Poroelastic model of the nucleus

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(1)PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE SCHOOL OF ENGINEERING. POROELASTIC MODEL OF THE NUCLEUS. NICOLÁS ANDRÉS PÉREZ GONZÁLEZ. Thesis submitted to the Office of Research and Graduate Studies in partial fulfillment of the requirements for the degree of Master of Science in Engineering. Advisor: DR. DANIEL HURTADO. Santiago de Chile, June 2015 c MMXV, N ICOL ÁS A NDR ÉS P ÉREZ G ONZ ÁLEZ.

(2) PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE SCHOOL OF ENGINEERING. POROELASTIC MODEL OF THE NUCLEUS. NICOLÁS ANDRÉS PÉREZ GONZÁLEZ Members of the Committee: DR. DANIEL HURTADO DR. LORETO VALENZUELA DR. DENIS WIRTZ DR. FERNAN FEDERICI DR. JORGE RAMOS. Thesis submitted to the Office of Research and Graduate Studies in partial fulfillment of the requirements for the degree of Master of Science in Engineering Santiago de Chile, June 2015 c MMXV, N ICOL ÁS A NDR ÉS P ÉREZ G ONZ ÁLEZ.

(3) To my parents Eric Pérez and Rosa González.

(4) ACKNOWLEDGEMENTS. These acknowledgements are my way to keep all of you eternal. My way to express gratitude and keep your names intertwined with my achievement. First of all, I want to thank my family. I am grateful of my parents Eric and Rossi for helping me get here. I would not have reached my dreams if they had not been there. My brother Javier has been a source of inspiration and dogged work, he showed me how this process of doing research is my own way of finding myself. Leah, who has been my pillar throughout this process, being there day and night when I needed to talk or find my north. I want to thank my professors, in particular to those who opened the gates to different opportunities. To my advisor, Dr. Daniel Hurtado, who has shown me the value of the field I am diving into. I will be always grateful for showing me the amazing opportunities in the field of biomedical engineering, biotechnology and biophysics. To Dr. Denis Wirtz, who has shown me that the versatility of physics is limitless. To my friends, thank you so much for always being there to listen to my ideas and dreams. David, Pablo, Paul, Karim, Camilo, Iván and Alexis, thank you for sharing moments full of happiness and silliness. You made me laugh during this journey. To all of you, thank you for helping me to keep going.. Nicolás Pérez. iv.

(5) TABLE OF CONTENTS. ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. iv. LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. vii. LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ix. FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. x. ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xi. RESUMEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xii. Introduction to Biophysics and Cell Mechanics . . . . . . . . . . . . . . . . .. 1. 1.1.. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2.. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 1.3.. A Crash Course on Cell Structure . . . . . . . . . . . . . . . . . . . . . .. 4. 1.4.. Existing Techniques in Cell Mechanics . . . . . . . . . . . . . . . . . . .. 7. 1.5.. Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. Nuclear Cell Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 2.1.. The Structure of the Nucleus . . . . . . . . . . . . . . . . . . . . . . . .. 11. 2.2.. The experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 2.3.. Poroelasticity and Mixture theory . . . . . . . . . . . . . . . . . . . . . .. 13. 2.4.. Our Poroelastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14. 3. Article: Poroelastic model of the nucleus . . . . . . . . . . . . . . . . . . . . .. 16. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 2 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. 1.. 2.. v.

(6) 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30. 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. Further Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39. 4.1.. Determination of constants, verification of the model . . . . . . . . . . . .. 39. 4.2.. What structure holds the external pressure? . . . . . . . . . . . . . . . . .. 40. 4.3.. Limitations of the model . . . . . . . . . . . . . . . . . . . . . . . . . .. 41. 4.4.. Future improvements of the model . . . . . . . . . . . . . . . . . . . . .. 41. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. 4.. 5.. vi.

(7) LIST OF FIGURES. 1.1 Example of an adherent cell. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.1 Simplification of the nucleus of the cell for mechanical modeling. . . . . . . . .. 15. 2.2 General scheme of the poroelastic model. . . . . . . . . . . . . . . . . . . . .. 19. 2.3 Time evolution of the nuclear volume upon trypsinization. . . . . . . . . . . .. 25. 2.4 Evolution of the solid phase, fluid phase, and total pressure inside the nucleus after trypsinization. After an initial stimulus, all pressure curves asymptotically converge to a steady state. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27. 2.5 Fluid expelled during trypsinization. The black solid line shows the time evolution of the fluid velocity at the nuclear envelope, where an exponential decay is observed after trypsinization. The red dashed line depicts the total expelled fluid volume at every time instant.. . . . . . . . . . . . . . . . . . . . . . . .. 28. 2.6 Time evolution of the nuclear volume during trypsinization in cells treated with Nocodazole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29. 2.7 Evolution of the solid phase, fluid phase, and total pressure inside the nucleus after trypsinization for cells treated with Nocodazole. After an initial stimulus, all pressure curves asymptotically converge to a steady state.. . . . . . . . . . . .. 30. 2.8 Fluid expelled during trypsinization for cells treated with Nocodazole. The black solid line shows the time evolution of the fluid velocity at the nuclear envelope, where an exponential decay is observed after trypsinization. The red dashed line depicts the total fluid volume expelled by the nucleus at every time instant.. . .. 31 vii.

(8) 2.9 This graph shows variation in the value for the apparent permeability. We have considered for different values ranging from 0.2 times the reference value to 5 times the reference value. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 2.10This graph shows variation in the value for the added pressure during trypsinization. We have considered reductions down to 6 kPa from the original 18 kPa that resemble the experimental data. . . . . . . . . . . . . . . . . . . . . . . . . .. 35. viii.

(9) LIST OF TABLES. 2.1 Summary of parameter values and references for the poroelastic model . . . . .. 26. ix.

(10) FOREWORD. Biomechanics has seen an explosive growth in the last few decades since scientists have started to see its potential in clinical developments. More and more students are going into Biomedical Engineering in order to use the tools provided by engineering to develop new technology and improve the healthcare system. Now engineers participate in this system, acting as observers and finding innovative solutions to old problems. This paradigm change is transforming the way we see our health. In this context is important to remember that no advance can be made if we forget basic research. In order to develop new technologies is necessary to understand the elementary principles underlying all phenomena we are interested in. In this context, I look forward to see the professionals that will be educated in this field in the new program developed by the Pontifical Catholic University of Chile. The integration of new professors with background in Biomechanics, Biophysics, Bioengineering and Nanobiotechnology will be crucial to keep this university at the frontier of science. I hope this thesis inspires students and future scientists to look further and try to discover a novel piece of knowledge. Because there is no more valuable secret to have in your hands.. x.

(11) ABSTRACT. Several diseases have been linked in the last few decades to structural malfunctioning at the cellular scale. Even though the understanding of the processes in the cell has increased in time, the field of biomechanics remain understudied, and only a few models have been developed to describe cell mechanics. In this work we pursue a clear description for the mechanics of the nucleus under changes of pressure across the nuclear envelope. The pressure differences that appear and generate nuclear volume changes have different origins within the cell such as osmotic pressure changes, the depolymerization of certain structural biopolymers or the cytoskeleton exerting pressure on the nucleus. In this work, we developed a byphasic model based on 4 general physical phenomena: force balance, mass balance, Darcy’s law and mechanical deformation. We developed 5 equations that use 7 parameters. From these, 2 were given by initial conditions. Other 2 were obtained from the literature while the last 3 were set to ressemble experimental data. The experimental data was obtained at Johns Hopkins University by collaboratos and focused in nuclear volume changes in adherent cells. From these experiments, we obtained two sets of data: the first one related to trypsinization of cells and the second one related to trypsinization of cells pre treated with nocodazole. The poroelastic model developed has been able to ressemble the experimental data and the asymptotic behaviour at the final state in both cases. Another novel finding of this research has been the establishment of a value for the permeability of the nucleus at about 10−13 µm2 . In addition, we have derived from the model, the pressure difference that appears exerted on the nuclear envelope once the trypsinization process happens. We believe that nuclear mechanics are better described by a model where the network and the fluid are considered as a mixture.. Keywords: Poroelasticity, nucleus, cell, biphasic. xi.

