Theoretical Investigation of Cell Polarity Initiation in the Early C. Elegans Embryo THESIS

Transcription

1 Theoretical Investigation of Cell Polarity Initiation in the Early C. Elegans Embryo THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Natalia Kravtsova Graduate Program in Mathematics The Ohio State University 2014 Master's Examination Committee: Professor Adriana Dawes, Advisor Professor Helen Chamberlin, Advisor

2 Copyright by Natalia Kravtsova 2014

3 Abstract Polarization is the process by which chemical species are unequally distributed throughout the cell creating biochemically distinct domains. In this study, mathematical modeling is used to theoretically determine what types of initial stimuli are able to initiate successful polarization. The model consist of five reaction-diffusion-advection partial differential equations for the main players in polarization process: ParA proteins (three forms), ParP proteins, and actomyosin protein complex. Equations are solved by method of lines, and data fitting is performed by linear and non-linear optimization methods (implemented in Matlab). Simulations of the model show that biochemical stimulus results in polarization of the embryo, while purely mechanical stimulus does not. In addition, regulation of actomyosin by inhibition from ParP and enhancement from ParA are studied, and it is found that inhibition from ParP plays more significant role in actomyosin regulation. Theoretical findings of this study suggest ways to direct biological experiments. ii

4 Dedication This document is dedicated to my family. iii

5 Acknowledgments I thank my advisor Dr. Adriana Dawes for all the knowledge I acquired in the field during our research and course work and her support in all my activities during graduate studies. I thank my advisor Dr. Helen Chamberlin for all the knowledge I acquired from our research meetings and coursework and her support during my graduate studies. I thank Department of Mathematics of The Ohio State University for providing me the opportunity to study in the program. iv

6 Vita June M. A. Musicology, Moscow State Conservatory 2012 B. S. Mathematics, The Ohio State University 2012 to present...graduate Teaching Associate, Department of Mathematics, The Ohio State University Fields of Study Major Field: Mathematics v

7 Table of Contents Abstract... ii Dedication... iiii Acknowledgments... iv Vita... v List of Tables... vii List of Figures... vviii 1. Introduction and Biological Background Polarity Initiation.2 2. Mathematical Model Previous Models Model Assumptions Model Variables and Equations Numerical Methods Results Discussion..26 References..29 vi

8 List of Tables Table 1. Parameter Values for Simulations... 8 vii

9 List of Figures Figure 1. Polarized Embryo Fluorescent Micrograph... 1 Figure 2. ParA/ParP interactions.3 Figure 3. Schematic Diagram of Polarization Process... 4 Figure 4. Oligomerization of ParA... 8 Figure 5. Domain Geometry Assumption... 9 Figure 6. Sample Initial Distributions...17 Figure 7. Results from Different Initial Stimuli 18 Figure 8. Meaning of ParA/ParP strength parameters..19 Figure 9. ParA vs. ParP strength combintations...20 Figure 10. Stimulus Time vs. Stimulus Strength..20 Figure 11. Stimulus Time vs. Spatial Extent 21 Figure 12. Relationships between Diffusion and Advections Coefficients..22 Figure 13. Actomyosin Regulation 23 Figure 14. Sensitivity Analysis..25 viii

10 1 Introduction and Biological Background Polarization is the process by which chemical species are unequally distributed throughout the cell creating biochemically distinct domains. Depending on the cell type, polarization can define anterior/posterior, apical/basal, dorsal/ventral and other types of distinct regions [18]. Figure 1 shows anterior/posterior domains in the zygote of nematode worm Caenorhabditis elegans (C. elegans). Establishment of anterior/posterior domains enables a cell to move in a certain direction, divide asymmetrically for further differentiation, and perform other important biological functions that depend on polarization [18, 6]. Therefore, studies of cell polarization are important for understanding how cells move and divide, and for identifying therapeutic targets for diseases such as metastatic cancer that rely on polarization. Figure 1: Fluorescent micrograph of a polarized early embryo of the nematode worm C. elegans (from [18]). Note the distinct spatial domains indicated by the green and red fluorescent proteins. Key players in polarization process are partitioning defective (PAR) proteins that are conserved over a variety of species [18]. PAR genes were first identified in nematode worm C. elegans in 1983 [6], and this organism still serves as a valuable model to study PAR protein distributions at the single cell level. In this work, we focus on polarization in C. 1

