest that AQPs and Kir channels localized in DLS follow each other, and the main
responder will depend on where the change in K+ occurs. The direction of K+ and water flux
depends on whether the change is due to increased or decreased extracellular K+. K+ and
water will be transported in the direction that leads to stabilized local K+ and water levels due
to the global cell homeostasis.
A
water
K+
B
water
Intuitive: Water
Non-intuitive: K+
(Regulated)
K+
Increased ECS
K+ level
Decreased ECS
K+ level
C
K+
water
D
Intuitive: K+
Non-intuitive: Water
(Regulated)
Increased DLS
K+ level
K+
water
Decreased DLS
K+ level
Figure 16. Functional couplings between K+ and water channels localized in DLS.
29
Modeling biophysical mechanisms underlying cellular homeostasis
30
Padideh Kamali-Zare
Modeling
Biology
Physics
Discussion and Conclusion
The work in this thesis is performed at the interface between physics and biology. The aim
has been to create models where the number of parameters and components are limited.
Accordingly, each biological component has to interact with a physical perturbation, and each
physical factor has to sufficiently influence a biological component in order to be taken into
account in the models. This type of modeling, in which the number of key parameters are
low, although may not show the full picture of a biological system, it can formulate the
direction of the system towards its homeostasis and physiological function.
From morphology to homeostasis
The body is a very crowded place where cells and extracellular matrix in tissues are tightly
attached to each other, and diffusion of material is highly restricted. However culture cells as
our model systems to study biology are surrounded by a sea of extracellular fluid where a
homogenous concentration of ions is maintained. One example of cells in tissue with special
geometry is a kidney principal cell. Figure 17 shows where in the kidney these cells are
located and how the diffusion barriers are distributed.
The results presented in this thesis suggest that this network of extracellular domains
surrounding the cells, together with the restrictions defined by membrane invaginations, can
create a local trans-cellular domain that has its own regulatory functions. This domain can
play an important role in the transport mechanisms for water and ions. In kidney principal
cells of the collecting duct, one such regulatory domain is created in DLS involving
aquaporins, inwardly rectifying K+ (Kir) channels and Na+,K+-ATPase. The deep and narrow
invaginations in the basal membranes of these cells define semi-closed extracellular spaces
where local gradients can be formed. These gradients can regulate the function of colocalized transporters and therefore DLS can be considered as a kind of microdomain,
important for cell physiology.
31
Modeling biophysical mechanisms underlying cellular homeostasis
Figure 17. Starting from top left: tile-scanned image of a kidney slice with 20 µm thickness. The
image is taken by 5x Objective and consists of 48 single Confocal images that are mapped together.
The red image on the black background is from the fluorescent channel. The next is a combined
image of both fluorescent and transmission light channels. It shows where the kidney glomerulus is
located. Below left is a cartoon picture showing the location of kidney collecting duct and the
principal cells in the kidney. Next picture exemplifies the geometry of these cells in the tissue. Then
the geometry used in the model of DLS is shown. Finally, a cartoon picture of co-localized proteins in
the basal membrane forming a microdomain is shown.
From homeostasis to morphology
One of the extreme cases of physical events in the body is in the blood stream that contains a
high level of ions and materials. An excess of materials get released from the blood to the
extracellular spaces of epithelial cells. These cells, which face imbalanced conditions in their
extracellular space at steady state, may urgently need a homeostatic effort to maintain their
internal content. As is also explained in previous sections, membrane invaginations formed in
kidney principal cells create a DLS where normal extracellular concentrations (clear from the
excess of material) can be maintained. This is similar to narrow intercellular geometries
formed in between cells e.g. neurons and astrocytes in the brain. Thus, the results presented
in this thesis suggest that formation of complex geometries helps cells to maintain their
homeostasis.
Modeling perspectives
Biophysical modeling derived from the right hypothesis can be a powerful method to
understand and explain biological systems. This type of modeling can collect the peaks of our
knowledge about a system that are provided by scattered observations, and link them together
in order to shift our level of understanding (Figure 18). Modeling can also test different
hypotheses to figure out what the underlying mechanisms for certain phenomena can be, how
32
Padideh Kamali-Zare
they perform and why they may be advantageous compared to other mechanisms and
ultimately selected by the nature.
Our understanding
Modeling can take these peaks, link them together and
shift up the level of our understanding about the system
Peaks are
made by
observations
System
Figure 18. The level of our understanding about a system.
Example 1. DLS modeling
Models presented for DLS in the kidney and brain followed the approach presented above.
Their goal was not to simulate reality but to explain reality in a simple way. Experimentally it
is difficult to measure K+ concentration in theses narrow geometries in vivo, and also, it is
most likely impossible to find a system, e.g. a knock-out lacking the invaginations that form
the DLS. Also we cannot compare a cell in vivo to a cultured cell, as the situation differs in
cell culture. Therefore, geometrical modeling is a good approach to address these problems.
