Phase Response Optimization of the Circadian Clock in Neurospora crassa

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2 Phase Response Optimization of the Circadian Clock in Neurospora crassa A dissertation submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requierements for the degree of Doctor of Philosophy in the Deparment of Mathematical Sciences of the McMicken College of Arts and Sciences by Jacob Bellman M.S. University of Cincinnati, Cincinnati OH 2015 B.S. Miami University, Oxford OH 2010 December 2015 Committee Chair: Dr. Sookkyung Lim

3 Abstract A circadian rhythm is a fundamental biological process observed in many organisms. Circadian oscillations play a vital role in maintaining the daily activities of 24 hours. Dysfunctions of this process can be dangerous to an organism, and even life threatening. In this research, analysis is performed on mathematical models of the circadian clock in order to reveal unknown features of the clock, in particular, its response to external stimuli. Both simulations and experiments are performed for cross-validation using the model organism, Neurospora crassa. A novel approach of this project is the use of the phase response curve to identify chemical reactions responding to external inputs such as light administration. The results found in this study may provide useful information for potential treatments for circadian related diseases such as sleep disorders. ii

4 c Copyright by Jacob Bellman 2016 iii

5 Dedicated to my lovely wife Michelle iv

6 Preface The following dissertation is the result of a three-year study committed to correctly modeling the dynamics of the circadian clock in response to light administration. Circadian clocks are extremely important for the well-being of an organism and they are sensitive to environmental cues, especially light. In order to understand how to optimize the performance of the clock, it is essential to understand precisely how the clock responds to the environmental cues capable of entraining it. Motivation Previous studies have modeled the circadian clock, but few have focused on the dynamics of phase response to light cues. Of the studies that highlight these dynamics, there are contradictory findings and missing links. This study seeks to correct these inconsistencies, and lay out a well-motivated, well-supported explanation of the inner workings of circadian phase response. Outline The six chapters of this dissertation collectively focus on developing a mathematical model of the Neurospora circadian clock capable of accurately capturing the phase response of the clock. The first chapter introduces the biological and mathematical background that will serve as a foundation for the concepts developed in this dissertation. The three sections of this chapter describe the biological details of the molecular mechanisms of the circadian clock, the mathematical basis of modeling these dynamics and mathematical techniques utilized in phase response analysis. Chapter two is an introduction to phase response analysis using extremely simple models. After an introduction, the proceeding three sections of this chapter highlight the possible phase responses from applying negative and positive pulses of variables in a 2D oscillating model. These results can often be applied when studying the phase v

7 response of more complex models. In chapter three, phase response analysis is performed on simple two and three-variable circadian models, a complex circadian model consisting of 9 variables and more than 24 parameters, and a unique four-variable model developed in this study. One of the most important results from this research is concluded in this chapter; the correct phase response curve cannot be achieved by only pulsing frq transcription. Chapter four describes both of the phase response experiments performed in this study. The first experiment reproduces previously acquired data of Neurospora s phase response to light, stressing the phase response curve shape pursued mathematically in this study. The second study is the first experiment to capture the Neurospora s phase response curve in response to pulsing frq transcription. Chapter five focuses on constructing the correct phase response curve as an optimization problem. The first two sections construct the theoretical framework of the mathematical problem, as well as infinitesimal approximations of the phase response curve. The third section of this chapter approaches the optimization problem using various strategies on multiple Neurospora circadian models. Chapter six reviews the results and findings of this study and revisits the conclusions made. The second section of chapter six proposes future endeavors in line with these studies. vi

8 Acknowledgments First and foremost, I would like to thank my advisors Dr. Sookkyung Lim and Dr. Christian Hong for their continued support and guidance throughout my studies. I would like to thank Dr. Jaekyoung Kim for his unrelenting support in these efforts when he was needed most. I would also like to thank Dr. Vaughan for always helping me with my scientific computation and numerical analysis insufficiencies. I would like to thank DARPA, the NIH, and the University of Cincinnati for providing me with much needed financial support throughout my time as a graduate student at UC. Finally, I would like to thank my family and friends for always giving me the confidence and drive necessary to complete my studies. Cincinnati, OH March 28, 2016 Jacob Bellman vii

9 Table of Contents List of Tables List of Figures x xi 1 Introduction The Circadian Clock Molecular Network The Phase Response Curve Mathematical Modeling Computational Methods Direct Method iprc Method Toy Models Introduction Pulsing Linear Degradation Model Definitions Phase Response Pulsing Nonlinear Degradation Pulsing Synthesis Phase Response Curves of Circadian Oscillators Simple Circadian Models Model Descriptions Phase Response A More Complex Model: the csp-1 Model Model Description Phase Response A New Simple Model: Model Model Description Parameter Selection Phase Response viii

10 4 Experimental Phase Response Curves Motivation Results Phase Shifting by Light Pulsing frq Transcription Materials & Methods Strains and Media Experimental Methods Phase Response Optimization Problem Statement Approximating the PRC with the IPRC Defining the IPRC PRC Approximation Optimization Applying the iprc Method A Geometric Approach to PRC Optimization Simulated Annealing Conclusions and Future work Conclusions Future work Bibliography 63 A Supporting Tables and Figures 67 A.1 Model Information A.2 Model 5 One Parameter Bifurcations A.3 Supplemental PRCs ix

11 List of Tables 5.1 Optimized Parameters for the vvd Model Light Response A.1 Parameters for Model A.2 Parameters for Model A.3 Parameters for Model A.4 Parameters for Model A.5 Parameters for Model A.6 Parameters for the csp-1 Model A.7 Parameters for the vvd Model A.8 Parameters for the vvd Model (Continued) x

12 List of Figures 1.1 The crucial molecular components of the circadian clock including reactions to light Type 1 and Type 0 PRCs Effect of applying negative linear pulses depends on the timing of the pulse Shifting a stable limit cycle towards an axis can produce a dead zone in a PRC The dead zone can be increased by slowing the angular velocity of the oscillator near the appropriate phases The parameters of a nonlinear degradation function affect the shape of the PRC Synthesis pulses invert shape of the PRC (compared to degradation pulses) Wiring diagrams for 4 basic circadian oscillators PRCs of the simple models pulsing frq transcription (multiplicative) PRCs of the simple models pulsing frq transcription (additive) Wiring Diagram of the csp-1 Model PRCs of the csp-1 Model pulsing frq transcription PRCs of the csp-1 Model pulsing WC-1 degradation Wiring diagram for a unique circadian oscillator PRCs of Model 5 pulsing frq transcription PRC of Model 5 pulsing WC-1 degradation The phase response curve of Neurospora crassa in response to light The phase response curve of Neurospora crassa in response to frq induction WC-1 Profile from an Optimal Pulse xi