11 elegans zygote. During this process, PAR-3, PAR-6, and atypical protein kinase C (apkc) accumulate at the anterior part of the cell, so in this work we will call these proteins ParA. Another set of proteins, PAR-1, PAR-2, and lethal giant larvae-like 1 (LGL-1) accumulate at the posterior pole, and we will call these proteins ParP. Both sets of proteins are associated with the cortex, a thin contractile layer under the cell membrane composed of actin polymers crosslinked by the contractile protein myosin ([9, 16]). ParA and ParP domains are complementary to each other because of mutual phosphorylation of domain components. It has been shown experimentally that PAR-1 is phosphorylated by apkc on threonine 595 residue, which results in dissociation of PAR-1 from anterior cortex [10], PAR-2 is excluded from anterior cortex by apkc-dependent phosphorylation [7] and LGL is phosphorylated by PAR-6/aPKC complex on multiple residues [17]. It has been also proposed that PAR-2 inhibits PAR-3 [7], and LGL-1 causes anterior PAR complex to leave the cortex [8]. Figure 2 schematically depicts the relationships between ParA and ParP proteins. Polarization is subdivided into two distinct phases: establishment and maintenance [2]. At the beginning of the establishment phase, ParA occupies the whole cortex, while ParP is mostly in cytoplasm. At fertilization, sperm provides a cue that excludes ParA from the cortex of the posterior pole; as a result, complementary ParP accumulates at the posterior. Maintenance phase is characterized by maintaining polarity due to mutual inhibition of ParA and ParP proteins [2]. 1.1 Polarity Initiation Establishment of polarity happens after the sperm enters the egg. The sperm provides microtubule-organizing center (MTOC) and a male pronucleus to the existing egg with a 2

12 Figure 2: ParA/ParP interactions: inhibition from ParA causes ParP to dissociate from the cortex, and vice versa (from [4]) female pronucleus. Approximately 30 minutes after fertilization, the MTOC positions near the cortex and contacts it [1]. This event initiates PAR proteins polarity establishment. The sequence of events during establishment of polarity is depicted on Figure 3. However, the nature of this initiating stimulus provided by sperm MTOC is not completely understood: Is it a mechanical weakening of the cortex or is it biochemical interactions of players that initiate polarity? Mechanical weakening of the cortex was proposed to be a polarity initiating mechanism [16]. It was based on the observation that when the sperm enters the egg, MTOC from the sperm disrupts previously symmetric cortical meshwork, which causes local weakning of the cortex. This weakening leads to flow of the cytoplasm away from regions of low tension thus creating tension gradient that propagates through the domain [16]. In contrast, biochemical nature of the stimulus that results in local depletion of ParA 3

13 Figure 3: Schematic diagram of polarization process from the sperm entry until the first cell cleavage (from [3]) and/or enhancement of ParP was proposed ([15], [11]). It was observed that GFP-tagged ParA proteins (PAR-6 and PAR-3) were initally excluded from the cortex near the site of sperm entry, and then migrated toward the anterior part of the embryo ([15]). It was proposed that biochemical signal that triggers ParA redistribution may come from the protein CDC-42 ([11]). Later studies suggest another mechanism of biochemical nature that relies on ParP initial enhancement rather than ParA depletion ([14]); this mechanism suggests the following sequence of events. PAR-2 binds to microtubules, and high microtubule density near the cortex protects PAR-2 from phosphorylation by apkc. This results in initial recruitment of few PAR-2 molecules to the cortex, and then additional recruitment facillitated by some positive feedback. PAR-2 then recruits PAR-1, which phosphorylates PAR-3. Phosphorylation results in exclusion of PAR-3/aPKC complex from the cortex and expanding of PAR-1/PAR-2 domain. Cortical flow displaces ParA from the posterior, and 4