Another advantage was that the model could, step by step, explore the role of individual
parameters. It has previously been proposed in many experimental studies that aquaporins
and K+ transporters can form a functional microdomain in order to facilitate K+ transport
mechanisms. Resulting from this question, the key elements of the model were then selected
to be aquaporins and Kir channels located in DLS. The final conclusion of the models was
that DLS can be described as a geometrical domain for local coupling between aquaporins
and Kir channels.
Example 2: ECS modeling
Several lines of evidence suggest a functional coupling between aquaporin-4 (AQP4) and
Kir4.1 localized to the neural synapse. A delayed K+ clearance from the synapse has been
shown in the case of mis-localization of aquaporin-4 [73]. It has been shown in two separate
studies that this coupling is likely not due to a physical interaction between the two proteins
AQP4 and Kir4.1. neither in astrocytes nor in retina muller cells [109, 110]. The question was
then whether AQP4 may influence Kir4.1 in another way in order to increase the K+
33
Modeling biophysical mechanisms underlying
u
cellular homeostasis
conductance of the channel.. Following
F
a recent study showing that the extracellular
ex
space is
increased in AQP4 KO mice [107], we hypothesized that this morpholog
logical change in the
+
ECS may play a role in K kin
inetics. Several geometrical models were perfo
rformed, taking ECS,
AQPs and Kir channels as the key elements. The conclusion of the mode
odels was that AQP4
+
can play two biophysical roles
les in K kinetics; one is due to an initially reduced
red
ECS and the
+
other due to a dynamic transp
sport of water following the K uptake by the cell and dynamic
morphological changes of the ECS.
Conclusion
The work in this thesis sugges
gests that narrow morphological features suchh as
a invaginations in
epithelial cell membranes orr synapses
sy
between neurons and astrocytes coul
ould be considered as
domains isolated both from the
th bulk extracellular space and intracellularr compartments.
c
The
small volume and limited diffusion
d
inside these domains facilitatess coupling between
different transport proteins without
w
the need for a direct molecular interaction.
in
Another
implication of the result prese
sented in this thesis is that the environment in the DLS-domains
could be regulated by only
ly a few components and parameters and yet
y have important
influences on cellular homeost
ostasis.
From water to water, life makes
m
the difference!
Figure 19. Sea water with air bubbles
bub
on the top
Living cells
34
DLS
Water
ter.
Padideh Kamali-Zare
Acknowledgements
And now the end of my PhD story, having a great chance to thank many people who have
helped me to accomplish this thesis, and I’m sure I cannot make it short!
First of all I would like to thank my supervisor Prof. Hjalmar Brismar, for being so unique
and special in his thinking, understanding and approaching biological physics, for sharing
with me those intellectual questions in biology where physics could strongly contribute, and
for patiently guiding me through them. Thank you Hjalmar for all the non-classical
discussions we had, for your novel ideas, supports, trust, and for making Cell Physics such a
nice and fun place to be; it was in fact a DLS for me these years that enhanced my efficiency!
(For those who are not in the field: DLS is where diffusion into a structure is normal but
diffusion out is limited, exactly how Cell Physics corridor was for me! ☺) I’m very grateful
for everything I learned during these years ranging from how to think deeply about problems,
to how to fix broken things with hands, and specially that a.m, p.m’s are just formalities and
for a scientist nights should be equally treated as days! All together it was a pleasure being a
PhD student at Cell Physics, thank you Hjalmar for offering me that!
I would also like to thank my co-supervisor Assistant Prof. Björn Önfelt, for all his help and
input to my thesis, and for playing such an important and valuable role in forming my
manuscripts. Thank you Björn, for spending so much time on my DLS projects, for all the
fruitful discussions we had, for teaching me how to formulate things in a simple and well
controlled manner and for never giving up your role specially when seeing a sentence with
152 words! ☺ (how could I write that?!). This thesis would have been never possible without
you!
Then I would like to thank Assistant Prof. Aman Russom for introducing me to the field of
microfluidics, for the idea of having a paper together, and for always spreading enthusiastic
ideas and energy around at Cell Physics. It was really fun working with you Aman, I believe
you as a senior always in my life ☺ and thank you for being such a reliable and supportive
friend after all.
I would like to acknowledge our collaborators:
Prof. Anita Aperia at Karolinska Institute for KTH/ KI collaboration that provided us with the
medically relevant questions, for believing in modeling and being so encouraging about novel
35
Modeling biophysical mechanisms underlying cellular homeostasis
skills of physicists. Prof. Lennart Bergström at Stockholm University, for the fruitful
collaboration with Physical Chemistry group at SU, for designing two interesting projects
regarding mesoporous spheres, and for always being supportive. And Prof. Geoffrey Manley
at University of California San Francisco for the collaboration regarding ECS project, for
nicely incorporating my geometrical models into the project and for providing discussions
with the UCSF group.