13 5.2 An Optimal iprc An Inaccurate iprc Optimizing the PRC of the csp-1 model Variable Profiles from an Optimal Pulse Optimal PRC of the csp-1 model vvd Model Wiring Diagram [32] Phase response inaccuracy of the vvd model Phase response of the vvd model deconstructed SA Results of the vvd Model A.1 One parameter bifurcation analysis for k 1 of Model A.2 One parameter bifurcation analysis for k 2 of Model A.3 One parameter bifurcation analysis for k 3 of Model A.4 One parameter bifurcation analysis for k 4 of Model A.5 One parameter bifurcation analysis for k 5 of Model A.6 One parameter bifurcation analysis for k 6 of Model A.7 One parameter bifurcation analysis for k 7 of Model A.8 One parameter bifurcation analysis for k 8 of Model A.9 One parameter bifurcation analysis for k 9 of Model A.10 One parameter bifurcation analysis for K of Model A.11 One parameter bifurcation analysis for m of Model A.12 PRCs from each parameter of Model A.13 Pulsing frq transcription in each model (additive) A.14 Pulsing frq transcription in each model (multiplicative) xii

14 Chapter 1 Introduction 1.1 The Circadian Clock The circadian clock is a molecular mechanism within many organisms which acts as an internal timekeeping device. Oscillating with a period close to 24 hours autonomously, the clock can be entrained by external cues to run exactly on 24 hour cycles [1]. The circadian clock is sensitive to environmental stimuli such as light, and its rhythm is known to interact with other biological processes of an organism including DNA damage response [24], metabolism [26], and the cell division cycle [19]. Circadian disruptions can cause ailments that range from sleep disorders [23] to cancer [24]. Understanding the mechanisms which govern circadian response to light is crucial in the development of treatment for circadian related disorders Molecular Network The fungus Neurospora crassa provides an excellent prototype for studying the dynamics of circadian rhythm. The basic interactions of crucial oscillating molecules are well understood in the Neurospora clock, and these molecules are analogous to components of clocks in other organisms, most importantly mammals. For instance, a key gene in the core clock of Neurospora is the frequency (frq) gene, whose protein is analogous to the complex formed from the period (per) and cryptochrome (cry) genes. The frq gene is transcribed into frq mrna with the aid of the transcription factor White 1

15 Collar Complex (WCC), analogous to Bmal1/Clock in mammals, and is subsequently translated into the Frequency (FRQ) protein. FRQ inhibits WCC from transcribing frq, acting as a negative element and closing what is referred to as a negative feedback loop (Figure 1.1). This negative feedback is the core mechanism of the circadian clock in many organisms (including Neurospora and mammals) that produces robust oscillations with a period close to 24 hours [6]. The free running period (the period in constant darkness) of the Neurospora clock is approximately 22 hours. wc-1 mrna WC-1 WC-2 WCC frq mrna vvd mrna FRQ VVD Figure 1.1: The crucial molecular components of the circadian clock including reactions to light. The wiring diagram depicts the vital molecular components in the Neurospora crassa circadian clock and their interactions. Red rectangles represent mrnas, green ovals represent proteins, red arrows represent transcription, green arrows represent translation, black lines ending in bars represent transcriptional inhibition, joined blue arrows represent complex formation, purple lines represent degradation and dashed yellow lines represent light dependent reactions. In constant darkness, WC-1 and WC-2 proteins form the heterodimer WCC, which promotes frq transcription. FRQ protein inhibits WCC dependent transcription, closing the negative feedback loop responsible for robust 22 hour oscillations. Upon receiving light, the clock responds by rapidly degrading WC-1 and inducing WCC dependent upregulation of frq, wc-1, and vvd transcription. The VVD protein inhibits WCC dependent transcription. Photoadaptation is the process which allows the clock to adapt to low light conditions by inhibiting initial strong reactions to light. This gives the clock the ability to continue to oscillate in low levels of light while remaining sensitive to increasing intensities of 2

16 light input. In Neurospora, it is known that the Vivid (VVD) protein is responsible for photoadaptation [30]. When Neurospora receives light, the transcription factor WCC becomes unstable/degraded [31] while also rapidly overexpressing genes including vvd [15]. After being translated, VVD binds to WCC and inhibits its ability to regulate transcription, suppressing the initial response to light The Phase Response Curve One of the most informative tools for measuring circadian response to light is the phase response curve (PRC). The light PRC shape for many organisms is very similar: a dead zone (region of little to no advances/delays) in the subjective day followed by a delay region in the early subjective night, and then an advance region in the late subjective night. PRCs are generally classified as either Type 1 (small phase shifts and no discontinuities), or type 0 (large phase shifts accompanied by a jump discontinuity). Figure 1.2 shows the typical type 1 and type 0 PRC shapes displayed by most organisms. A compilation of experimental light PRC curves for various organisms is organized in [27]. Figure 1.2: Type 1 and Type 0 PRCs. A phase response curve plot measures the phase shift (y-axis) in response to a pulse applied at different circadian times (x-axis) and can be classified as either type 1 or type 0. (left) A typical type 1 PRC is characterized by a region of little to no phase shifts (a dead zone) in the subjective day followed by a delay region in the early subjective night and an advance region in the late subjective night. (right) A type 0 PRC generally has much larger phase shifts than a type 1 PRC and a jump discontinuity. 3