(12) RESUMEN. Muchas enfermedades han sido relacionadas en las últimas décadas a un mal funcionamiento estructural a la escala celular. En este trabajo buscamos una clara descripción para la mecánica del núcleo bajo cambios de presión en la membrana nuclear. Las diferencias de presión que aparecen y generan cambios en el volumen del núcleo tienen diferentes origenes dentro de la célula tales como cambios en la presión osmótica o la despolimerización del citoesqueleto. En este trabajo, desarrollamos un modelo bifásico basado en cuatro fenómenos fı́sicos: balance de fuerzas, balance de masas, la ley de Darcy y deformación mecánica. Hemos desarrollado 5 ecuaciones que usan 7 parámetros. Dos fueron dados por condiciones iniciales. Otros dos fueron obtenidos desde la literatura mientras que los últimos tres fueron determinados minimizando error en el ajuste de la curva. Los datos experimentales fueron obtenidos en Johns Hopkins University por colaboradores que se enfocaron en cambios volumétricos del núcleo en células adherentes. De estos experimentos hemos obtenidos dos sets de datos: El primero que se relaciona a tripsinización de células y el segundo que se relaciona a la tripzinicación de células pretratadas con nocodazole. El modelo poroelástico ha sido capaz de reproducir la curva experimental y el comportamiento asimptótico en el estado final en ambos casos. Otro hallazgo de esta investigación ha sido el establecimiento de un valor para la permeabilidad del núcleo con un valor de 10−13 µm2 . Además, hemos derivado de este modelo la diferencia de presión que aparece en la membrana nuclear una vez que el proceso de tripzinización ocurre. Creemos que la mecánica celular es mejor descrita por un modelo donde los polı́meros y el fluido son considerados una mezcla.. Keywords: Poroelasticidad, nucleo, célula, bifásico. xii.

(13) 1. INTRODUCTION TO BIOPHYSICS AND CELL MECHANICS. What is the structure a human cell holds? Is the nuclear shape relevant in disease?. Questions like these have been started to be answered quantitatively in the last decades even though they were answered a century ago in a qualitative way (Pelling & Horton [2000]). In the last few years physicists, biologists, mathematicians and other have worked together to understand in a deeper way how cells and tissues work in our body, in a joint work to defeat diseases such as cancer.. 1.1. History The study of cell mechanics can be tracked to the discovery of the cell itself. A cell constitutes the elemental unit of tissues, and the name was coined by Robert Hooke in “Observation XVIII” of the Micrographia, where he wrote: ...I could exceedingly plainly perceive it to be all perforated and porous, much like a Honey-comb, but that the pores of it were not regular... these pores, or cells, ... were indeed the first microscopical pores I ever saw, and perhaps, that were ever seen, for I had not met with any Writer or Person, that had made any mention of them before this... (Hooke [1665]). The cell was discovered and many studies started relating this elemental brick of nature to the functions performed by differents tissues in our bodies. The development of better microscopes helped scientists to observe their behavior and what are they made of. Many years passed but little interest was generated in understanding the physics behind all of these biological processes. In the last few decades, physicists and engineers have turned to biology, discovering that many elementary questions remain unanswered (Pelling & Horton [2000]). Even 1.

(14) though the combination of physics and biology is really useful, when it comes to biophysics, it is quiet difficult to find a definition. In 1940, J.R. Loofbourow from MIT said that there is “no clear agreement, even among biophysicists, as to what the term biophysics means” (Loofbourow [1940]). Four decades later, in 1984, a similar conclusion was reached in the Eighth International Congress of Biophysics saying that “there probably never was an adequate definition of biophysics. Perhaps there never will be” (Clarke [1984]). This confirms that the field of biophysics is so broad and that the approaches vary so much that even the fact of creating a definition for the field is a difficult task. Despite the difficulties when trying to define the field of biophysics, today is undeniable the impact that it has had in biology. In recent years, our understanding of different phenomena has been quantified and analyzed through the keen eyes of mathematicians, physicists and engineers. These scientists have helped biologists to complete the picture with new pieces of the puzzle. One of the most important contributions of physics to biology was the x-ray structure determination which lead to the discovery of DNA structure in 1953 by Watson and Crick (Watson & Crick [1953]). Other techniques have revolutionized the field as well such as synchrotron radiation or NMR among others (Frauenfelder et al. [1999]). It is also impressive that the field of quantum mechanics is mixing with biology in what has been presented as quantum biology in a new attempt to explain the physics behind life itself (Lambert et al. [2000]). We have mentioned before that our interest is on one particular field within biophysics, which is cell mechanics. This field focuses in the analysis of biological structures and its mechanical behavior. It tries to understand issues like how cells perceive mechanical cues or how cells generate physical forces (Rodriguez & Sniadecki [2013]). In some cases cells change their structure and form stronger adhesions (?) and in other cases they can reduce their stiffness in order to respond better to environmental changes (Krishnan et al. [2009]).. 2.

(15) F IGURE 1.1. Example of an adherent cell.. 1.2. Motivation In the last few years there has been a rising interest to fund research with new approaches in cell biology with particular interest in certain diseases such as cancer. Among these new approaches, we find those based in the physical sciences. Different research center have started looking for physicists, mathematicians and engineers that can bring a fresh perspective when dealing with cell biology and the fundamental descriptions of their behavior. Cells have been extensively explored from a molecular perspective but only recently they have been explored from the perspective of biophysics and biomechanics. How do they move? How does the cytoplasm behave? Is the cytoplasm equivalent to a fluid or does the cytoskeleton play a fundamental role? These questions need to be answered as part of the quest of understanding the cell in health and disease. The lack of models that describe why cells or tissues behave in the way they do only shows how this phenomena remains understudied. This is why we believe that the introduction of our new approach with more versatility for nuclear mechanics could explain behavior from fundamental principles. 3.

(16) As we mentioned, we want to focus in the mechanical behavior of the nucleus. In particular, our collaborators have done experimental analysis on mouse embryonic fibroblasts (MEF) which are adherent cells such as the one shown in figure 1.1. Some researchers propose that the cell behave as a gel (Pollack [2002]). Others propose that it behaves as a viscoelastic continuum (Karcher et al. [2003], Wagner et al. [1999]). So far there are few models for certain cells and there is no effective model that resembles the mechanical behavior of all type of cells. Even fewer models haven been developed to understand the mechanical behavior of the nucleus and so, our journey starts in the next chapter by understanding the state-of-the-art of nuclear cell mechanics.. 1.3. A Crash Course on Cell Structure Understanding the structure of cells is the first step to propose a model able to explain its behavior. This is why we need to know the composition of cells and which elements are relevant when considering interactions with the nucleus. The main structural elements of the cell can be categorized in membranes, organelles, cytoskeleton and filaments. These elements perform different mechanical functions within the cell, generating complex relationships that define cell behavior. Cellular membranes First we mentioned membranes. In cells is possible to find multiple membranes that are relevant for mechanics, one of them is the bipid layer that defines the limit of the cell and another example is the nuclear envelope that defines the limit of the nucleus separating the genetic material from the rest of the cell. These membranes have different properties such as surface tension or permeability given that the first one is a simple bilipid layer whereas the latter is a double bilipid later. These properties will be key in our model of the nucleus of the cell. 4.