Special thanks to my co-authors:
Dr. Jacob M. Kowalewski: for all his help, support and encourages during the years of
working together at Cell Physics, for sharing with me his vast knowledge in mathematics and
computer science, for being a fantastic teacher, and for always smiling at my face as equally
as if he would face an apple sign! ☺. Thank you Jacob for whatever we shared during these
years including the heavy responsibility of Cell Physics course as together we made it so fun
to do. I wish academic life treats you very kind and you always be successful and happy in
your scientific career.
Dr. Marina Zelenina: for being my teacher in biology at KTH during my master program and
for encouraging physicists to approach biology some time in their lives! Thank you Marina
for your patience, for the papers and posters we made together and for AQPs meetings and
discussions.
Dr. Jovice Boonsing Ng: for our nice papers, for being both an excellent collaborator and a
true friend; thank you Jovice for always trusting my view, for being so hard working and goal
oriented in the projects and for thinking about all the coming problems before they really
come! As I said in your PhD party, I’ll never forget people with whom I shared challenging
and stressful moments of my life which you were in fact one of those.
Dr. Xiaoming Yao and Dr. Zsolt Zador at UCSF: for their patience in understanding my
models far from distances and via phone conferences, and for guiding me through my models
in order to make them as realistic as possible. It was really fun talking and discussing things
with you. Thank you for all your trust in modeling.
Sahar Ardabili: for being my co-author, my officemate and a close friend. Thank you Sahar
for all the hard work we did together specially during the last few months, (I guess we have
survived till now as we promised each other to do! ☺), for all late night works, for everything
we shared and for all your trust, understanding and nice talks. I wish PhD life treats you very
kind and gives you a lot of feedback and motivation while you are getting further into it as
you really deserve.
Special thanks to: Dr. Mahmood Amiry-Moghaddam at Oslo University for all the nice
discussions we had, for believing in modeling and encouraging me all this time to stick with
my thinking, and for the nice time in the Persian community reception in 2009-SfN-meeting
in Chicago. Prof. Rick Rogers and Dr. Rosalinda Sepulvedas from Harvard University, for
their support and encourages. Thank you Rick for being always open and positive for outside
the box discussions, for believing in my modeling skills and thinking. Thank you Rosy, for
36
Padideh Kamali-Zare
everything we send each other far from distances including energy, support and inside the
box gifts! ☺
Many thanks to my former and current colleagues at Cell Physics for making such a friendly
environment at work: Susanna, Andreas, Gordon, Erland, Maria, Victor, Bruno, Karolin,
Hattie, Thomas F, Ali, Jonas H, Robert, Linda, Thomas L, Mohammad, Heike, Storbjörn,
Athanasia, Jonas L, Ronak, Sara, Mariam, Petra and Reza. Special thanks to: Victor who was
my first officemate at Cell Physics for teaching me the rules of Cell Physics! Thomas
Liebman (what can I say Thomas about you!) thank you for calling me a source of invited
disruptions in the office, for all the fun things we did together, for accepting my reasoning to
sometimes sit under my desk and read my favorite philosophical books ☺, and finally for
proofreading this thesis. Hattie for being such a nice, calm, kind and gentle officemate; I
really wish I could be like you Hattie! ☺ Gordon for all the hopes you were giving me in the
moments of disappointments and for the beautiful Orchide in the office. Ronak, Mariam and
Sara for all your help with the glomeruli analysis. Erland, for teaching me cell culturing and
Bruno for all your nice support and advice.
And thanks to people at Karolinska: Yanhong, Nina, Raija, Juan, Alexander B, Eli, Lena,
Nermin, Alexander R, Sergey, Zuzana, Youthong, XiaoLi, Ann-Christine, LillBitt, Eivor,
Ulla, Markus, Susanne, Georgiy and Rachel. Special thanks to Nina and Zuzana for the nice
time in Chicago.
I would also like to thank Torbjörn Nordling from electrical engineering department at KTH,
for being a model modeler, for our discussions in ICSB 2008 and the nice continuation in
2009!
My sincere gratitude to Assoc. Prof. Göran Manneberg, for believing in my teaching skills
and encouraging me to teach. My never-ending appreciation to Prof. Kjell Carlsson who was
my program coordinator during my master studies at KTH, and after that a fantastic advisor.
It is because of you Kjell I managed to do my higher education in Sweden. I would also like
to thank Prof. Jerker Widengren for being involved in the biological physics master program,
for FCS workshops and for being our nice corridor-mate for a long time. Thanks also to the
rest of biomolecular physics group (former and current members): Tor, Gustav, Andry,
Heike, Hans, Per, Stefan, Linda, Shadi, Sofia, Johan, Evagelos, Sara and Lei. Thanks to Prof.