17 1.2 Mathematical Modeling When developing a mathematical model of the circadian clock, there are a few important tools that must be utilized. Kinetic equations aid in developing a system of ordinary differential equations (ODEs) capable of capturing the behavior of the core oscillating clock proteins. These equations contain parameters related to rates of physical interactions in the system. These parameters are usually unknown, and the kinetic equations introduce nonlinear terms into the model which make the system difficult to analyze. Bifurcation theory is a useful tool for studying the effect of parameters on a dynamical system. As unknown parameter values are unavoidable in the development of a realistic ODE model, bifurcation theory is essential when selecting a set of parameters to appropriately define a model. It is important that a circadian clock model maintains a period which is relatively stable to parameter perturbations in order to simulate the robustness of the circadian clock. Bifurcation analysis can aid in this process by making it possible to determine regions of oscillations and determining the period of the model for various ranges of parameter values. When modeling a circadian clock it is important that the clock responds appropriately to temporary changes (pulses) of parameter values. Pulses can be used to simulate drug administration by altering the corresponding parameters for the duration of the dosage release. Pulses may also be used to simulate environmental changes such as finite changes in light exposure. After pulsing a parameter that simulates reaction to light exposure the system will experience a phase shift. The plot of the change in phase against the time of the applied pulse is known as a phase response curve (PRC). Phase response curves resulting from light pulses have been measured experimentally for various organisms [14] and have been simulated by circadian models [10, 17, 32]. 4

18 1.3 Computational Methods Two methods of computing PRCs are used in this research: the direct method, and a method using infinitesimal phase response curves (iprcs). Both methods have advantages as well as limitations and both are used in this research to accomplish different goals. The direct method is computationally slow, but accurate for any type of pulse, whereas the iprc method is fast, but can be inaccurate for large pulse strengths as it is an approximation acquired by dropping high order terms relating to the pulse [21] Direct Method The direct method of calculating a PRC is the most basic approach as it directly follows the definition of a PRC. Using this method, each point of the PRC is calculated by integrating the system with and without a pulse (starting on the limit cycle). After allowing the pulsed trajectory to return to a small neighborhood of the limit cycle, the phases of the pulsed and unpulsed solutions are compared, determining the phase shift. The direct method is very straightforward as the phase can be easily defined on the limit cycle. The phase of any point on the limit cycle can be defined as the time it takes a limit cycle traversing trajectory to first reach the point. In order for the phase to be uniquely defined, the initial conditions of the limit cycle trajectory must be predetermined. Because the direct method allows for a pulsed trajectory to return sufficiently close to the limit cycle there is no need to define the phase apart from the limit cycle. This method is very accurate, especially if the attraction of the limit cycle is strong, forcing the pulsed trajectory to quickly return to the limit cycle. All phase response curves via the direct method were calculated in the following manner. Using the program XPP-AUT [8], MATLAB, or Python, the ODE system was solved with the inclusion of a parameter pulse, defined as simultaneous square wave increases of specified parameters. The pulsed trajectory was compared to the limit cycle solution, where both solutions were solved with identical initial conditions. This process 5

19 was automated in a loop, re-solving the ODE whenever the time of pulse was changed, allowing for a plot of the phase shift against the time of the applied pulse. A significant amount of cycles were allowed prior to measuring phase shifts in order to allow for transients to be surpassed. Although the direct method of calculating a PRC is accurate and straightforward, the method has limitations. For instance, every value of the PRC requires an integration of sufficient length to ensure the pulsed trajectory becomes close enough to the limit cycle. This can be very costly in terms of computation time especially if the attraction of the limit cycle isn t very strong. In particular, the direct method is unsuitable for calculating PRCs resulting from pulses of extremely small amplitudes. The size of the phase shifts will be reduced as the pulses are reduced in amplitude/length, eventually requiring an extremely small step size to accurately capture the phase shift. This particular numerical issue can be avoided by approximating the PRC with infinitesimal PRCs (iprcs) iprc Method The iprc method of calculating PRCs is based on assuming a small pulse strength, and using local analysis to approximate phase changes. Although this method is slightly more complex than the direct method, it is much faster because each PRC value is calculated using an integral with the same length of the pulse. The obvious limitation of the iprc method is accuracy. While the iprc converges to the actual PRC as the pulse strength (amplitude/length) decreases, the method can be inaccurate when pulses are strong enough to drive trajectories far from the limit cycle. This method relies on disregarding high order terms related to the pulse strength which will add to the error as the pulse strength increases. More details of this method can be found in Chapter 5. 6

20 Chapter 2 Toy Models 2.1 Introduction It is well known that the circadian clock is sensitive to certain external environmental cues such as light. It is assumed that the parameters of the kinetic equations change in a specific manner for the duration of the external signal. To study the effect of pulsing kinetic parameters in circadian oscillators we have developed basic 2-D oscillators with circular stable limit cycles in the first quadrant. With these toy circadian models we have simulated PRCs in response to pulses which resemble degradation and synthesis, modeled with linear and nonlinear functions. 2.2 Pulsing Linear Degradation Model Definitions Toy models below are defined in polar (Model T1, Model T3) and Cartesian (Model T2 Model T4) coordinates where a linear pulse is assigned to Model T2 and Model T4: dr 1 dt = r 1(r 1 1), dθ 1 dt = 1, (Model T1) 7

21 x 1 = αf 1 ( x 1 s 1, y 1 s 2 ) p 1 (t)x 1, α β y 1 = βg 1 ( x 1 s 1, y 1 s 2 ) p 2 (t)y 1, α β dr 2 dt = r 2(r 2 1), dθ 2 dt = cos(θ 2) ɛ, 0 < ɛ << 1 x 2 = f 2 (x 2 s 1, y 2 s 2 ) p 1 (t)x 2, y 2 = g 2 (x 2 s 1, y 2 s 2 ) p 2 (t)y 2, (Model T2) (Model T3) (Model T4) The pulse functions p i (t) := a i H(t)H(σ i t), i = 1, 2 are defined using the Heaviside step function H, where a i is the amplitude of the pulse, and σ i is the duration of the pulse. f 1 and g 1 are defined such that Model T2 is a Cartesian representation of Model T1 (x i = r 1 cos(θ i ), y i = r 1 sin(θ i ), i = 1, 2) that has been stretched by factors α (horizontally) and β (vertically), and shifted s 1 in the positive x 1 direction and s 2 in the positive y 2 direction. Similarly, f 2 and g 2 are defined such that Model T4 is a Cartesian representation of Model T3 that has been shifted s 1 in the positive x 2 direction, and s 2 in the positive y 2 direction. Here, the unit circle is a stable limit cycle for Model T1 and Model T3. The limit cycle of Model T1 has unit angular velocity whereas the limit cycle of Model T3 has a varying angular velocity which oscillates between a small value ɛ, and 2 + ɛ. By shifting these models to the first quadrant using s i, these models more accurately resemble circadian oscillators. The pulses administered in Model T2 and Model T4 are of the same form as the pulses applied to linear degradation of circadian oscillators Phase Response At first glance, it seems reasonable to predict the shape of the PRC based on the shape of the solution of the variable whose degradation is being pulsed. It can be expected that pulsing the degradation of a variable that is decreasing would cause a phase advance, and 8