(17) To understand the mechanics of these membranes is important to comprehend how they form and the kind of interaction that is present in their structure. In general membranes are formed by different amphiphilic molecules (Boal [2001]). When this molecules are in water, they tend to associate and form higher order structures in order to minimize their energy. Depending on the molecular geometry of these amphiphilic molecules, the are able to form micelles or other structures. When their geometric structure is close to a cylinder, they tend to form membranes like the one that surrounds the cell or like the nuclear envelope that encloses the genetic material. Cytoskeleton Another important element within a cell is the cytoskeleton. This organelle is formed mainly by three types of filaments. These filaments are known as microtubules, intermediate filaments and actin microfilaments. Each type of filament has its own function but overall they work together conforming the skeleton of the cell. The cytoskeleton carries out many functions: it organizes the contents of the cells, it recruits chromosomes during mitosis, it organizes transports of elementes within the cell, etc. In addition, it connects the cell with the external environment and it generates coordinated forces that enable the cell to move and change shape (Fletcher & Mullins [2010]). These three types of filaments have evolved differently and so their mechanical properties are very different. This allows them to perform different functions in the cell. The smaller filament is made out of actin which is the most abundant protein in most eukaryotic cells (Dominguez & Holmes [2011]). This filament is formed by polymerizing monomeric actin (G-actin) to filamentous actin (F-actin) and it plays a fundamental role in most of the cells functions. One particular role that is highlighted in biology for this particular filament is the interaction with myosin, which constitutes the principle for muscle contraction and cell motility (Dominguez & Holmes [2011]).. 5.

(18) The second filament is the intermediate filament which also constitutes an important percentage of the cytoskeleton. These filaments have been seen as the element in the network that absorbs mechanical stress and that integrates the other two filaments Herrmann et al. [2011]. Their presence in cell-cell junctions is also highlighted given the importance of connecting multiple cells within the extracellular matrix. Finally, we have microtubules which constitute the third main network in the cytoskeleton. This polymeric network is composed of α and β tubulin heterodimer subunites assembled into protofilaments (Conde & Caceres [2009]). One single microtubue is made out of 10 to 15 protofilaments that associate together forming a hollow cylinder. Its functions are multiple such as mitosis, cell motility, intracellular transport, secretion, cell shape, etc. It has gained a lot of attention in the last few decades for its dynamic behavior that allows fast adaptation to changing conditions. These three polymers conform the cytoskeleton and so their interaction is very attractive from a mechanical point of view given its complexity and dynamic behavior. Mechanical properties of polymeric network Now, what is known about the mechanical behaviour of this polymeric network? This is where we change our perspective. We stop thinking just about the biological function and we start thinking of the mechanical relevance of this cytoskeleton. In 2002, Stamenovic and Inger examined models of cytoskeletal mechanics of adherent cells. In their work, they advocate for the idea that adherent cells sense their mechanical environment, which regulates their functions (Stamenović & Ingber [2002]). It is relevant to analyze the mechanical behaviour of this polymeric network since it has been established that mechanical loads applied to cells generate changes in cell shape and affects functions related to movement, growth differentiation and proliferation (Stamenović & Ingber [2002] Harris et al. [1980] Dembo [1989]). 6.

(19) Understanding the mechanical properties of these filaments is important when considering the biphasic nature of the cytoplasm. Doing this requires to look at each filament individually given that they all show different functions and properties. As we mentioned before, the actin network of the cytoskeleton is composed by thin filaments made up by actin monomer (globular G actin) which polymerizes to form the filaments (Cooper [2000]). The diameter of these thin filaments is 5-10 nm with a Young’s modulus in the order of 1 GPa (Gittes et al. [1993],Mofrad & Kamm [2006]). Intermediate filaments’ functions are less understood and their diameter is ≈ 10 nm. Microtubules are tubular biopolymers (outer and inner diameter of ≈ 24 and ≈ 12 nm respectively) with a Young’s modulus in the order of 1 GPa (Gittes et al. [1993]). Notice that is expected that actin filaments are stronger than microtubules. There are two classes of structural mechanical models of the cytoskeleton that have been well studied to understand mechanisms by which the cell generates stresses in order to resist deformation (Stamenović & Ingber [2002]). The first type are known as open-cell foams. In this models, after deformation of individual cytoskeletal filaments, a general stress arise. In the second type, there is a pre-existing stress that resists applied loads. The latter is known as stress-supported structures (N. Wang et al. [2002]). For many years, it was costumary to use a model were the cortical membrane of the cell is the element that provides mechanical stablity (Stamenović & Ingber [2002]). Even though this model is useful in suspended cells, it does not perform well in the study of adherent cells (endothelial cells, fibroblasts, etc).. 1.4. Existing Techniques in Cell Mechanics In order to understand how cells behave and charaterize them in the context of cell mechanics, first is necessary to be able to measure different physical properties. With that in mind, different techniques have been developed such as micropippete aspiration, 7.

(20) which is oftenly used to analyze the cell as a whole entity by pulling part of the cell into a glass pippete after applying negative pressure. Other techniques use the compression of cells between plates to analyze cell response under pressure (Cheng et al. [2009]). Many other techniques have been developed to understand cell mechanics such as Microneedle manipulation, AFM, Optical tweezers, optical stretcher, magnetic tweezers, microfluidic devicese, acoustic tweezers, etc (Rodriguez & Sniadecki [2013]). Techniques like the ones just mentioned using cell indentation and manipulation are now quantifying mechanical properties that were described qualitatively 100 years ago (Pelling & Horton [2000]). Modelling the mechanics of the cell The first models of cells considered them to be an homogeneous gel or a viscoleastic fluid (Bingham [1933], Heilbrunn [1927]). These initial ideas are still being used in combination with complex mathematical and computational techniques. Today, the viscoelastic nature of cells has been tried to be proven using finite element techniques in which a dense meshwork is used to resemble experimental data (Charras et al. [2005], Mofrad & Kamm [2006], Y. L. Wang & Discher [2007]), but this is definitely not the only theory out there. Other models are presented in the following list: (i) Elastic behavior: Linear models. (ii) Elastic behavior: Non-linear models. (iii) Viscoelastic behavior: Maxwell models. (iv) Power-law structural damping models. (v) Biphasic behavior: Poroelastic models. (vi) Poro-viscoelastic models. (vii) Active behavior: Active poroelastic gels. Among these models those that consider the biphasic nature of the cell have been capturing more attention lately. In particular, the poroelastic nature of the cytoplasm has 8.

(21) been discussed in the past by Mitchison et al (Mitchison et al. [2008]) and it has been recently proven to fit experimental data by Moeendarbary et al recently (Moeendarbary et al. [2013]). Despite the fact that many models have been suggested, analyzing viscosity, elasticity, viscoelasticity, plasticity and poroelasticity, the scientific community has not obtained a complete theoretical description of cell mechanics that is both, time dependent and predictive (Pelling & Horton [2000]) and therefore, the true nature of the cell remains in the shadows. 1.5. Thesis Outline We start Chapter 2 with a detailed description of Nuclear Cell Mechanics starting with a biological description of its structure and finishing with a first description of the poroelastic model we pursue to develop in this thesis. In Chapter 3 we have included the article associated with this research that has been submitted for publishing. In Chapter 4 we discuss further the implications of our model and in Chapter 5 we finish with our final conclusions.. 9.