Fredrik Laurel, our current corridor-mate for being always energetic and fun to talk, and to
his group as well as SU group in the same corridor, and to Prof. Hans Hertz, Lektor Peter
Unsbo, Els-Mari Kristiansson, Agneta Christianson, Agneta Falk and Björn Pettersson, for all
their administrative help at AlbaNova.
I would like to acknowledge my former advisors in physics who have built up my basic
training: Prof. Hashem Rafii-Tabar, Assoc. Prof. Kerasoos Ghafoori-Tabrizi, Prof.
Hamidreza Sepanji, Prof. Hadi Salehi Kermani and Prof. Nasser Mirfakhrai, at Shahid
Beheshti University (SBU) in Tehran and Institute for studies in theoretical physics and
mathematics (IPM), for teaching me the deep and fundamental concepts of physics during my
undergraduate studies and for being involved in my unconventional thinking. Thank you for
37
Modeling biophysical mechanisms underlying cellular homeostasis
never disappointing me whenever I ask you to read tens of my pages full of unusual thoughts
and models and to give me feedback!! Now I understand how challenging it must be to treat
such an unusual student who wanted to change the world so quickly!
I have also a great pleasure to thank my father’s colleagues at Iranian audit organization and
Iranian association of certified public accountants, specially Mr. Teymoori, Mr. Ariya, Mr.
Arbabsoleymani and Mrs. Afshar, for all their support, advice and encourages. Thank you
Mr. Teymoori for all your nice help and discussions for my bachelor thesis.
My special thanks to my art-mentor Ms. Farah Kiainejad for teaching me the secrets of not
only Persian traditional music but also life in general! For all your love and support since I
was 12, I feel proud when you call me your daughter; and thanks for teaching me not to scare
from thunderstorms of either the sky or life!
I would like to thank Dr. Mohammad Kamali for being a nice relative, a real artist and for
always encouraging me to pursue an academic career. And thanks for your violin songs,
always as delicate as your soul!
I have a great chance to deeply thank Forough and Ghazal for our wonderful classes, and for
everything you did for me to feel home in your place. With you, I never missed my culture
and traditions. Thanks for all your support, nice advice and never-ending exciting discussions
on music, poetry and philosophy!
Fariya! Thank you for being my long time friend in Sweden, for everything we shared these
years starting from a master program in biological physics to all the moments of laughs and
cries! Thanks for always being a wonderful company, for all the funny things we did together
and for all your love, support and understanding! You have now become a part of my family.
Thanks also to Andreas for being so caring, for the beautiful gifts, and the nice time in
Uppsala.
Mohammad! Thank you for being a super nice neighbor both at work and at home, a true and
reliable friend with whom I shared many things during the last two years, for never letting me
give up in the most challenging moments of my life this time and never leaving me alone in
the hard times! Thank you Mohammad for being who you are!
Special thanks to Mrs. Firoozeh Sadjadi for all the help and support during all years of my
education. Many thanks to: Mrs. Jaleh Salajegheh and her family for all their support, help
and never-ending music interest. Mr. Hasan Moghaddam, for being my Tar teacher in
Stockholm. My music students with whom I had always a lot of fun and it was the only time I
could forget modeling for a while. Farah, Taghi, Kaveh, Tina and Pardis, for giving me and
Afshin a warm Persian home in Sollentuna when we needed it the most. Athanasia for being
my nice roommate for a year, for all the fun moments we had together during the time you
were doing your master thesis at Cell Physics, for all the nice talks we had when we both
needed it the most and for your fantastic Greek foods and songs, and Ida for whatever we
shared, for being a nice and understanding company specially when going to and getting back
from my Tar-classes and for all the beautiful poems you have sighted for me.
38
Padideh Kamali-Zare
Thanks also to: Cinderella team (you know who you are!): Atoosa, Ahmad, Amin, Meysam
and Mohammad, for such a memorable trip! Special thanks to Ahmad for the nice support
during the time of writing my thesis, and the rest of my friends who are now spread out all
over the world, Parisa, Said, Behnaz, Mostafa, Davood, Babak, Mahdieh, Mojgan, Vina,
Elham, Mojdeh, Shima, Nima, Neda, Mohammad, Sara, Shadi, Samira, Bita, Negin,
Soudabeh and Alireza, thank you all for encouraging me all this time and filling my time with
joy and fun.
I would also like to express my sincere gratitude to Ashouri-family for believing in me and
my goals, and for always giving me respect and support.
Thanks to my dear uncles, their families, and the rest of my relatives back in Iran. Special
thanks to Mrs Ezzat Nayyeri and Dr. Mehdi Soraya for all their support as my second
grandparents.