22 pulsing the degradation of a variable that is increasing would cause a delay (Figure 2.1, left and right). Additionally, assuming a variable becomes relatively small, it is reasonable to predict that little to no advances/delays will appear when pulsing the degradation of the same variable when it is near its minimum (Figure 2.1, center), creating a dead zone. This is due to the fact that the degradation pulsing function is conventionally a linear or a Michaelis Menten function, monotonically dependent on the variable concentration. This suggests that the PRC created from pulsing the degradation of an oscillating variable that becomes small enough will have the same pattern (dead zone, delay, advance) as the experimentally obtained light PRC exhibited by most organisms [27]. Figure 2.1: Effect of applying negative linear pulses depends on the timing of the pulse. In the figure above, a negative linear pulse is applied to x 1 of Model T2 where s 1 = 1.001, s 2 = 2, and α = β = 1. The pulsed time dependent x 1 solution is shown in red while the unpulsed solution is shown in black. While x 1 is sufficiently large the pulse causes a phase advance when ẋ 1 < 0 (left), and a phase delay when ẋ 1 > 0 (right). When x is sufficiently small, a negative linear pulse has little effect on x 1, and consequently little effect on the phase (center). Here, square pulses with amplitude 1 and length were applied as negative linear pulses to x 1 (a 1 = 1, a 2 = 0, and σ = 0.262) at t = 1, t = 3.7, and t = 5 (from left to right) PRC simulations were created with the simple 2-D system derived from an oscillator whose stable limit cycle is the unit circle traced with angular velocity 1 (polar differential equations are provided in Model T1). This oscillator was shifted to the first quadrant to further resemble the nonnegative nature of a biochemical oscillator (Model T2). PRCs were then obtained by applying a pulse resembling that of linear degradation for each variable. Operating under the assumption that a dead zone will appear between an advance and a delay region if the variable being pulsed becomes small enough, PRCs were computed for the oscillator before and after shifting the limit cycle toward the axes, 9

23 anticipating a more pronounced dead zone as the minimum value of the variable being pulsed is reduced. With the intention of amplifying the dead zone, these simulations were repeated with two different oscillators, one whose limit cycle is an ellipse traced with angular velocity 1 (Model T2, α β), and one whose limit cycle is a circle traced with varying angular velocity that slowed as it neared one of the axes (Model T3, Model T4). The pulse administered (p 1 or p 2 ) for each model resembles a linear degradation pulse in a biochemical oscillator. As we will now discuss simulated PRCs, a few things should be mentioned. Firstly, there is no concrete definition of a dead zone as this is a concept intended to describe a phenomenon qualitatively rather than quantitatively. Subsequently, the length, and for that matter the existence of a dead zone is somewhat subjective and only approximated here. Secondly, unless otherwise noted, the PRCs presented here stretch vertically as the pulse strength is increased, leaving the PRC virtually unchanged with a proper rescaling. This phenomenon is justified by the fact that PRCs with relatively small pulse lengths are proportional to the infinitesimal PRC [27]. This causes the choice of the pulse amplitude and consequently the PRC amplitude to be somewhat arbitrary. When plotting a PRC, the maximum advance or delay of a PRC cannot exceed half of the free running period (FRP), and as the strength of the pulse increases, so will the scaling of the PRC until reaching the [-FRP/2, FRP/2] boundary, in which case the PRC will transition to Type 0 (if not already). For this reason, a modest amplitude value of 1 was chosen for each pulse amplitude a i, and a circadian hour (FRP/24) was chosen for the length of each pulse. As can be seen in Figure 2.2, shifting the original model, with a stable limit cycle centered at (2, 2), towards the y 1 -axis causes the PRC resulting from a negative linear pulse in the x 1 variable to acquire an originally absent dead zone. There is clearly a difference in scales between the two PRCs. This can be explained by the fact that the function p 1 (t) is defined with the same length and amplitude in both models. As the pulsing function (p i ) is being multiplied by x 1, shifting the limit cycle towards the y 1 -axis 10

24 Figure 2.2: Shifting a stable limit cycle towards an axis can produce a dead zone in a PRC. Using Model T2, the above figure shows side by side comparisons of PRCs defined by a negative linear pulse in the x 1 variable, where the circular limit cycle (α = β = 1 in both cases) with radius 1 is centered at (2, 2) (Left) and (1.001, 2) (Right). Positive values correspond to phase advances, and negative values correspond to phase delays. In both cases, the pulse is defined with the same amplitude and length (a 1 = 1, a 2 = 0, σ = 0.262). Phase planes are positioned above the corresponding PRCs. will result in lower values of x 1 causing the response to the pulse to decrease, and hence the PRC to decrease in amplitude. For this reason, and the fact that we are focused only on the qualitative shape of the PRCs, PRCs will often be normalized by their largest phase shift. With the intention of intensifying the dead zone, the geometry of the limit cycle was changed to an ellipse with a major:minor axis ratio of 10:1. Again, negative linear pulses on the x 1 -variable were administered to this system in order to acquire PRCs before and after shifting the flattened side of the limit cycle towards the y 1 -axis. In Figure 2.3 (Left) it can be seen that compared to Figure 2.2, shifting the ellipse towards the y 1 -axis 11