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(23) 2. NUCLEAR CELL MECHANICS. The nucleus is the core of the cell. It contains the genetic material that codifies all the instructions that run our bodies. Time ago, it was thought that the expression was a chemical phenomena, but now, the field of mechanotransduction has shown how mechanical cues can modify the expression of certain proteins. Thus, the science of epigenetics is going through a paradigm shift in which structure and chemical composition are both relevant while studying diseases. This is one of the multiple reasons why it is important to understand the mechanics of the nucleus.. 2.1. The Structure of the Nucleus A simple way to look at the nucleus is imagining a thick layer that encloses fluid in the same way that a balloon works. This surface has multiple channels that connect the external media with the internal media. Within the nucleus we have the genetic material and the nucleoplasm (which constitutes the fluid phase). Understanding how these elements interact is difficult given the complexity of nature. In order to model biological structures, the first step is to simplify the model and preserve certain symmetries. It is a mistake to think that modelling individual elements is enough because their interaction could be more relevant that their individual contribution to the system (Fischer [2008]). In the past different models have analyzed nuclear mechanics trying to prove its viscous, elastic or viscoelastic nature as a whole (Guilak et al. [2000], Guilluy et al. [2014]). Other researchers have focused in particular elements of the nucleus to understand mechanical behavior and measure their properties. A recent work that used a computational 11.

(24) model based on viscoelasticity has been able to simulate and resemble data from experiments (Vaziri et al. [2006] Vaziri & Kaazempur [2007]). Even though this models characterize the nucleus, we believe that they do not describe the true biphasic nature of the cell.. 2.2. The experiment In order to correctly model the nucleus we jointed efforts with our collaborators at Johns Hopkins University where nuclear volume changes were analyzed in Human Embryonic Fibroblasts. This cell line was cultured and studied according to the protocols in (Kim et al, submitted). The experiment involved two different sections. In the first section, nuclear volume changes were measured over time after trypsinization of the cultured cells. The second section also involved trypsinization but in this case the measurements were performed in cells pretreated with nocodazole. To understand in detail the experiment then is necessary to understand how trypsin and nocodazole work. The first one of these substances is trypsin which constitutes a protease that cleaves peptide chains mainly at the carboxyl side of amino acids lysine or arginine. It is commonly used in biotechnological applications and is broadly used in cell culture. In this particular context, trypsin is used to digest the proteins that bind the adherent cell to the dish where they are being cultured (Gori [1964]). Once they detach from the dish, the cell goes from a flat shape to a round shape. In this process of reshaping, the total cell volume changes and the nuclear volume changes. The second substance we mentioned is nocodazole which is an agent that disrupt microtubule function by supressing microtubule dynamics or inducing microtubule depolimerization (Blajeski et al. [2002]). This agent was used to analyze how does the nuclear volume changes under trypsinization when we also depolymerize microtubules and therefore weaken the cytoskeleton. 12.

(25) Using trypsin and nocodazole our collaborators were able to measure the volume of the nucleus over time. These results prompted us to develop a model that accounts for the structure within the nucleus and that is able to ressemble the experimental data (more details in the experimental data is presented in chapter 3).. 2.3. Poroelasticity and Mixture theory We have mentioned that a variety of models have been developed in the last two decades to determine an appropriate model that describes the mechanics of the cell. Among these models, we are interested in poroelasticity and mixture theory given that they describe biphasic compositions. We think that the biphasic nature of the nucleus could be accurately described by these theories by assuming that DNA forms a polymeric mesh in which the nucleoplasm can flow. In addition, we believe that the combination of the polymeric mesh within the nucleus, the nucleoplasm and the nuclear envelope are mechanically relevant altogether. Mixture theory has been applied in the past in the biological context to analyze the mechanical response of cells under external loads. Previous work using mixture theory in the context of cells has considered a non-deformable solid matrix and a fluid phase that moves throughout the polymeric mesh (Ateshian et al. [2007, 2006, 2013]). In addition to the biphasic nature of the composition, mixture theory consider k solutes in the fluid phase and it considers the modelling of the flow across the membrane. This theory is particularly interesting when analyzing the relationship between permeability changes (for the kth solute) and how this affects the mechanical response of the cell. The origin of poroelasticity can be tracked to the theory developed by Terzaghi in 1923 that described a one-dimensional consolidation process for soil deformation (Terzaghi [1923]). This theory accounted for the presence of fluid and its effect in the overall deformation. In 1935, Biot published a linear theory of poroelasticity and in 1936 Rendulic 13.

(26) generalized Terzagui’s theory to three dimensions (M. A. Biot [1935], Rendulic [1936]). Biot kept improving his model and published results that were compatible with Terzagui’s model and Rendulic’s model (M. Biot [1941]). Poroelasticity is a biphasic theory that considers a solid, deformable and porous matrix that is filled with fluid. Under pressure, this composition tends to deform considering the interaction of both phases. We are particularly interested in poroelasticity because we believe that the compaction of DNA is relevant in when considering changes in the nuclear volume. In this sense, the nuclear deformation is not only due to water leaving the nucleus but also due to a compaction process of the chromatin mesh.. 2.4. Our Poroelastic Model In order to model the nucleus we have considered three relevant structural elements. First, we considered the chromatin mesh within the nucleus. In second place, we considered the fluid phase in the nucleus. Finally, we considered the nuclear envelope. The theory has been developed in full detail in chapter 3 so here we will discuss the physical phenomena behind the model. First, as mentioned before, we have considered the chromatin mesh as a porous solid media uniformly distributed across the nucleus (see figure 2.1). This polymeric mesh is deformable and compressible under pressure. Our approach accounts for the idea that DNA will compact when an external force is applied instead of just reorganizing in a smaller space. Second, we consider a fluid phase given by the nucleoplasm. In the model we have simplified this component to be an incompressible fluid with no solutes. The interaction of the solid mesh with the fluid phase leads to a redistribution of the pressure among these two phases. 14.

(27) F IGURE 2.1. Simplification of the nucleus of the cell for mechanical modeling.. Finally, since we are interested in nuclear mechanics, the nuclear envelope is particularly relevant for our model. By including the nuclear envelope we have a clear structure for boundary conditions that has an associated permeability that modules the velocity with which water can leave the system. In addition, membranes in a medium can be understood mathematically via a balance of forces that leads to Laplace’s law.. The physical principle behind this system can be summarized as follows, • Mass balance • Force balance • Flow through permeable membrane (Darcy’s law) • Deformation These principles and the equations that they lead to are detailed in chapter 3.. 15.

(28) 3. ARTICLE. Poroelastic model of the nucleus 1 Nicolás Pérez1,6 , Alfredo Celedón, Denis Wirtz1−5 , Daniel E. Hurtado6 . 1. Department of Chemical and Biomolecular Engineering, The Johns Hopkins University, Baltimore, Maryland,. 2. USA. Johns Hopkins Physical Science Oncology Center, Baltimore, The Johns Hopkins University, Maryland, USA.3 Department of Pathology, The Johns Hopkins University School of Medicine, Baltimore, Maryland , USA. 4 Department of Oncology, The Johns Hopkins University School of Medicine, Baltimore, Maryland, USA. sive Cancer Center, The Johns Hopkins School of Medicine, Baltimore, Maryland, USA.. 6. 5. Kimmel Comprehen-. Department of Structural. and Geotechnical Engineering and Biomedical Engineering Group, Pontificia Universidad Catolica de Chile, Santiago, Chile. Keywords: Poroelasticity, nucleus, cell, porous, material.. Abstract The cell nucleus plays a fundamental role on the genetic expression and biological function of the cell. Mechanical stimulus and the ensuing biological response of the nucleus have been recognized as fundamental in gene regulation and biological signaling. Despite its importance, the mechanical behavior of the nucleus remains understudied, mainly due to the difficulty to assess pressure and deformation during cellular processes. In this work, we propose a poroelastic mechanical model of the nucleus that allows us to estimate the mechanical properties and nuclear pressure differences based on cell imaging. In particular, we study the nuclear volume evolution under two scenarios, trypsinization of mouse embryonic fibroblasts and trypsinization of mouse embryonic fibroblasts pretreated with Nocodazole. The proposed model captures the decrease of the nuclear volume while giving estimates for the range of pressure differences inside and outside the nucleus. In particular, a pressure increase acting on the outer surface of the nuclear envelope of up to 18 kPa was associated to process of trypsinization while a pressure of 15 kPa was obtained for the cells under trypsinization that were pretreated with Nocodazole. Other novel findings of this work are estimates for the nuclear envelope permeability, found to be in the order of 5 × 10−13 µm2 . We envision that the novel application of poroelasticity to explain the mechanical behavior of the nucleus can establish a new basis for understanding the deformation of the nuclei as well as quantitatively characterizing the structural interaction with the cytoskeleton and other organelles of the cell.. 1. Submitted. 16.