Thanks to my mother, Farzaneh Vaziri for being my first teacher in life. Thank you for
dedicating your life to me and my sister, for always encouraging me to be on the right track
and to be fair and rational specially in my important decisions. Thanks to be the first to
introduce me the meaning of being a mother and how to dedicate my life to whatever I love
as my child, no matter whether its biological or not.
Thanks to my father, Ali Kamali-Zare, for his everyday calls, for staying awake late nights to
talk to me and for reminding me about myself! Thank you for always getting happy when I
am happy, and sad when I am sad, and for never leaving me alone in ups and downs of my
life. I never felt we are far from each other since the power of your talks always filled the
gaps and distances between us. Thank you for teaching me how to think about the reality of
nature without following any fanaticism, for all our philosophical discussions and for being
such a wonderful mentor for me.
Thanks to my only sister Golzar, for being such an incredible sister. Thank you for sharing
with me whatever you experienced in your life just because I was younger and I needed to
learn new things! Thanks for always taking care of me both at school and at home, for
coming to Sweden during my master thesis when you knew how much I needed you, and for
such a big heart that you have. Whatever I achieve in my life in fact does involve a lot of
your contribution!
Afshin! Thank you for being the hero of a decade of my life! I cannot find a more competent
word to express my gratitude to you. Thank you for whatever we did for each other many
years with love and patience! This thesis owes you a lot just as your Monero does! Thank you
for being the first one entering my dream world and respecting its structure so that your wills
and desires were always the next priority for you. Thank you for always pushing me to follow
my goals although it was sometimes the most difficult thing for me to do. We in fact grew up
together, I always wish life treats you super kind and brings you whatever you desire with
love and happiness!
39
Modeling biophysical mechanisms underlying cellular homeostasis
My special thanks to the memory of my grandmother, Parvin Soraya, who grew me up, for
being my first friend in life, thank you for explaining me the meaning of love when I asked
you for the first time at the age of 5; and your definition was “not to dye” but instead “live”
for something! And yes! You proved it so well by living for me and fighting against your
heart disease until you made sure I’ll survive on my own! Thank you for teaching me the artoriented aspects of life. You’ll be always alive in my heart and I will always love you!
Many thanks also to the happy red roses in Cell Physics who never dye and always smile, to
the small green plant which has survived now for four winters, and to my flower shoes which
have kept me all this time happily on my track!
And finally thanks to nature who creates us all these exciting questions in science and make
us so curious to find their answers!
Padideh Kamali-Zare
Stockholm, 2010
40
Padideh Kamali-Zare
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45
Modeling biophysical mechanisms underlying cellular homeostasis
Appendix A
Full description of the DLS model in the kidney
Structures
Name
Type
Inside
Outside
Cell
Feature
N/A
Cellmembrane
ECS
Feature
N/A
N/A
Cellmembrane
Membrane
Cell
ECS
Geometry of the model
46
Padideh Kamali-Zare
Kinetics Parameters
Name
Expression
Role
Unit
J
(Cl_ECS * H2O_initial / H2O_ECS)
reaction rate
uM.s-1
150000.0
user defined
uM.s-1
H2O_initi al
Kinetics Parameters
Name
Expression
Role
Unit
I
- (C * delVk * Gk *
Kchannel_Cellmembrane)
inward current density
pA.um-2
J
(I / (1.0 * _F_nmol_))
reaction rate
uM.um.s-1
C
1.0
user defined
1000000.0 item-1
delVk
(Voltage_Cellmembrane - Vk)
user defined
mV
Gk
22.0
user defined
1.0E-14 S
Vk
(25.6 * log((K_ECS / K_Cell)))
user defined
mV
47
Modeling biophysical mechanisms underlying cellular homeostasis
Kinetics Parameters
Name
Expression
Role
Unit
J
(Vm - Vm_initial)
reaction rate
uM.s-1
Vm
(25.6 * log((Keffective_ECS / K_Cell)))
user defined
uM.s-1
Vm_initial
-92.87
user defined
uM.s-1
(K_ECS * H2O_initial / H2O_ECS)
user defined
uM
150000.0
user defined
uM
Keffective _ECS
H2O_initi al
Kinetics Parameters
Name
Expression
J
Role
Unit
reaction rate
uM.um.s-1
(c * Pf * AQP4_Cellmembrane *
(delK_ECS delK_Cell + delCl_ECS delCl_Cell - (Beta * delH2O_ECS)))
I
0.0
inward current density
pA.um-2
c
1.0
user defined
1
Pf
0.0070
user defined
1.0E-18 m3.s-1
(K_ECSinitial - K_ECS)
user defined
uM
(K_Cellinitial - K_Cell)
user defined
uM
(Cl_ECSinitial - Cl_ECS)
user defined
tbd
(Cl_Cellinitial - Cl_Cell)
user defined
tbd
0.014
user defined
tbd
(H2O_ECSinitial - H2O_ECS)
user defined
tbd
4000.0
user defined
uM
delK_EC S
delK_Cell
delCl_EC S
delCl_Cel l
Beta
delH2O_ ECS
K_ECSini tial
48
Padideh Kamali-Zare
Initial Conditions
Species
Structure
Initial Conc.