14 microtubule binding is not required anymore. Additional recent evidence for biochemical nature of the stimulus was shown in [20]: it was noted that PAR-2 locates on cortex and excludes PAR-3 before myosin flows are visible, suggesting that myosin asymmetry is not required for polarity initiation. In this study, mathematical modeling is used to theoretically determine what types of stimuli are able to initiate successful polarization. Simulations of the model show that biochemical stimulus results in polarization of the embryo, while purely mechanical stimulus does not. In addition, regulation of actomyosin by inhibition from ParP and enhancement from ParA are studied, and it is found that inhibition from ParP plays more significant role in actomyosin regulation. Theoretical findings of this study suggest hypotheses to be tested experimentally. 5

15 2 Mathematical Model 2.1 Previous Models Two mathematical models were built previously to specifically address polarization in C. elegans. Both models consist of partial differential equations, and both attempt to explain how polarity is established. A model proposed by [19] consists of four reaction-diffusion equations describing the dynamics of cortical and cytoplasmic ParA and ParP. Actomyosin network is modeled as 1D Hookean spring that extends from the anterior pole (x = 0) to some point x = x 0, and its density is uniform at that region (represented by constant a). At each time step, the value of a is calculated and incorporated into ParA equations serving as enhancement for ParA cortical binding. However, this model cannot account for establishment of the boundary between ParA/ParP domains that is essntial for polarization process. Model used in this study, in turn, is capable of explaining how the boundary between domains can be held at the specific place. Two-dimensional version of the current model is publisehd previously (see [3]); it is shown (in two-dimensional case) that the boudnary between ParA/ParP domains sets at a specific place because of certain domain geometry characteristics. Onedimensional version used in this study ignires this aspect, but all other features of the model are preserved. Another model is proposed by [5] and uses reaction-diffusion-advection system of two equations (for ParA and for ParP). Cortical flow is represented by a linear advection term where advection velocity is constant. For instance, in ParA equation, advection term is (νa). The drawback of this approach is that advection is imposed to the system. Indeed, structure of the advection term is not mathematically justified, but rather it is said that 6

16 the simplest case was considered ([5]). This creates a problem of obtaining model results dictated by the model construction rather than the process the model describes. That might be a reason why simulations of the model were not used as predictive tool in [5]. Model used in this study is very different in that aspect: advection term is based on local actomyosin gradient, which makes cortical flow to be endogenous property of the model. This allows to use model simulations as predictive tool rather than only construct that matches the observed data. Model presented in this work is a slightly modified version of the model proposed in [4] and further developed in [3]. 2.2 Model Assumptions 1 1. ParA proteins oligomerize [4]. It is assumed for simplicity the largest oligomer to be a dimer. Figure 4 schematically depicts ParA dimerization. 2. ParA and ParP proteins are mutually inhibitive (see Introduction and Biological Background section for details) 3. ParP inhibits recruitment of myosin to the cortex [16] and ParA enhances actomyosin recruitment [20]. These interactions are represented by non-linear functions of ParA and ParP in kinetic part of actomyosin equation (5). This makes the current model different from [3] where only ParP inhibition was considered 4. Cytoplasmic concentrations of all proteins are large compared to cortical concentrations, and thus are assumed to be at quasi steady state [4] 5. All kinetic interactions follow mass action kinetics [4] 1 For further justification of assumptions and equation construction see [4, 3] 7

17 A1 A10 A11 Figure 4: ParA is simulated in the following 3 forms: A 1, cortical monomer, A 10, cortical dimer bound by one side, and A 11, cortical dimer bound by 2 sides. Diagram represents kinetic interactions between these 3 forms reflected in the kinetic terms of the model 6. In this work, the egg is assumed to be 1D object that extends from the anterior (x = 0) to the posterior (x = L) (Figure 5) 7. All proteins diffuse on the cortex; ParA and actomyosin diffuse with equal rates, while ParP diffuses sightly slower [4] 8. All proteins advect with advection velocity linearly proportional to cortical tension, while tension being linearly proportional to actomyosin gradient [3]. That is, v adv = ν (actomyosin), where ν is a constant 2.3 Model Variables and Equations Model variables are: 8