25 has a nearly indistinguishable effect compared to shifting the circle near the y 1 -axis, and does not create a dead zone larger than that seen in Figure 2.2. This is, in part, due to the fact that these two limit cycles differ in geometry, but not in angular velocity. This means that although the values on the flat side of the ellipse may be shifted closer to an axis, the phase of these values is nearly unchanged from the case of the circle. This demonstrates the fact that the PRC shape cannot be determined/manipulated from the shape of the limit cycle alone. Model T3 exhibits the same circular limit cycle geometry as Model T1, while the angular velocity of the solution varies periodically. Specifically, this system has an angular velocity that is a positive sinusoidal function of the angle, reaching very small values near the minimum value achieved by x 2. This causes the trajectory of the limit cycle solution to slow as x 2 approaches its minimum. Negative linear pulses on x 2 were administered to the Cartesian form of this system shifted to the first quadrant (Model T4) in order to acquire PRCs before and after shifting the slow side of the limit cycle towards the y 2 -axis (Figure 2.3 (Right)). Shifting Model T4 successfully extends the dead zone to about 50% of the FRP, compared to the dead zone accounting for less than 25% of the FRP when shifting Model T2. This is clearly attributed to the fact that the limit cycle solution slows near the minimum of x 2, allowing for the dead zone to account for a larger range of the phase. One clear takeaway from this analysis is that the shape of the limit cycle is not sufficient information to determine the PRC shape, no matter the choice in pulsing function, as the limit cycle provides very little information about the timing of the phase. Also, linear degradation pulsing can yield the correct PRC shape if the variable being pulsed becomes relatively small. Once a dead zone has been acquired, it is clear that decreasing the angular velocity of the solution around the minimum value of the variable being pulsed can increase the length of the dead zone. The question becomes whether or not it is possible to choose parameters that manipulate the limit cycle in the desired manner. 12

26 Figure 2.3: The dead zone can be increased by slowing the angular velocity of the oscillator near the appropriate phases. (Left) Linear degradation PRCs pulsing x 1 of Model T2 (stretching factors α = 1, β = 10) before and after shifting the elliptical limit cycle center from (2,11) to (1.001,11). (Right) Linear degradation PRCs pulsing x 2 of Model T4 before and after shifting the circular limit cycle with varying angular velocity (ɛ = 0.01) from center (2,2) to (1.001,2). In both cases, the pulse is defined with the same amplitude (a 1 = 1, a 2 = 0) and the same length of one circadian hour. Limit cycle positions (Top) and corresponding PRCs (Bottom) are differentiated by color. Each PRC is normalized by the absolute maximum phase shift. 2.3 Pulsing Nonlinear Degradation Next, this analysis was extended to incorporate pulses which affect nonlinear degradation. Specifically, a Hill-type degradation function replaced the linear function as this is another standard function used to simulate degradation. This choice in degradation function slightly complicated the analysis as this added two parameters to the model (the hill coefficient n and the threshold parameter K i ). Model T5 and Model T6 are polar and Cartesian representations of these models, respectively: 13

27 dr 3 dt = r 3(r 3 1), dθ 3 dt = 1, (Model T5) x n 3 x 3 = f 1 (x 3 2, y 3 2) p 1 (t), x n 3 + K1 n y 3 = g 1 (x 3 2, y 3 2) p 2 (t), y3 n + K2 n y n 3 (Model T6) where the pulses are p i (t) := a i H(t)H(σ i t), i = 1, 2, x 3 = r 3 cos(θ 3 ), and y 3 = r 3 sin(θ 3 ). Here, Model T6 is a Cartesian representation of Model T5 shifted to center at (2, 2) with degradation functions x n 3 x n 3 +Kn 1 and y3 n. Notice that Model T5 and Model T6 are the y3 n+kn 2 same as Model T1 and Model T2 apart from the fact that the pulse in Model T2 resembles linear degradation and the pulse in Model T6 resembles nonlinear degradation. When simulating Model T6, the limit cycle position was fixed, and K 1 and n were allowed to vary due to the fact that changing K 1 and/or K 2 has a similar effect on the PRC as shifting the limit cycle. The threshold constants (K 1 and K 2 ) in each degradation function can be difficult to measure experimentally, and are generally unknown. This parameter plays a pivotal role in determining the shape of the PRC. In Figure 2.4 (Left), K 1 is allowed to vary while n is fixed at 50. A large value for n was chosen for exaggerated demonstration, although qualitatively similar results occur for all values of n. Increasing K 1 shifts the pulsing function to the right, and, depending on what values are traversed by the limit cycle, the pulse can have very different effects. For instance, Model T6 traverses all values on the interval [1, 3] in both x 3 and y 3 directions when there is no pulse applied, but for a pulse with K 1 = 1 the degradation function y y 50 3 will mostly take on values in the interval [2, 4]. This creates phase shifts at nearly all phases, and hence no dead zone appears. Although, when larger values are chosen for the threshold constant such as K 1 = 3, the degradation function y y 50 3 yields a large region of extremely small values. The phases causing this pulsing function to be small contribute to a very pronounced dead zone as 14

28 the pulsing function will be nearly indistinguishable at these phases, having little to no affect on the phase of the oscillator. Increasing n is also biologically significant as it represents the number of steps in a chemical reaction which can be very difficult to measure. In Figure 2.4 (Right), n is allowed to vary while K 1 is fixed at 2. Increasing the hill coefficient causes the degradation function to transition from approximately linear to sigmoidal, approaching a unit step function in the fixed domain [1, 3]. This transformation in the degradation function causes a larger portion of the range of the function to approach 0, causing the pulse to have a small effect on the system for these values, and hence causing a dead zone to appear in the PRC. Figure 2.4: The parameters of a nonlinear degradation function affect the shape of the PRC. (Left) Using Model T6 with a fixed hill coefficient n = 50, nonlinear degradation PRCs in the x 3 variable were produced for various values of the threshold constant (K 1 = 1, K 1 = 2, K 1 = 3). (Right) Using Model T6 with a fixed degradation threshold constant K 1 = 2, nonlinear degradation PRCs in the x 3 variable were obtained for various values of the hill coefficient (n = 1, n = 5, n = 50). In both cases, the pulse is defined with the same amplitude (a 1 = 1, a 2 = 0) and the same length of one circadian hour. Degradation functions (Top) and the corresponding PRCs (Bottom) are differentiated by color. Each PRC is normalized by the absolute maximum phase shift. 15