(29) 1 Introduction Cell and nuclear morphology have long played a fundamental role in biology and medicine. Champy and Carleton explored in 1921 how cells change it shape and how is related to the structures inside, and suggested a possible relation between the shape of some cells and their respective nuclei (Champy & Carleton [1921]). Some types of cancer are routinely diagnosed on a nuclear morphology basis (Zink & Nickerson [2004]), while other diseases can be diagnosed based on similar evidence (Mounkes & Stewart [2003] Lammerding et al. [2004]). Despite of the biological relevance and medical implications of such findings, the fundamental mechanisms that control morphological changes of the nuclei remain an understudied and challenging field of research (Vaziri et al. [2007]). The mechanical behavior of cells has long been studied with experiments such as micropipette aspiration or microindentation (Vaziri et al. [2006]) and theoretical models that resemble its behavior (Ateshian et al. [2006] Ateshian et al. [2007] Ateshian et al. [2013]). Hartmann & Delgado studied numerical simulations of the mechanical behavior of a yeast cell using Cauchys equation of motion for a continuous medium (Hartmann & Delgado [2004]). Other authors suggest that the cell has a gel like structure (Pollack [2002]), while some authors have developed models where the cell has a viscoelastic behavior (Wagner et al. [1999] Karcher et al. [2003]). More recently, by analyzing the motion of magnetic nanorods inserted into 3T3 fibroblasts, the rheological behavior of the cytoplasm has been characterized as a Bingham fluid (Castillo et al. [2003]). Moreover, mixture theory has been used in chondrocytes to explore the application of multiphasic theories to the cytoplasm and extracellular matrix (Ateshian et al. [2006] Ateshian et al. [2007] Ateshian et al. [2013]). While many mechanical models have been introduced for the whole cell and cytoplasm behavior, fewer models have been proposed to describe the nucleus behavior, mainly due to the fact that probing the cell nucleus non-invasively is, and remains experimentally challenging. A viscoelastic continuum-based computational model of an isolated nucleus has simulated micropipette aspiration resembling data from experiments (Vaziri 17.

(30) et al. [2006] Vaziri & Kaazempur [2007]). A recent work showed evidence for viscoelastic behavior (Guilluy et al. [2014]) and other models have focused in modeling nuclear blebbing, by simulating the structural change in the nucleus of the cell (Funkhouser et al. [2013]). Recently, Kim et al (submitted, 2015) have studied the volume-pressure dependence in the nucleus driven by osmotic-pressure changes when subject to trypsinization. To explain the observed nuclear shaping, they postulate that the nuclear envelope is the main contributor to the cell mechanical response. This model is biophysically motivated by the structural role of lamin meshwork at the nuclear periphery. In their formulation, the nucleus interior is not considered to play a significant role in the structural behavior nucleus. Their data shows volume reductions in the nucleus upon trypsinization for two cell lines, adherent mouse embryonic fibroblast (MEF) and human foreskin fibroblast (HFF). When trypsin was applied, these cells detached from substrate changing from a spread and flat shape adhered to the substrate to a round shape that is suspended over the substrate. By using real-time monitoring over single cells, it was possible to analyze nuclear volume variations up to 50% (Kim et al. [2015]). In this work, we hypothesize that not only the nuclear envelope, but also the nucleus interior, i.e., chromatin, nucleoskeleton and nucleoplasm, play an important role in determining the morphological changes of the nucleus. To assert this hypothesis, we propose for the first time a poroelastic theory of the nucleus to study its mechanical behavior (Chandran & Barocas [2012], Guéguen [1994], H. Wang [2000]). Poroelastic models have been previously employed to describe cells (Pena et al. [1998]) and their migration (Mitchison et al. [2008]). Microindentation tests performed on different types of cells have been recently performed to validate the poroelastic response of the cytoplasm (Mitchison et al. [2008], Moeendarbary et al. [2013]). Here, we focus on the cell nucleus and consider the nucleus interior to behave as a continuum poroelastic material enclosed by an extensible. 18.

(31) membrane, through which a fluid phase, namely the nucleoplasm, passively flows under the presence of pressure gradients. We incorporate the effect of mass exchange across the nuclear envelope by assuming a porous permeable membrane that mediates flow in and out the nucleus. To validate the proposed model, we compare our results against cellular volume changes associated to changes in the pressure acting on the nuclear envelope by trypsinization in mouse embryonic fibroblasts and another set of the same cells pretreated with Nocodazole (Kim et al. [2015]).. 2 Materials and methods 2.1 Theoretical framework - A poroelastic model of the nucleus We consider the biphasic nature of the nucleus contents, where we identify a solid phase made up of chromatin and the nucleoskeleton, and a fluid phase, the nucleoplasm, mainly composed by water. We further assume that the solid phase behaves elastically, with homogenized material properties that characterize its behavior, while the fluid phase is considered to be incompressible. A sketch of the mechanical model adopted in this work is shown in figure 2.2.. F IGURE 2.2. General scheme of the poroelastic model.. 19.

(32) Under the assumptions mentioned above, the mechanical behavior of the nucleus interior can be modeled in a continuum fashion as a poroelastic material (M. Biot [1941]). To account for the fluid transport in and out the nucleus, we model the nuclear envelope as a deformable porous membrane, where flux is triggered by pressure gradients across the envelope, and the membrane permeability is controlled by nuclear pore complexes. We further assume that passive transport mechanisms are responsible for most of the mass flux across the nuclear membrane, and neglect any active-transport contributions. To simplify the analysis, a spherical shape of the nuclear envelope is assumed at all times. The balance of linear momentum for a spherical vessel yields Laplaces law, namely pint (t) − pout (t) =. 2σ R(t). (2.1). where pint (t) is the total pressure exerted by the nucleus interior on the nucleus envelope, pout (t) is the external pressure exerted on the outer side of the nuclear membrane by the cytoskeleton and the cytoplasm, σ is the membrane surface tension (which we have assumed to be constant in the entire process of volume reduction), and R(t) is the radius of the nucleus. Let φf (t) be the fluid volume fraction, i.e., the ratio of the volume of fluid phase over the total volume of the nucleus, and φs (t) be the solid-phase volume fraction, i.e., the ratio of the volume of solid phase over the total volume of the nucleus. Since we only consider a two-phase solid-fluid medium, it holds that φf (t) + φs (t) = 1. (2.2). The total pressure is then decomposed into a fluid-phase pressure pfint (t) and a solidphase pressure psint (t) such that pint (t) = φf (t)pfint (t) + (1 − φf )(t)psint. (2.3). where the condition φs (t) = 1−φf (t) has been implicitly assumed in the last equation.. 20.