18 Anterior pole Posterior pole X=0 X=L Figure 5: Illustration of domain geometry assumption: Three-dimensional embryo (an ellipsoid) is represented by one-dimensional object (a line). This way, anterior-posterior directions are preserved, while other directions are omitted (model does not consider movement of proteins along other axes) A 1 : Cortically bound ParA monomer A 10 : Singly cortically bound ParA dimer A 11 : Doubly cortically bound ParA dimer P : Cortical ParP M: Cortical actomyosin Cytoplasmic concentrations of ParA monomers, ParA dimers, ParP, and M are represented by A y, A 2y, P y, M y, respectively. Based on the assumptions in 2.2, system of five reaction-diffusion-advection equations is constructed [4, 3]: 9

19 A 1 = k A t ona y koff A A 1 2k + d A k d A 11 k + d A ya 1 + k d A 10 r A P A 1 + D a A 1 ν (A 1 M) (1) A 10 t = k A10 on A 2y k A off A 10 k A on11a 10 + k A off A 11 k d A 10 + k + d A ya 1 r A P A 10 + D a A 10 ν (A 10 M) (2) A 11 t = k + d A2 1 k d A 11 k A off A 11 + k A10 on 2r A P A 11 + D a A 11 ν (A 11 M) (3) P t = kp onp y k P off P r p (A 1 + A A 11 )P + D p P ν (P M) (4) M t = konm M k p y ( K p + P + k aa K a + A ) km off M + D m M ν (M M) (5) Since proteins do not escape from the egg, no flux boundary conditions at x = 0 and x = L are imposed for all variables. Model is nondimensionalized using the following scalings: τ = 1, x = L, koff A Ā1 = A 10 = A 11 = A y, P = P y, M = M y. Since A y, P y, M y are the largest possible embryonic concetrations of ParA, ParP, and actomyosin, respectively, all output values in the solution would have values between 0 and 1. Also, koff A, constant of dissociation of ParA from the cortex, is assumed to be the largest rate constant since, once bound, proteins prefer to stay bounded to the cortex; therefore, time nondimensionalized time variable τ represents a fast time. Nondimensionalized space variable x also has values values between 0 and 1 as a result of scaling by egg length L. Nondimensionalized model equations are: 10

20 A 1 = b 1 A 1 2b 2 A b 3 A 11 b 2 A 1 + b 3 A 10 b 4 A 1 P t + D 1 A 1 µ (A 1 M) (6) A 10 t = b 5 A 10 b 3 A 10 + A 11 b 6 A 10 + b 2 A 1 b 4 A 10 P + D 1 A 10 µ (A 10 M) (7) A 11 t = b 2 A 2 1 b 3 A 11 A 11 + b 6 A 10 2b 4 A 11 P + D 1 A 11 µ (A 11 M) (8) P t = b 7 b 8 P b 9 (A 1 + A A 11 )P + D 2 P µ (P M) (9) M t b 11 = b 10 ( b 12 + P + b 13A b 14 + A ) b 15M + D 3 M µ (M M) (10) Values of parameter grouping are presented in the table below. 11

21 Table 1: Parameter values for simulations Parameter Dimensional Grouping Parameter Value b 1 k A ona y /koff A b 2 k + d A y/koff A b 3 k d /ka off b 4 r A P y /koff A b 5 b 6 k A10 on A 2y /k A off k A11 on /k A off b 7 k P on/koff A b 8 k P off /ka off b 9 r p A y /koff A b 10 k M on/koff A 0.1 b 11 k p /P y 0.1 b 12 K p /P y 0.01 b 13 k a 0.1 b 14 K a /A y 0.01 b 15 k M off /ka off D 1 D a /koff A L D 2 D p /koff A L D 3 D m /koff A L µ νm y /koff A L Numerical Methods Simulations of the model were performed using MATLAB software. 12