29 It is apparent that pulsing degradation rates of a model can yield a PRC with the correct shape, but the presence and size of the dead zone is dependent on many factors. For instance, we have seen that the dead zone is dependent on the position of the limit cycle in the phase plane, the angular velocity of the system, and the type of degradation simulated by the model. It is possible to control each of these factors, but this may come at the price of sacrificing the accuracy of the model. For example, it is easy to modify the hill coefficient and threshold constant in a degradation function in order to cause a dead zone to appear, but in doing so, the model may become unrealistic biologically. Furthermore, it may be very difficult to adjust the angular velocity of a model in order to control the size of the PRCs dead zone, especially in a system of higher dimension. 2.4 Pulsing Synthesis When simulating molecular synthesis such as transcription or translation, kinetic equations generally either call for linear or Hill-type equations, just as with degradation, only affecting variables in a positive, rather than negative manner. This makes our analysis very simple because we can predict that the results from pulsing synthesis will generally produce phase shifts opposite of those produced with degradation. This argument is again justified by the fact that PRCs with relatively small pulses scale in proportion to the amplitude of the applied pulse. In order to support this claim, we have produced PRCs for Model T7 before and after shifting the limit cycle towards the axes as before. Model T7 is given below: x 1 = αf 1 ( x 1 s 1, y 1 s 2 ) + p 1 (t)x 1, α β y 1 = βg 1 ( x 1 s 1, y 1 s 2 ) + p 2 (t)y 1. α β (Model T7) Notice that Model T7 is identical to Model T2, but the pulses are now being added instead of subtracted from the right hand side of the ODEs, creating pulses which resemble 16

30 synthesis instead of degradation. Figure 2.5 provides PRCs where the pulse given is representative of an increase in synthesis in a biochemical oscillator. Notice that as the limit cycle approaches the x 1 -axis the dead zone again appears, but the order of dead zone, delay region, advance region is now reversed with a positive pulse. Similar results could be constructed with a nonlinear synthesis-like pulse. Figure 2.5: Synthesis pulses invert shape of the PRC (compared to degradation pulses). Using Model T2, the above figure shows side by side comparisons of PRCs defined by a positive linear pulse in the x 1 variable, where the circular limit cycle (α = β = 1 in both cases) with radius 1 is centered at (2, 2) (Left) and (1.001, 2) (Right). Positive values correspond to phase advances, and negative values correspond to phase delays. In both cases, the pulse is defined with the same amplitude and length (a 1 = 1, a 2 = 0, σ = 0.262). Limit cycles are provided above corresponding PRCs. 17

31 Chapter 3 Phase Response Curves of Circadian Oscillators 3.1 Simple Circadian Models The goal of this research is to develop a mathematical model of the circadian clock that is as simple as possible while still being able to provide valuable and otherwise unattainable information about phase response. In particular, we are focused on accurately simulating phase response to light in the circadian clock of Neurospora crassa. When simulating a PRC with a mathematical model, the PRC shape depends on which parameters are perturbed, as well as the strength of the pulse (controlled by the amplitude and duration of the pulse). In Neurospora, it has been observed experimentally that frq mrna concentration experiences a rapid increase in response to a pulse of light, regardless of when the pulse is applied in the circadian day [3]. This has led to the hypothesis that simulating a light pulse in a Neurospora clock model can be done accurately by increasing the rate of transcription of frq. Surprisingly, when applying a positive pulse to the transcription of frq to current Neurospora circadian models, the resulting PRC does not resemble the experimentally observed PRC. This will be shown with each of the simple Neurospora circadian models discussed below. 18

32 3.1.1 Model Descriptions Figure 3.1 shows the wiring diagrams of 4 basic circadian oscillators that have been developed to simulate frq mrna and FRQ protein concentrations in the Neurospora circadian clock. Two of these models consist of three variables, differentiating between active and inactive forms of the FRQ protein. Each model shares the circadian characteristic of the negative feedback loop responsible for oscillations. These models are variations of models developed and analyzed in [2]. The ODE systems for each of the four basic models can be written as follows: da dt = da dt = ν 1 + ( B K a ) m k 1A + k 0, db dt = k 2A + k 3AB n Kb n + k 4B, Bn da dt = ν 1 + ( B K a ) m k 1A + k 0, db dt = k 2A k 3B 1 + ( B K b ), n da dt = ν 1 + ( C K m ) n k 1A + k 0, db dt = k 2A k 3 B, dc dt = k 4B k 5 C, ν 1 + ( C K a ) n k 1A + k 0, db dt = k 2A k 3 B k 4 B + k 6 C k 7BC m Kc m + C, m dc dt = k 4B k 5 C k 6 C + k 7BC m Kc m + C. m (Model 1) (Model 2) (Model 3) (Model 4) In each model, A represents frq mrna concentration and B represents FRQ protein concentration. In Models 3 and 4, B and C differentiate between inactive and active protein forms, respectively. The negative feedback in each model is modeled with a Hill function in the transcription of frq mrna, inhibited by the FRQ protein. Other than the 19

33 Figure 3.1: Wiring diagrams for 4 basic circadian oscillators. Models 1 and 2 include two variables, mrna (A) and protein (B). Both of these models include rates of transcription (ν), translation (k 2 ) and degradation (k 1 and k 2 in Model 1, k 1 and k 3 in Model 2). There is a negative feedback incorporated into both of these models and the difference between the two models is that the protein autoregulation is positive on translation in Model 1 and negative on degradation in Model 2. Models 3 and 4 include protein in inactive (B) and active (C) forms. Model 3 is an adaptation of the Goodwin Model [13] and Model 4 is an extension of Model 3 including protein inactivation and protein autoregulation. Hill function and autoregulation functions in Models 1, 2, and 4, each kinetic reaction is modeled with linear dynamics. The defining difference between Model 1 and Model 2 is the way in which positive autoregulation of FRQ protein concentration is modeled. In Model 1, this regulation is a positive interaction on translation, whereas in Model 2 it is an inhibition of degradation. The autoregulation in both of these models is necessary for oscillations, and increases robustness of the period of the oscillators. Model 3 and Model 4 are adaptations of the Goodwin oscillator [13]. The difference between these two models is the existence of FRQ autoregulation. The inclusion of this mechanism in Model 4 increases period robustness [2]. Each of the four simple models have been assigned parameter values that cause the systems to attract to stable limit cycles with period close to 22 hours. Furthermore, 20