Diffusion Const.
Cl
Cell
4000.0 uM
100.0 um2.s-1
F
K
Cell
150000.0 uM
100.0 um2.s-1
F
H2O
Cell
1600000.0 uM
10.0 um2.s-1
F
H2O
ECS
150000.0 uM
10.0 um2.s-1
F
Cl
ECS
106000.0 uM
100.0 um2.s-1
F
K
ECS
100.0 um2.s-1
F
AQP4
Cellmembrane
((1400.0 * (x < 0.77) * (y < 8.35) *
(x > 0.38) * (y > 7.65)) + (140.0 *
(x < 0.67) * (y < 7.65) * (x > 0.21)
* (y > 4.0)) + (1400.0 * (x < 1.58)
* (y < 8.35) * (x > 1.07) * (y >
7.65)) + (140.0 * (x < 1.58) * (y <
7.65) * (x > 1.07) * (y > 4.0)) +
(1400.0 * (x < 2.02) * (y < 8.35) *
(x > 1.68) * (y > 7.65)) + (140.0 *
(x < 1.97) * (y < 7.65) * (x > 1.68)
* (y > 4.0)) + (1400.0 * (x < 2.94)
* (y < 8.35) * (x > 2.35) * (y >
7.65)) + (140.0 * (x < 2.97) * (y <
7.65) * (x > 1.97) * (y > 4.0)))
molecules.um-2
0.0 um2.s-1
F
CLC
Cellmembrane
(1.5 * (y < 3.25)) molecules.um-2
0.0 um2.s-1
F
Kchannel
Cellmembrane
0.0 um2.s-1
F
(4000.0 + (2000.0 * (y > 9.0)))
uM
(5.0 * (x < 2.95) * (x > 0.1))
molecules.um-2
Fixed
(T/F)
Electrical Mapping - Membrane Potential
Membrane
Calculate V (T/F)
V initial
Specific Capacitance
Cellmembrane
T
-92.87 mV
0.1 pF.um-2
49
Modeling biophysical mechanisms underlying cellular homeostasis
Constant Name
Expression
_F_nmol_
9.648E-5
_K_GHK_
1.0E-9
_N_pmol_
6.02E11
_R_
8314.0
_T_
300.0
Beta
0.014
C
1.0
c
1.0
C_Cellmembrane
0.1
Cl_Cell_diffusionRate
100.0
Cl_Cell_init
4000.0
Cl_Cellinitial
4000.0
Cl_ECS_diffusionRate
100.0
Cl_ECS_init
106000.0
Cl_ECSinitial
106000.0
GClC
0.1
Gk
22.0
10.0
H2O_Cell_diffusionRa te
H2O_Cell_init
1600000.0
10.0
H2O_ECS_diffusionR ate
H2O_ECS_init
150000.0
H2O_ECSinitial
150000.0
150000.0
H2O_initial_Cleffectiv e
150000.0
H2O_initial_Del_volta
ge_AQP4_
I_Cellmembrane
0.0
K_Cell_diffusionRate
100.0
K_Cell_init
150000.0
50
Padideh Kamali-Zare
Constant Name
Expression
_F_
96480.0
Constant Name
Expression
K_Cellinitial
150000.0
K_ECS_diffusionRate
100.0
K_ECSinitial
4000.0
K_millivolts_per_volt
1000.0
KMOLE
0.0016611295681063123
Pf
0.0070
-92.87
Vm_initial_Del_voltag e_AQP4_
-92.87
Vm_initial_Del_voltag e_withoutAQP4
-92.87
Voltage_Cellmembran e_init
Function Name
Expression
AQP4_Cellmembrane
AQP4_Cellmembrane _init
K_AQP4_Cellmembrane_total
((1400.0 * (x < 0.77) * (y < 8.35) * (x > 0.38) * (y > 7.65)) + (140.0 * (x < 0.67) * (y
< 7.65) * (x > 0.21) * (y > 4.0)) + (1400.0 * (x < 1.58) * (y < 8.35) * (x > 1.07) * (y >
7.65)) + (140.0 * (x < 1.58) * (y < 7.65) * (x > 1.07) * (y > 4.0)) + (1400.0 * (x <
2.02) * (y < 8.35) * (x > 1.68) * (y > 7.65)) + (140.0 * (x < 1.97) * (y < 7.65) * (x >
1.68) * (y > 4.0)) + (1400.0 * (x < 2.94) * (y < 8.35) * (x > 2.35) * (y > 7.65)) +
(140.0 * (x < 2.97) * (y < 7.65) * (x > 1.97) * (y > 4.0)))
CLC_Cellmembrane
K_CLC_Cellmembrane_total
(1.5 * (y < 3.25))
CLC_Cellmembrane_i nit
delCl_Cell
(Cl_Cellinitial - Cl_INSIDE)
delCl_ECS
(Cl_ECSinitial - Cl_OUTSIDE)
delH2O_ECS
(H2O_ECSinitial - H2O_OUTSIDE)
delK_Cell
(K_Cellinitial - K_INSIDE)
delK_ECS
(K_ECSinitial - K_OUTSIDE)
delVk
(Voltage_Cellmembrane - Vk)
51
Modeling biophysical mechanisms underlying cellular homeostasis