22 Parameters for simulations were chosen in two steps. First parameter sets for kinetic terms only were tested to find parameters that result in bistable kinetics. Matlab root finding function fsolve was used to find steady states with values between 0 and 1. Parameter set was proposed to be a good candidate for bistable kinetics if the following three roots existed simultaneously: ˆ high ParA, low ParP, high actomyosin ˆ middle values of ParA, ParP, actomyosin ˆ high ParP, low ParA, low actomyosin Code for this procedure is provided in Matlab Codes section, code 1 (Matlab codes can be obtained from the author). Second step was to check whether the candidate set produces travelling wave solution. This is done using the main code of this study described below. Equations were solved by method of lines, where space was discretized by central differencing for both diffusion and advection terms, and resulting system of ODE s was passed to MATLAB function ode23s for integration with respect to time variable. No flux boundary conditions are implemented by the following formula (for generic variable u): u(end + 1) = u(end 1) for the right boundary and u( 1) = u(2) for the left boundary, where u(end + 1) and u( 1) are extended values for discretized vector u(1 : end). Code 2 in Matlab Codes shows the function file that solves the system (note: since Matlab is not very fast in loops, everything is vectorized). Holding the stimulus at the last 10 percent of the domain was implemented by setting time derivative to zero at that portion of the domain for a variable of interest, while solving regular system on all other portions of the domain. For example, to represent holding the 13

23 stimulus for all ParA variables, on the last 10 percent of the domain system is defined as A 1 t A 10 t = 0 (11) = 0 (12) A 11 t = 0 (13) P t = b 7 b 8 P b 9 (A 1 + A A 11 )P + D 2 P µ (P M) (14) M t b 11 = b 10 ( b 12 + P + b 13A b 14 + A ) b 15M + D 3 M µ (M M) (15) On the first 90 percent of the domain regular system (6) - (10) is solved. This requires additional boundary condition at the interface between the site of stimulus action and the rest of the domain. This right boundary condition for the variable u responsible for the stimulus is u(x 0, 0) = const where const equals to the value held as a stimulus. The code that calculates the solution when stimulus is held for a certain amount of time works in two steps: 1. calculate the solution for some time t hold representing time to hold the stimulus. During this time, temporal derivative for the last 10 percent of the domain is 0, while for the rest 90 percent of the domain it is equal to regular spatially discretized system as in Code Solution resulting form previous step is stored as new initial conditions, and then used for the rest of simulation time when soling regular system as in Code 2. 14

24 Code 3 provides functions for step 1 of this procedure (where system (11) - (15) is solved on the last 10 percent of the domain while the on rest of the domain system (6) - (10) is solved). Since holding time and strength of the stimulus was variable, the model can be rewritten with heaviside functions representing turn on and turn off the stimulus at the last 10 percent of the domain for a variable of interest u: u t = M hold + (M reg M hold )H(t t ) In this notation, t represents the final time until stimulus of interest is held, M reg stands for model (1) and M hold stands for model with holding stimulus such as (11) - (15). Several times during this study the data obtained from simulations was fitted to a certain functional form. The parameters for fitted function (function proposed by a user) are found by non-linear optimization using Matlab function lsqcurvefit. Sample code is provided in Code 4. Sensitivity analysis was used in this study to quantify how much actomyosin gradient would be affected by random change in parameter values for ParA and ParP term (see Results section). Procedure to perform sensitivity analysis is described in [13]. Procedure is based on Latin Hypercube method and is outlined below. Suppose there are two parameters of interest in the model. Then, range of interest for these parameters is divided into N subintervals. For each simulation result, subinterval was chosen randomly for each parameter, and actomyosin gradient was calculated for randomly chosen parameter value in that subinterval. In this particular case, there are two random parameter values for each actomyosin gradient output, and other parameter values are kept fixed. Then, correlation coefficient was calculated between each vector of random parameter values and actomyosin gradient vector (thus, two correlation coefficients per simulation, one 15

25 for each parameter). From known distribution of correlation coefficient, statistical test was conducted to determine how significant each coefficient is. Caclulating correlation coefficient and it s p-value is left for Matlab function corrcoef, while other parts are coded manually and presented in Code 5. Finally, the main code that defines parameter values and initial conditions and calls the solver to solve the system can be found in Code 6. 16