(33) To account for the increase of cytoplasmic pressure upon the trypsinization right after time t=0, we define the external pressure as pout (t) =. pcytoskeleton + pcytosol − ptrypsinization if t ≤ 0 pcytoskeleton + pcytosol. (2.4). if t > 0. where pcytoskeleton represents the pressure exerted on the nuclear envelope by the cytoskeleton, predominantly by microtubules, pcytosol is the pressure exerted by the cytosol, and ptrypsinization correspond to the variation of pressure exerted on the membrane once the trypsinization occurs, which is assumed to take on positive values such that this process increases the total external pressure. We assume that mass is conserved at all times, and that only the fluid phase (nucleoplasm) can enter or leave the nucleus by being transported across the nuclear envelope, while the solid components remains inside the nucleus at all times. We describe the outward fluid velocity by q(t). Then, the statement of mass balance for the nuclear domain yields 4πR(t)3 d φf (t) q(t)4πR (t) = − dt 3 2. (2.5). The fluid velocity across the membrane is related to the fluid pressure gradient across the porous nuclear envelope through a Darcy-type linear constitutive equation of the form . k q(t) = (pfint − pcytosol ) µd. (2.6). where k is the intrinsic permeability of the nuclear envelope, µ is the dynamic viscosity of the fluid and d is the membrane’s thickness. For simplicity, in the sequel we define the apparent permeability as γ = k/µd, and assume that µ = 10−3 P a · s and d = 5 · 10−8 m, which are typical values for the cytosol dynamic viscosity (Liang et al. [2007] Liang. 21.

(34) et al. [2009]) and nuclear envelope thickness (Bajer & Mole-Bajer [1969]). The definition (6) implies a positive fluid velocity when the fluid is being transported outwards the nucleus to the cytoplasm. In order to complete the model we introduce constitutive equations that describe the pressure-volume deformation relation. We consider the nucleus at t=0 as the reference configuration, and express all variations of pressure and volume with respect to this reference state using the increment operator, which for an arbitrary function is defined as ∆f (t) = f (t) − f (0). Let V (t) = 4πR3 (t)/3 be the current total volume, and Vf (t) = φf (t)V (t) the current fluid volume inside the nucleus. Then, the constitutive laws for a porous elastic medium read M. Biot [1941], Detournay & Cheng [1993], Geertsma [1957], Schutjens et al. [2000, 2004], R. Zimmerman [1991], S. W. Zimmerman R. & King [1986]. 1 ∆V (t) f = − ∆pint (t) + α∆pint (t) V (0) K ∆Vf (t) 1 f = − α∆pint (t) + (α + φ(α − 1))∆pint (t) Vp (0) φK. (2.7) (2.8). where K is the apparent bulk modulus of the porous medium, Ks is the bulk modulus of the solid phase, and α = 1 − K/Ks . The governing equations just presented are complemented with initial conditions. In our case, we assume that at time t=0 the initial nucleus radius R0 = R(t = 0) and initial fluid volume fraction φf 0 = φf (t = 0) are known. Further, we assume that velocity across the nuclear envelope is negligible, i.e. q(0) = 0. (2.9). Thus, combining (1)-(8) we arrive at an ordinary differential equation for the nucleus radius of the form. 22.

(35) γ∆pint (R) Ṙ(t) = − R dφf + φf (R) 3 dR. (2.10). where 1 1 1 R3 = ptrypsinization + 2σ − −K 1− 3 α R R0 R0 1 1 ∆pint (R) = ptrypsinization + 2σ − R R0. ∆pfint (R). (2.11) (2.12). and p φf 0 R03 α−1 f φf (R) = ∆pint (R) + g(R) 1+ 2R3 K. (2.13). and g(R) is given by, g(R) =. 2 4αR3 α−1 f ∆pint (R) − (∆pint (R) − ∆pfint (R)) 1+ 3 K Kφf 0 R0. (2.14). It should be noted that, after considering the increments of the pressure instead of their absolute values, the current model equations do not need to specify initial values for the total and fluid-phase pressure, neither it is necessary to know the cytosol pressure pcytosol or the cytoskeleton pressure pcytoskeleton . Thus, the analysis of the variations of the nucleus total pressure and volume depends only on variations of the external pressure, i.e., the pressure increase due to trypsinization. In particular, the initial condition (9) together with (6) results in q(t) = γ∆pfint (t). (2.15). that is, the velocity of the fluid transported across the nuclear membrane depends only on the increment of the fluid-phase pressure. The fluid volume flow rate across the nuclear membrane is Q(t) = 4πR2 (t)q(t). (2.16). 23.

(36) Due to the complexity of the differential equation (10), a closed-form solution cannot be found. Therefore, numerical integration packages (Scipy, Python) were used to determine the time evolution of the nuclear radius. 2.2 Poroelastic model parameters There are 7 constants involved in the poroelastic model introduced in section 2.1. From elasticity, the apparent bulk modulus K can be estimated from the elastic modulus and the Poisson ratio, which were set to 4400P a and 0.48, respectively, according to values reported in the literature (Caille et al. [2002], Dahl et al. [2008], Guilak et al. [2000], Kha et al. [2004], Vaziri et al. [2006], Vaziri & Mofrad [2007]), yielding a bulk modulus value of to 36.66kP a. The surface tension of the nuclear membrane was set to 0.01P a · m, following (Boal [2001]). The initial value of the nucleus radius R0 was set equal to 4.9µm, as observed in experimental measurements. Finally, in the literature the porosity of the nucleus of different cells is found to be in the range of 70% to 90% (Century et al. [1970], Nagy et al. [1981], Yen-Chow et al. [1974]) and therefore we assumed an initial porosity φf 0 of 0.8, which implies that in normal conditions, 80% of the nucleus volume is considered to be water. Estimates for the remaining three parameters, namely the apparent permeability of the nuclear envelope γ, the Biot-Willis coefficient α and the pressure drop due to trypsinization ptrypsinization could not be found in the existing literature. Therefore, we consider these parameters to be free, and proceed to determine their values through data fitting for the case of trypsinization of MEF. For the case of trypsinization of MEF treated with Nocodazole, we assume the parameters and obtained in the previous case, and only fit the pressure variation due to trypsinization.. 3 Results 3.1 Trypsinization of mouse embryonic fibroblasts 24.

(37) F IGURE 2.3. Time evolution of the nuclear volume upon trypsinization.. Figure 2.3 shows the time evolution of the nucleus volume observed from experiments, reported by the mean value and standard error. Following (Kim et al. [2015]), we assume that the nucleus volume remained approximately constant after t = 2500 sec, showing some deviation from the mean but otherwise no departure from an asymptotic volume of roughly 380µm3 . A marked decay of the nuclear volume is observed during the first five hundred seconds, which we associate to a stage of complete trypsinization. The volume evolution, as dictated by the proposed poroelastic model after fitting the three free parameters is shown in Figure 2.3, while the parameter values are reported in Table 1. Once the nuclear radius as a function of time was computed, the increment in the fluid-phase pressure, the increment of the total pressure, the fluid volume fraction and solid volume fraction were determined from equations (11), (12), (13) and (2), respectively. 25.

(38) TABLE 2.1. Summary of parameter values and references for the poroelastic model. Parameter Bulk modulus K. Lipid bilayer surface tension Radius (Initial condition) φf (0) Porosity (Initial condition) Apparent permeability γ Biot Willis’ coefficien α Pressure increase Ptrypsin. Parameter value 36.6kP a. 0.01P a · m 5.65 · 10−6 m 0.8 10−12 m/P a · s 0.8 18 kPa. Source Caille et al. [2002] Dahl et al. [2008] Guilak et al. [2000] Vaziri & Mofrad [2007] Kha et al. [2004] Vaziri et al. [2006] Boal [2001] Assumed from data Assumed from data Fit to data Fit to data Fit to data. The evolution of the incremental total, fluid-phase and solid-phase pressure are reported in Figure 2.4. At time t=0 sec, we see a steep pressure increase in fluid-phase and total pressure, which is explained by the sudden increase of the external pressure initially accommodated solely by the nucleus solid and fluid phases, yielding an instantaneous total pressure increment of 18 kPa. As time evolves, the fluid-phase pressure drops and the solid-phase pressure rises, resulting in a monotonically increasing total pressure. As the system reaches steady state, the increment in fluid-phase pressure returns to zero, a necessary condition to reach a zero-flow state at the nucleus membrane, while the solid-phase pressure increases in 91.5 kPa. The total pressure increment at steady state is 18 kPa. The fluid velocity can be estimated from the increment in fluid pressure, as stated in equation (14). Figure 2.5 shows the resulting fluid velocity as a function of time. As expected, an external pressure increase results in positive values of the fluid velocity, i.e. fluid is transported outside the nucleus upon trypsinization. Initially, a sudden increase in fluid velocity is observed, followed by a smooth decay, which asymptotically takes the system back to an equilibrium state where the velocity across the membrane is zero In addition, the total fluid volume expelled by the nucleus at every time instant is shown in 26.