Function Name
Expression
F_Cellmembrane
( - I_Cl_flux - I_K_Flux)
I_Cl_flux
- (GClC * CLC_Cellmembrane * (Voltage_Cellmembrane - Vncl))
I_K_Flux
- (C * delVk * Gk * Kchannel_Cellmembrane)
J_Cl_flux
(I_Cl_flux / - _F_nmol_)
J_Cleffective
(Cl * H2O_initial_Cleffective / H2O)
(Vm_Del_voltage_AQP4_ - Vm_initial_Del_voltage_AQP4_)
J_Del_voltage_AQP4 _
(Vm_Del_voltage_withoutAQP4 - Vm_initial_Del_voltage_withoutAQP4)
J_Del_voltage_withou tAQP4
J_K_Flux
J_water_Flux
(I_K_Flux / _F_nmol_)
(c * Pf * AQP4_Cellmembrane * (delK_ECS - delK_Cell + delCl_ECS delCl_Cell - (Beta * delH2O_ECS)))
AQP4_Cellmembrane_init
K_AQP4_Cellmembra ne_total
CLC_Cellmembrane_init
K_CLC_Cellmembran e_total
K_ECS_init
(4000.0 + (2000.0 * (y > 9.0)))
Kchannel_Cellmembrane_init
K_Kchannel_Cellmem
brane_total
K_Kchannel_Cellmembrane_total
Kchannel_Cellmembr ane
(5.0 * (x < 2.95) * (x > 0.1))
Kchannel_Cellmembr ane_init
Keffective_ECS
(K * H2O_initial_Del_voltage_AQP4_ / H2O)
Vk
(25.6 * log((K_OUTSIDE / K_INSIDE)))
(25.6 * log((Keffective_ECS / K)))
Vm_Del_voltage_AQ P4_
(25.6 * log((K / K)))
Vm_Del_voltage_with outAQP4
Vncl
- (25.6 * log((Cl_OUTSIDE / Cl_INSIDE)))
52
Padideh Kamali-Zare
PdeEquation H2O
Rate
0.0
Diffusion
H2O_ECS_diffusionRate
Initial
H2O_ECS_init
PdeEquation Cl
Rate
0.0
Diffusion
Cl_ECS_diffusionRate
Initial
Cl_ECS_init
PdeEquation K
BoundaryYm
0.0
BoundaryYp
6000.0
Rate
0.0
Diffusion
K_ECS_diffusionRate
Initial
K_ECS_init
PdeEquation Cl
Rate
0.0
Diffusion
Cl_Cell_diffusionRate
Initial
Cl_Cell_init
PdeEquation K
Rate
0.0
Diffusion
K_Cell_diffusionRate
Initial
K_Cell_init
JumpCondition H2O
InFlux
J_water_Flux
OutFlux
- J_water_Flux
53
Modeling biophysical mechanisms underlying cellular homeostasis
Overriden Parameters
Name
Actual Value
Default Value
Pf
0.0
0.0070
PdeEquation H2O
Rate
0.0
Diffusion
H2O_Cell_diffusionRate
Initial
H2O_Cell_init
MembraneRegionEquation Voltage_Cellmembrane
UniformRate
MembraneRate
Initial
(1000.0 * (I_Cellmembrane - ( - I_Cl_flux - I_K_Flux)) /
C_Cellmembrane)
-92.87
JumpCondition Cl
InFlux
J_Cl_flux
OutFlux
- J_Cl_flux
2.1.6.1. No water permeability
54
Padideh Kamali-Zare
0.0
Time Bounds - Starting
Time Bounds - Ending
1.0
Time Step - Min
1.0E-8
Time Step - Default
1.0E-6
Time Step - Max
1.0
Error Tolerance - Absolute
1.0E-9
Error Tolerance - Relative
1.0E-9
Keep Every
1000
Keep At Most
1000
Use Symbolic Jacobian (T/F)
F
Solver Name
Semi-Implicit Finite Volume Compiled,
Regular Grid (Fixed Time Step)
Time Bounds - Starting
0.0
Time Bounds - Ending
10.0
Time Step - Min
1.0E-8
Time Step - Default
1.0E-6
Time Step - Max
1.0
Error Tolerance - Absolute
1.0E-9
Error Tolerance - Relative
1.0E-9
Keep Every
10000
Keep At Most
1000
Use Symbolic Jacobian (T/F)
F
Geometry Setting
Geometry Size (um)
(3.0, 10.0, 0.01)
Mesh Size (elements)
(90, 180, 1)
55
Modeling biophysical mechanisms underlying cellular homeostasis
Advanced Settings
Solver Name
Semi-Implicit Finite Volume Compiled,
Regular Grid (Fixed Time Step)
Time Bounds - Starting
0.0