26 3 Results 3.1 Type of stimulus affects existence of travelling wave Two main types of initial conditions representing initial stimuli are shown in Figure 6. Figure shows sample initial depletion of a protein (ParA or actomyosin) and sample initial enhancement of a protein (ParP). Initial depletion of ParA and initial enhancement of ParP represents biochemical stimulus, while initial actomyosin depletion represents mechanical stimulus as referred later in this work protein level protein level egg length egg length Figure 6: Sample initial distributions used as initial conditions in simulations. Initial depletion of a protein (a) is used to represent local depletion of actomyosin (mechanical stimulus) and ParA (biochemical stimulus). Initial enhancement of a protein (part (b)) is used to represent local enhancement of ParP (biochemical stimulus) Simulations show that only biochemical stimuli are sufficient to initiate travelling wave. Figure 7a shows no wave propagation resulting from pure physical stimulus of local actomyosin depletion, and Figure 7b shows wave propagation resulting from biochemical stimulus that locally depletes ParA or locally enhances ParP. Biochemical stimuli are subdivided into 2 categories: ParA depletion and ParP enhance- 17

27 protein level time space (a) No wave propagation from initial actomyosin depletion (actomyosin profile) protein level space time (b) Wave propagation from initial ParA depletion (ParA profile) Figure 7: Results from physical and biochemical stimuli ment. Parameter p is introduced to represent the relative strength of the stimuli. For instance, p a = 1 used in ParA initial distribution means that ParA protein levels are depleted to their low steady state, and p = 0.5 means depletion to half-distance between high and low steady states (so the lower ParA level in the region of acting stimulus is described by equation high p a (high low)). Similarly for ParP, p = 0.2 would mean that ParP is 18

28 enhanced to 20 percent of the distance between low and high steady state (so the higher ParP level is the region of acting stimulus is described by low + p p (high low)) (Figure 8) Figure 8: Illustration of the meaning of p a and p p in initial conditions (from [12]) Simulations show that there exist a set of ParA/ParP strengths combinations under which travelling wave is possible; but, travelling wave is not initiated if each of ParA or ParP stimuli acts alone, in the absence of the other. Results are shown in Figure Longevity of stimulus affects existence of travelling wave It was observed that even a relatively weak biochemical stimulus can generate propagating wave if held for a certain amount of time at the site of action (last twelve percent of the domain in this simulation). Figure 10 shows minimum amount of time required for a stimulus of certain strength to be held in order to start travelling wave. Strength of the stimulus is 19

29 ParP strength no wave wave ParA strength Figure 9: Minimum ParP strength to initiate wave at given ParA strength represented by the value of parameter p introduced above pa data pa exp fit pp data pp exp fit 25 min tcue enhancement/depletion level Figure 10: Holding time for a stimulus depends on stimulus strength Simulation shows that minimum time to hold the stimulus (t cue ) decreases exponentially when strength of the cue increases. Best exponential fits for ParA and ParP data are tcue a = 0.54e pa 22.9 and tcue p = 0.56e 0.81 pp, respectively. This implies that time to hold the stimulus for both ParA and ParP decreases exponentially with strength of the stimulus. 20

30 3.3 Spatial extent of stimulus affects stimulus longevity required for existence of travelling wave It was observed that spatial extend of the cue can change holding time required for wave propagation. To quantify the effect, small cue strength was considered (pa = 0.76, the smallest possible ParA strength for wave propagation from previous simulation, and pp = 0.2, similar argument). For this strength, minimum holding time required for wave propagation was recorded for different spatial extents of the cue (x cue was ranging as last percents of the domain). Numerical data is shown in Figure 11. These data suggest exponential dependence of the time of cue on spatial extend of stimulus. Equations for ParA and ParP fits are, respectively, t cue = 17261e 59xcue t cue = 834.7e 33.9xcue ParA data ParA fit ParP data ParP fit 25 tcue xcue Figure 11: Minimum stimulus time varies with spatial extent of the cue Simulation shows that time to hold the stimulus can get smaller if the spatial extent of the stimulus gets larger. For a stimulus that depletes ParA holding time can be close to zero (but never actually zero); for a stimulus that enhances ParP, in turn, holding time stops 21