(39) F IGURE 2.4. Evolution of the solid phase, fluid phase, and total pressure inside the nucleus after trypsinization. After an initial stimulus, all pressure curves asymptotically converge to a steady state.. figure 2.5. We also observe that, after approximately 2500secs, the fluid volume expelled reaches a limit value of 300µm3 .. 27.

(40) F IGURE 2.5. Fluid expelled during trypsinization. The black solid line shows the time evolution of the fluid velocity at the nuclear envelope, where an exponential decay is observed after trypsinization. The red dashed line depicts the total expelled fluid volume at every time instant.. Trypsinization of mouse embryonic fibroblasts treated with Nocodazole Figure 2.6 shows the evolution in time of the nucleus volume observed from trypsinization experiments of cells treated with Nocodazole (Kim et al. [2015]), reported by the mean value and standard error. We observe a steady state with constant nucleus volume after t = 2500 sec, similarly to the case of cells not treated with Nocodazole. We see the same marked decay of the nuclear volume in the first 500 hundred seconds that we saw in the case without the treatment of Nocodazole. In addition, figure 2.6 shows the volume evolution obtained from the time integration of the poroelastic model, where a pressure drop value due to trypsinization of 15 kPa was found to give an excellent fit of the model to the experimental data. We remark here that the only parameter fitted in this case was 28.

(41) F IGURE 2.6. Time evolution of the nuclear volume during trypsinization in cells treated with Nocodazole.. the trypsinization pressure, whereas the values for the apparent permeability and the BiotWillis coefficient where the same as in the case of cells not treated with Nocodazole. Figure 2.7 shows the evolution of the fluid-phase and solid-phase pressure in the case of trypsinization of cells treated with Nocodazole. At time t=0, we see a fast increase in the total pressure, which can be decomposed as a fast increase in the fluid-phase pressure and a steady increase in the solid-phase pressure. As the system evolves we see that the pressure in the fluid-phase tends to zero to reach equilibrium with the fluid outside the nucleus. The solid-phase pressure increases because of the compression of the nucleus, reaching a value of approximately 78 kPa at steady state. Results for the water expelled in the case of trypsinization of cells treated with Nocodazole are reported in figure 2.8. Initially, we observe a sudden increase in the flux velocity 29.

(42) F IGURE 2.7. Evolution of the solid phase, fluid phase, and total pressure inside the nucleus after trypsinization for cells treated with Nocodazole. After an initial stimulus, all pressure curves asymptotically converge to a steady state.. across the membrane due to the exerted pressure, followed by a smooth decay towards a no-flux steady state.. 4 Discussion Mechanical interaction of the cytoplasm and the nucleus In this work, we have studied two different experiments (Kim et al. [2015]), and have proposed a mechanical model based on a poroelastic material assumption that captures the dynamic behavior of the nucleus shape. The first experiment consists on MEF that undergo trypsinization. In this process, trypsin degrades proteins eliminating the links between the cells and the substrate. Once the links are degraded, the tension that previously anchored the cells to the substrate no longer exists, letting them evolve to a more stable 30.

(43) F IGURE 2.8. Fluid expelled during trypsinization for cells treated with Nocodazole. The black solid line shows the time evolution of the fluid velocity at the nuclear envelope, where an exponential decay is observed after trypsinization. The red dashed line depicts the total fluid volume expelled by the nucleus at every time instant.. state. Cells evolve to a spherical state and have volume variations in the cytoplasm as well as the nucleus. Due to the elimination of the tension that anchored the cells, a new force balance is reached forcing variations in the osmotic pressure and the tensions in the cytoskeleton. In the second experiment, MEF were subjected first to a Nocodazole treatment, and afterwards follow a process of trypsinization. In this case, the microtubules are degraded generating a new balance of forces within the cell. With no microtubules, when Trypsin is applied, the transduction of forces on the nuclear membrane becomes hindered by a lack of internal structures. Kim et al (Kim et al. [2015]) argue that changes in osmotic pressure drive volume variations in the nucleus and proposed a model to explain these volume changes focusing in the mechanical role of the nuclear membrane to withstand the additional pressure. 31.

(44) A novel feature of the poroelastic model presented in this work is the assumption that the chromatin contained inside the nucleus plays a structural role in the mechanical behavior of the nucleus. Further, by construction, we explicitly consider the interaction of solid and fluid phases, and therefore we can account for a viscoelastic behavior not present in other nucleus mechanical models (Ateshian et al. [2007, 2006, 2013]). This allows us to consider that the final solid volume within the nucleus will contract relating to the physical phenomena of actin condensation. This process of actin condensation has been seen in several experiments in which the nucleus reduces its volume (Finan et al. [2011], Irianto et al. [2013]). Our model takes the condensation process has a compression that leads to a compaction of solid material and the reduction/elimination of pores in the genetic material. In addition to the compressibility of solid phase, our model differs from the models developed in the past with Mixture Theory because we focus in the mechanical consolidation that is seen after the load is applied on the nuclear envelope. Previous models use an approach from continuum mechanics similar to ours but have focused in studying the fluid response and final profile after osmotic shocks. Furthermore, this studies have been simulated the cytoplasm and the extracellular matrix ignoring the behavior of the nucleus (Ateshian et al. [2007, 2006, 2013]). Pressure variations inside and outside the nucleus One particular benefit from the model is the understanding of how would the pressure evolve within the system. This pressure evolution is shown in figure 3 and demonstrates that initially the fluid phase takes most of the load but as the system evolves, the solid phase takes most of the increase by deforming and contracting. All measurements are relative to a basal pressure. We remark here that the model only needs as input pressure differences, and therefore we cannot estimate absolute pressure. 32.

(45) values. However, it may be possible to combine these experiments with micropipette aspiration, which allows for the determination of absolute pressures in the cytoplasm. The value obtained for Ptrypsinization in the model is in the range of biological cells as can be found in the literature. In reference (Mahaffy et al. [2004]) is mentioned how the cell can withstand pressures of the order of 10 kPa by deforming in the large scale. In addition, the use of two data sets has provided an additional value: the pressure difference due to microtubule depolymerization with Nocodazole. In our analysis we have obtained 18 kPa for trypsinization MEFs and 15 kPa for trypsinization in MEFs pretreated with Nocodazole therefore, less pressure is needed to reach the stationary state. From this, we can infer that microtubules exert a pressure in the order of 3 kPa by contracting. Thus showing that the cytoskeleton dynamics affect strongly nuclear dynamics. Relevance of the initial state in the model. The initial values of the radius and the solid fraction are crucial to determine the rest of the parameters of the model. Once the initial solid fraction is established, the Biot Willis coefficient and the Young Modulus can be fitted to data to determine how flexible the genetic material is. Therefore, our selection of initial state is not relevant when analyzing orders of magnitude of properties or the behavior of the nucleus. In addition, an interesting feature of the model is that variations on the initial radius do not seem to be relevant under the constraint of low surface tension. All previous figures were generated with simulations that considered an average value of 5.65 · 10−6 m for the initial state of the nucleus. We generated the same figures varying the radius between the smallest nucleus in the data set and the biggest nucleus, and we found that no considerable impact appeared in the figures, thus supporting the hypothesis that the percentage of decrease in volume is independent of the nucleus size. Nuclear envelope porosity, and its role in the time scale of nuclear volume decrease.. 33.