34 the parameters in these models were chosen to maximize period robustness in response to local parametric alteration and intrinsic noise [2]. The parameter values used in this research for Model 1 - Model 4 are provided in Tables A.1 - A Phase Response PRCs were created for each pulsable parameter (all but Hill coefficients) of the four simple models, some of which are shown in Figures 3.2 and 3.3. One of the characteristics of the simulated PRCs is that increasing the strength of a square pulse via its duration or amplitude causes the amplitude of the PRC to increase. This should not be surprising as this is a known characteristic of the infinitesimal approximation of the PRC, which will be discussed in more detail in Chapter 5. Because of this, it can be assumed, unless otherwise noted, that the PRCs stretch vertically without changing the general shape, as the strength (amplitude and/or length) is increased. Each PRC in Figures 3.2 and 3.3 consist of phase shifts with similar orders of magnitude. Notice that this is not necessarily true for the amplitudes of the pulses. This is due to the fact that phase sensitivity is parameter-dependent; each parameter has a different level of control over the phase. It is worth mentioning that each PRC simulated for these models starts at circadian time 0 (CT0). This was accomplished by starting each PRC on the limit cycle at the point where frq mrna reaches its maximum, based on the experimental data supporting the fact that frq mrna peaks near CT0 [9]. When pulsing one parameter of a model, the shape of the PRC is often unchanged for various pulse amplitudes until the [-T/2, T/2] barrier is reached. At this point the PRC becomes (if not already) discontinuous due to the [-T/2, T/2] plotting convention. PRCs with large amplitudes and/or discontinuities are considered to be Type 0, as opposed to Type 1. Type 0 PRCs can be achieved by pulsing any parameter that allows for the PRC to increase to the [-T/2, T/2] barrier. For Models 1-4 not all PRCs were able to reach this barrier as not all parameters had enough effect on the phase of the system, even with pulses of extremely large amplitude. 21

35 We first applied pulses to the rate of frq transcription for each model, reproducing the known response to light in Neurospora. This can either be done through the multiplicative parameter ν, or the additive parameter k 0. It can be seen in Figure 3.2 that pulsing ν in each model consistently yields a PRC shape opposite to that of the experimental light PRC. That is, the dead zone, advance and delay regions are in the incorrect order. When pulsing k 0 in each of the basic models, there is no dead zone and the PRC shape is consistently sinusoidal (Figure 3.3). Each of these results are consistent when applying the same pulse to many published models such as [5] and [32] (see Figures A.13 and A.14). These results were predicted in Chapter 2 with the PRC shapes acquired by pulsing the synthesis of toy models. Figure 3.2: PRCs of the simple models pulsing frq transcription (multiplicative). Each PRC is in response to a 5 minute square wave multiplicative pulse of the transcription of frq. That is, the pulse increases the parameter ν for a length of 5 minutes. The amount added to ν for Models 1-4 is 15, 200, 200, and 300, respectively. We have completed phase response analysis for each of the four simple models by determining the PRC shape in response to pulsing each parameter individually. Surprisingly, when pulsing any parameter in the four basic models, the correct PRC shape is unattainable. 22

36 Figure 3.3: PRCs of the simple models pulsing frq transcription (additive). Each PRC is in response to a 5 minute square wave additive pulse of the transcription of frq. That is, the pulse increases the parameter k 0 for a length of 5 minutes. The amount added to k 0 for Models 1-4 is 1.5, 5, 0.02, and 3, respectively. 3.2 A More Complex Model: the csp-1 Model The correct PRC shape is unattainable when pulsing any single parameter in the simple circadian oscillators. This has led us to question the importance of the complexity of the mathematical model when properly modeling phase response. To this extent, we conducted PRC analysis on the complex Neurospora circadian model from [5] consisting of 9 variables and 24 pulsable parameters (all but Hill coefficients). By determining the PRC shape from pulsing each individual parameter of a more detailed model, we sought to determine if the correct PRC shape is more dependent on the complexity of the model, or the number of parameters affected by light. 23

37 3.2.1 Model Description The Neurospora model evaluated in this section includes the core circadian components (frq mrna, FRQ protein, wc-1 mrna, and WC-1 protein) as well as the metaboliccircadian link csp-1 (mrna and protein). The model incorporates negative feedback of FRQ through the formation of a WC-1:FRQ heterodimer, and distinguishes between nuclear and cytoplasmic forms of WC-1 and FRQ. The csp-1 model does not include WC-2 as WC-1 is the protein responsible for transcription and WC-2 remains relatively constant throughout the circadian day [32]. The wiring diagram for the csp-1 Model can be found in Figure 3.4. For the full details of this model, see [5]. Figure 3.4: Wiring Diagram of the csp-1 Model [5]. This diagram portrays each interaction of the Neurospora crassa circadian clock model from [5], denoted here as the csp- 1 Model. Ellipses represent protein concentrations (assigned with an n for nuclear and c for cytoplasmic), and rounded rectangles represent mrna concentrations. A black dashed line distinguishes the separation between nuclear and cytoplasmic elements, blue dashed arrows represent induction of transcription, and red dashed lines ending in bars represent transcriptional inhibition. Black lines represent degradation, shuttling, translation, and complex formation. The names of the rate constants label each of the interactions described by the model. 24