31 changing when reaching 12.3 seconds (of simulation time). That is, there exists a point in the domain after which ParP stimulus s spatial extend has no more effect on time to hold the stimulus for wave propagation. For ParA there is no such point: spatial extend of the stimulus keeps effecting holding time over the whole domain. 3.4 Existence of the left-moving wave depends on relationship between diffusion and advection coefficients To establish relationship between diffusion and advection, certain values of diffusion coefficient D were considered, and minimum values of advection coefficient µ required for left-moving wave were recorded (Figure 12). Diffusion coefficient of ParA (D1) was used for that purpose; diffusion coefficient for myosin was taken to be the same (D3 = D1), and diffusion coefficient of ParP was taken to be slightly smaller (D ), as proposed in original model [4] data exponential fit 0.04 mu D x 10 3 Figure 12: Relationship between D and mu for left-moving wave: for each value of diffusion coefficient, there exist a minimum value of advection coefficient that is necessary for travelling wave existence 22

32 Simulation shows exponential relations between D and µ, where the larger is D, the larger µ is required for the wave to propagate to the left (µ = e (556.9D) ). Below this value of µ, the wave may propagate to the right (or does not propagate at all, and system returns to it s symmetrical state). Theoretical foundation of this result is discussed in Discussion section. 3.5 Actomyosin is regulated by ParA and ParP Proposed regulations for actomysin are positive feedback from ParA and negative feedback from ParP. To determine which of these effects plays a greater role in actomysin regulation, influence of ParA and ParP terms in the model was varied, and effect on travelling wave existence was observed. Figure 13 shows values of constants in front of ParA and ParP terms in actomyosin equation for which travelling wave propagation is possible. Simulation shows that even weak wave ka no wave kp x 10 3 Figure 13: ParA and ParP influence on actomyosin: for each value of the coefficient in front of ParA term in actomyosin equation, there is a minimum value of ParP coefficient under which travelling wave is possible 23

33 ParP regulation of myosin is enough for polarity establishment. Regulation from ParA, in turn, must be several orders of magnitude higher than regulation from ParP in order for left-moving wave to exist. In fact, constant in front of ParA term in myosin equation (ka) can be zero, while constant in front of ParP term (kp) can never be zero for a wave to propagate to the left. There is simply no value for ka that would result in desired wave if kp is zero. 24

34 3.6 Myosin gradient is more sensitive to ParP than ParA Sensitivity analysis was performed to quantify how sensitive the model output is to the constants tested above. Method is described in Numerical Methods section. Parameters to vary are constants in front of ParA and ParP terms in actomysin equation. The range of ka and kp was taken such that travelling wave exists under any ka/kp combination from the range. Range of each parameter of interest was divided into 1000 subintervals, and each subinterval was sampled exactly once. Other parameters were kept fixed. Model output was created for each combination of parameters, and correlations between parameters of interest and response variable were calculated. Results are presented in Figure [corrcoef,pvalue]= [corrcoef,pvalue]= e actomyosin gradient actomyosin gradient ka values kp values (a) Actomyosin sensitivity to ParA (b) Actomyosin sensitivity to ParP Figure 14: Actomyosin sensitivity to ParA and ParP Sensitivity analysis shows that actomyosin gradient has almost perfect correlation with value kp (p-value for statistical test is ), while value of ka does not significantly affect actomoysin gradient (p-value is 0.05). Implications of this result are discussed in the Discussion section. 25