Time Bounds - Ending
1.0
1.0E-8
Time Step - Min
Time Step - Default
1.0E-6
Time Step - Max
1.0
Error Tolerance - Absolute
1.0E-9
Error Tolerance - Relative
1.0E-9
Keep Every
1000
Keep At Most
1000
Use Symbolic Jacobian (T/F)
F
Geometry Setting
Geometry Size (um)
(3.0, 10.0, 0.01)
Mesh Size (elements)
(90, 180, 1)
Advanced Settings
Solver Name
Semi-Implicit Finite Volume Compiled,
Regular Grid (Fixed Time Step)
Time Bounds - Starting
0.0
Time Bounds - Ending
10.0
Time Step - Min
1.0E-8
Time Step - Default
1.0E-6
Time Step - Max
1.0
56
Padideh Kamali-Zare
1.0E-9
Error Tolerance - Absolute
Error Tolerance - Relative
1.0E-9
Keep Every
10000
Keep At Most
1000
Use Symbolic Jacobian (T/F)
F
Overriden Parameters
Name
Actual Value
Default Value
Pf
0.0
0.0070
Simulation Name: water permeability: 7x
Overriden Parameters
Name
Actual Value
Default Value
Pf
0.049
0.0070
Advanced Settings
Solver Name
Semi-Implicit Finite Volume Compiled,
Regular Grid (Fixed Time Step)
Advanced Settings
Solver Name
Semi-Implicit Finite Volume Compiled,
Regular Grid (Fixed Time Step)
Time Bounds - Starting
0.0
Time Bounds - Ending
1.0
Time Step - Min
1.0E-8
Time Step - Default
1.0E-6
57
Modeling biophysical mechanisms underlying cellular homeostasis
1.0
Time Step - Max
Error Tolerance - Absolute
1.0E-9
Error Tolerance - Relative
1.0E-9
Keep Every
1000
Keep At Most
1000
Use Symbolic Jacobian (T/F)
F
Simulation Name: water permeability: x and 7x, short and long time
Simulation Description: For all the simulations: The model describes a kidney principal cell including both
basolateral and apical membranes. K+ transport from the basal extracellular space to the apical
extracellular space is simulated. In addition, the mechanism by which K+ can be recycled into the
extracellular space is described by the model of Kir7.1 currents in ECS and DLS. The role and function of
water channels in efficient clearance of DLS from the excess K+ is reflected in the term Keffective_ECS.
Overriden Parameters
Name
Actual Value
Default Value
Pf
0.049
0.0070
58
Padideh Kamali-Zare
Advanced Settings
Solver Name
Semi-Implicit Finite Volume Compiled,
Regular Grid (Fixed Time Step)
Time Bounds - Starting
0.0
Time Bounds - Ending
10.0
Time Step - Min
1.0E-8
Time Step - Default
1.0E-6
Time Step - Max
1.0
Error Tolerance - Absolute
1.0E-9
Error Tolerance - Relative
1.0E-9
Keep Every
10000
Keep At Most
1000
Use Symbolic Jacobian (T/F)
F
Overriden Parameters
Name
Actual Value
Default Value
Gk
0.0
22.0
Pf
0.0
0.0070
Geometry Setting
Geometry Size (um)
(3.0, 10.0, 0.01)
59
Modeling biophysical mechanisms underlying cellular homeostasis
Mesh Size (elements) (90, 180, 1)
Advanced Settings
Solver Name
Semi-Implicit Finite Volume Compiled,
Regular Grid (Fixed Time Step)
Time Bounds - Starting
0.0
Time Bounds - Ending
1.0
Time Step - Min
1.0E-8
Time Step - Default
1.0E-6
Time Step - Max
1.0
Error Tolerance - Absolute
1.0E-9
Error Tolerance - Relative
1.0E-9
Keep Every
1000
Keep At Most
1000
Use Symbolic Jacobian (T/F)
F
60