(46) F IGURE 2.9. This graph shows variation in the value for the apparent permeability. We have considered for different values ranging from 0.2 times the reference value to 5 times the reference value.. A novel finding of this work is the determination of the nuclear membrane permeability, which takes a value of 5 · 10−13 µm2 . This value was found to be within the range reported in the literature (Jafari Bidhendi & Korhonen [2012], Jiang & Sun [2013]). To analyze the sensitivity of the model to the membrane permeability we performed simulations for variations of up to 20% in the permeability parameter, see Figure 2.9. We observe that the main feature controlled by the permeability is the decay rate of the volume, but the final nuclear volume remains unaffected by changes in the permeability values. This is to be expected, since the poroelastic response in steady state will only depend on the pressure increments and equilibrium, while the permeability will only modulate the transient behavior of the system. Sensitivity in Biot Willis Coefficient and Added Pressure 34.

(47) F IGURE 2.10. This graph shows variation in the value for the added pressure during trypsinization. We have considered reductions down to 6 kPa from the original 18 kPa that resemble the experimental data.. The sensitivity analysis of the Biot Willis coefficient is more complex than the other parameters given that we have realized how intertwined this parameter is with the bulk modulus. In rock mechanics, the volumetric variation tends to be smaller, which this two parameters independent. In our case, if we consider a change in the Biot Willis coefficient, we are forced to change the value of the bulk modulus by an unknown relationship. This is unknown since the relationship that the nucleoskeleton bulk modulus establishes with the young modulus has not been established. To use lower values of the Biot Willis Coefficient and not consider changes in the young modulus is contradictory by definition. The effect of changing the Biot Willis coefficient is therefore absorbed by the pressure curves and hence, by the membrane permeability. 35.

(48) In the case of the last parameter, we explored different values for the added pressure showing that smaller values for the pressure result in smaller deformations of the nucleus driven by less water flowing towards the cytoplasm. Notice that the minimum pressure value considered in figure 2.10 leads to a significantly smaller volume reduction. This indicates that within the framework that we have considered, any volume reduction in the nucleus around 10%-40% is generated with pressures in this magnitude order.. 36.

(49) Limitations of this study Poroelastic model are constructed based on the assumption of small (infinitesimal) deformations. In our case, volumetric strain reaches 50%, which may be considered as a moderate strain, and depart from the small-deformation assumption. In future attempts to model the nucleus mechanical response, modeling efforts should aim at considering a large-deformation poroelasticity model. Such models, though, will lead to highly nonlinear evolution equations, which may pose additional hurdles inherent to non-linear systems. The same consideration applies to the constitutive models and parameters considered in this work. Throughout the model derivation, we have considered the Biot-Willis parameter to be constant. This assumption is typically done in the study of poroelastic materials subject to small deformations, like soil, bone and metallic foams, to name a few. However, for moderate to large deformations, the apparent bulk modulus will necessarily change due to a change in its microstructure and phase densities a highly compressed porous material will have smaller amounts of fluid inside. This should be reflected in the Biot-Willis parameter, which would be more realistic by having a dependence on the porosity. Another limitation of this study is the focus on the nucleus as an isotropic system, rather than a spatially heterogenous media. This assumption allows us to simplify the spatial dependence of the poroelastic equations. However, chromatin is known to be a highly heterogeneous material, with substructures like nucleoli where chromatin density is much higher than the average nucleus density. The consideration of heterogeneous media necessitates a more complex modeling effort, where the nucleus interior should be spatially considered, and material properties should have a spatial characterization.. 37.

(50) 5 Conclusions In this work we have shown in a quantitative way that there is a tight mechanical interaction between the nucleus, the nuclear envelope and the cytoplasm. In particular, our model supports the hypothesis that the nucleus interior, i.e., chromatin, nucleoskeleton and nucleoplasm, plays a fundamental role in determining the morphological changes of the nucleus. In particular, we have shown that under changes in the cytoplasmic pressure, the nucleus behaves poroelastically. In particular, in this experiment we have seen one particular scenario of nuclear volume reduction but there are many more scenarios where this condition is present. In addition is crucial for the model to get a better understanding of the flux through nuclear pore complexes. In conclusion, we believe that a poroelastic model of the nucleus could serve as a theoretical frame to analyze different properties and could open new possibilities to understand why nuclear deformation is related to many diseases as cancer or muscular dystrophy. Understanding nuclear mechanics could also help understand how it is related to nuclear migration processes and other organelles such as the actin cap (Khatau et al. [2009, 2010], Kim et al. [2012], Starr [2003]). The establishment of the poroelastic model in a simple and rigorous way allows the development of stronger theories associated to physical processes in the biological context. We consider this biphasic porous model as a key step into the modelling of nuclear deformation, mechanics and mechanotransduction.. 38.

(51) 4. FURTHER DISCUSSION. One particular benefit from this mechanical model is that in order to observe a reduction in the nuclear volume, we need to observe a positive pressure difference. Once this pressure difference has been established as a step function, the equations lead to a time evolution that reaches a new stationary state. This allows to estimate a pressure difference applied to the nucleus from the experimental data. This is helpful considering that in several experiments, the use of certain drugs such as trypsin affect not only one but many pathways, making difficult to track what structure exerted the pressure. The poroelastic approach has been considered in biological sciences in a broad spectrum of structures: cells, extracellular matrix, tissue, bones, etc. Though it has draw attention, these model appear unorganized and lacking a unified description. The poroelastic approach is especially interesting in biological sciences because it can describe a biphasic description taking into account the behavior of the polymeric network present in cells or other tissues. Our model has focused in this biphasic behaviour focusing in the study of the nucleus in contrast to other models alike that have focuse in the cytoplasm behaviour Ateshian et al. [2007, 2006, 2013]. The most important difference in the approach that we have taken from previous articles is the consideration of active compressibility from the solid phase. In previous research using mixture theory, the solid phase is taken as an incompressible material. The fact that we consider compressibility for the polymeric network accounts for the condensation of chromatin, a phenomena observed experimentally when the nucleus reduces its volume Lanerolle [2011].. 4.1. Determination of constants, verification of the model In the article we have explained in more detail the procedure to obtain the constants for this model. To summarize this, initially, we obtained the 3 free parameters from the model 39.

(52) (α, Ptrypsinization and γ). Once these parameters were fitted to the first data set obtained from experiments, we proceeded to use our second data set corresponding to trypsinization on cells treated with Nocodazole. In this cells, the reduction in nuclear volume is less steep than it is in cells with their microtubules intact. Given that this variation is smaller, we varied the value of Ptrypsinization manually and reduced it to the value of 15kP a. Using the variation, we were able to resemble the experimental data with high precision and it allowed us to verify the solution obtained with the first data set. One discussion that was not included in detail in the article was the analysis over the fitted model and the last 3 points of our experimental data. At first, it seems possible that a better adjustment could be obtained, thus generating a better set of parameters to represent this behavior. Given the complexity of the model this is not that simple to obtain since there is an unknown relationship between the young modulus E and the Biot-Willis coefficient α. This relationship hinders the optimization process to obtain better fittings.. 4.2. What structure holds the external pressure? In our model we have considered 2 structures, the nucleoskeleton and the nuclear envelope. Once the nucleus’ volume starts decreasing, part of the pressure is taken by the membrane and part is taken by the solid phase within the nucleus. While analyzing the available data sets, we observed that the best fit is obtained at small values of σ (nuclear envelope’s surface tension). This result shows that in the poroelastic model that we have developed, the additional pressure is finally balanced in the steady state by an increase in the solid pressure within the cell. In figure XX, we show a summary of how different variables associated with the nucleoskeleton change in this model during compression. This argument opens a new possibility when trying to understand nuclear mechanics given that other current research claim that the nucleus’ interior is not relevant for its mechanics and behavior. 40.