35 4 Discussion Simulations of mathematical model were performed to suggest the nature of a stimulus that results in successful polarization of early C. elegans embryo. In addition, model was used to theoretically determine the most effective mode of actomyosin regulation. These theoretical results lead to hypotheses that can be tested biologically. Model suggests that purely mechanical stimulus is not enough to initiate polarization (Figure 7) thus arguing against the hypothesis of [16] about mechanical weakening of the cortex as a primary force that drives polarity initiation. In contrast, simulation supports findings about biochemical stimulus that locally decreases ParA ([11], [15]) or locally increases ParP ([14], [20]). Hypothesis about mechanical vs. biochemical nature of the stimulus can be experimentally tested in the following way: local inhibition of actomyosin activity can be implemented while keeping other protein expressions constant. Then, polarity of an egg can be analyzed. It is natural to assume that both biochemical stimuli might act together with different strengths: Figure 9 shows minimum relative strengths possible for each biochemical stimulus in order for polarity to be established. Simulation shows that ParP enhancement alone is not able to initate polarity, but together with small ParA depletion such combined biochemical stimulus is effective for successful polarization. This hypothesis can be tested experimentally by locally depleting ParA and/or enhancing ParP and observing their effects on polarity establishment. However, such an experiment might be time and resource consuming, so simulation of the model can help to direct experimental efforts. Model suggests that polarization is possible even when the strength of the stimulus is small, under the condition that the stimulus acts for a certain amount of time (different for ParP and ParA stimuli, Figure 10). This prediction is consistent with [1]. They find 26

36 that the signal coming from centrosome is needed for only a certain period of time, after which deactivating the centrosome via laser ablation does not stop polarization. Current study identifies minimum simulation time needed for a stimulus to persist and provides a fitted functional form of the dependence between strength of the stimulus and minimum stimulus time (Figure 10). To obtain this dependence experimentally, a large amount of experimental work and resources would be needed. The model of the current study allows to reduce the cost of this experimental work by providing a computer simulation that suggests a dependence between minimum time and strength of the stimulus. Model suggests that there exist a relationship between diffusion and advection coefficients. Figure 12 shows that diffusion must be smaller than certain threshold in order for advection to dominate and the wave to propagate to the left. This is intuitive from the biological perspective as well, since cortical flow must be strong enough to overcome the proteins to diffuse out of the point of interest. Mathematically this relationship is analyzed in [12] where precise threshold for diffusion coefficient is determined. In that work, current 5-variable system is reduced to 1-variable; then, transformation to moving coordinate z = x + ct is performed, and resulting second order ODE is rewritten as a system of two first order ODE s. From this system, the eigenvalues of the Jacobian at the steady state were found in terms of D and µ, and since eignevalues must be strictly negative for a stable node to exist, relationship between D and µ was derived. It was found that D < cµ, where constant c is one of steady states of 1-variable model. These results are confirmed numerically in this study. Additional results of this study are concerned with actomyosin regulation, which is described by kinetic terms in equation (10). Previous works focus on either positive feedback from ParA [16] or negative feedback from ParP [20], but never consider which feeddback is 27

37 more essential for creating and maintaining actomyosin gradient. Simulations of the current work suggest that inhibition of actomyosin by ParP is more important than enhancement by ParA. This conclusion is based on two simulation results presented in Sections 3.5 and 3.6. Figure (13) shows that even without positive feedback from ParA (ka = 0), polarization is possible, while negative feedback from ParP is necessary for initiation of polarity (kp cannot be zero). This result is consistent with the next simulation (14) that shows actomyosin gradient to be sensitive to ParP negative feedback and not sensitive to ParA positive feedback. The proposed hypothesis from these results is that only one feedback loop (negative regulation by ParP) is sufficient to initiate polarity. This hypothesis can be tested by blocking ParA - actomysin feedback loop and observing whether polarization happens. The main limitation of this study is outlined below; it proposes a direction for future work. There are other proteins associated with ParA and ParP that can affect their function and play a role of upstream regulators of proteins from ParA and ParP complexes. Previous works focus on ECT-2, RHO-1, and CDC-42 that are shown to interact with Par proteins ([15]). For instance, ECT-2 was shown to be depleted at the posterior part of the cortex and enriched at the anterior part shortly after the beginning of polarization; it was proposed that ECT-2 was acting upstream of PAR-3 ([15]), which is a part of ParA group in denoted in this study. RHO-1 was also placed downstream of ECT-2 along with ParA proteins ([15]). CDC-42 was shown to be required for ParA localization at the cortex ([11]). Relationships between Par and other proteins might be important for understanding polarization, thus future works might need to include these interactions. 28

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