38 The system of ODEs for the csp-1 Model are provided below: d[m w 1] [W n ] m = k 1 dt K + [W n ] k 4[M m f ] + q Mf, (3.2.1) d[m w 2] K1 r = k 7 dt K1 r + [C] k 10[M r f ] + q Mw, (3.2.2) d[m f ] K1 r = k 7 dt K1 r + [C] k 10[M r f ] + q Mw, (3.2.3) d[w 1 c ] = k 2 [M f ] (k 3 + k 5 )[F c ], dt (3.2.4) d[w 2 c ] = k 2 [M f ] (k 3 + k 5 )[F c ], dt (3.2.5) d[f n ] = k 3 [F c ] + k 14 [F :W n ] [F n ](k 6 + k 13 [W n ]), dt (3.2.6) d[w c ] [F c ] p [M w ] = k 8 dt K 2 + [F c ] p K 3 + [M w ] (k 9 + k 11 )[W c ], (3.2.7) d[w n ] = k 9 [W c ] [W n ](k 12 + k 13 [F n ]) + k 14 [F :W n ], dt (3.2.8) d[f :W n ] = k 13 [F n ][W n ] (k 14 + k 15 )[F :W n ], dt (3.2.9) d[m c ] K4 r = k 16 [W n ] dt K4 r + [C] k 17[M r c ], (3.2.10) d[c] = k 18 [M c ] k 19 [C]. dt (3.2.11) The parameter values used for the csp-1 Model are in Table A.6, and the following notation is used for the variables: M f - frq mrna, F c - FRQ protein in the cytoplasm, F n - FRQ protein in the nucleus, M w - wc-1 mrna, W c - WC-1 protein in the cytoplasm, W n - WC-1 protein in the nucleus, F :W n - FRQ:WC-1 complex, M c - csp-1 mrna, and C - CSP-1 protein Phase Response We started the phase response analysis of the csp-1 Model by pulsing each parameter corresponding to parameters in the basic models. Agreeably, all of the PRC shapes from the csp-1 Model were qualitatively similar to those acquired from the basic models. Figure 3.5 shows the PRC shapes resulting from pulsing frq mrna (additive and multiplicative) as was done with the basic models, resulting in similar PRC shapes. 25

39 Figure 3.5: PRCs of the csp-1 Model pulsing frq transcription. Both PRCs are in response to a 1 hour square wave pulse which increases the time derivative of frq mrna. The multiplicative pulse (left) increases the parameter that multiplies the synthesis function of frq mrna (k 1 ) by 1 and the additive pulse (right) adds a constant amount of 0.1 to the time derivative of frq mrna. After pulsing every parameter in the csp-1 Model, it was determined that only a few pulse choices yield the correct PRC shape. Among these parameters were the linear degradation rates of the transcription factor and photoreceptor WC-1 in the cytoplasm and the nucleus (k 11 and k 12, respectively). This discovery is significant because it has been shown experimentally that WC-1 becomes unstable/degraded upon light administration [31]. The PRCs in response to pulsing WC-1 degradation can be found in Figure 3.6. Figure 3.6: PRCs of the csp-1 Model pulsing WC-1 degradation.the PRCs are in response to a 1 hour square wave pulse of WC-1 degradation in the cytoplasm (left) and the nucleus (right). The pulse of cytoplasmic degradation increases the corresponding parameter (k 11 ) by 30, and the pulse of nuclear degradation increases the corresponding parameter (k 12 ) by 0.3. Phase response analysis of the csp-1 Model suggests that WC-1 degradation might have a significant impact on phase response. It was shown with our toy models that linear degradation can cause the correct PRC shape. These connections led us to hypothesize 26

40 that WC-1 was a necessary addition to the simple models in order to achieve the correct PRC pattern. 3.3 A New Simple Model: Model 5 When developing a mathematical model intended for simulating a physical event, deciding what variables to include is extremely important. Including a large number of variables can increase the qualitative accuracy of the model by incorporating all known dynamics, but this will be at the cost of diluting the model with unknown parameter values, decreasing the quantitative accuracy of the model. Furthermore, having too many variables can complicate a model, making it difficult to analyze. On the contrary, if a model is too simple it will likely be unable to provide any enlightening information about the physical situation in question. Monitoring this balance is very important, and was crucial in the development of the following model Model Description When building a Neurospora circadian model, it is always natural to start with the 2 most important components of this system, frq mrna and FRQ protein concentrations. As it was decided that WC-1 protein concentration would be included, it was also necessary to include the interaction between the WC-1 and FRQ proteins. It is known that FRQ protein inhibits frq transcription [6], so an inactive complex formed between WC-1 and FRQ was also introduced. This complex provides the negative feedback in the Neurospora circadian oscillator, which is responsible for oscillations. It was decided that this model would not simulate the White Collar Complex (WCC), which is a complex formed between WC-1 and WC-2. We chose to omit WC-2 altogether as WC-1 is the component that is responsible for transcription. Furthermore, we have excluded wc-1 mrna in order to preserve simplicity. In its place, a constant rate of WC-1 protein synthesis sufficiently simulates translation as it is known that wc-1 remains 27

41 relatively constant throughout the circadian day [22, 25]. The wiring diagram for Model 5, incorporating the above dynamics, is shown in Figure 3.7. Figure 3.7: Wiring diagram for a unique circadian oscillator. Model 5 includes four variables: frq mrna (F m ), FRQ protein (F p ), WC-1 protein (W ), and a heterodimer formed between WC-1 and FRQ proteins (W :F p ). Each of the kinetic interactions in the wiring diagram are color coded: constant protein synthesis (red), transcription (blue), translation (green), dimerization (gray), and degradation (orange). The interactions are labeled with the associated parameters: k 1 k 9, K, and m. k 0 is excluded from the diagram because it is set at 0, only to be increased during a pulse. Starting from the basic wiring diagram of the four variables (Figure 3.7), a system of ODEs was developed and is provided below: d[f m ] [W ] m = k 3 dt K m + [W ] k 4[F m m ] + k 0, d[f p ] = k 5 [F m ] k 6 [F p ] + k 7 [W :F p ] k 8 [W ][F p ], dt (Model 5) d[w :F p ] = k 8 [W ][F p ] k 7 [W :F p ] k 9 [W :F p ], dt d[w ] = k 1 k 2 [W ] + k 7 [W :F p ] k 8 [W ][F p ]. dt Model 5 was developed with mass action kinetics for each reaction excluding transcription of frq, which employs a Hill function as to incorporate a saturation effect. 28

42 3.3.2 Parameter Selection Parameter selection for Model 5 was based on first finding a parameter set yielding oscillations, rescaling to ensure the period was approximately 22 hours, and then using bifurcation analysis to measure the robustness of the model. If the period of the limit cycle oscillations was relatively sensitive to the alteration of any of the parameter values, corresponding modifications were made to the parameter set. This process was repeated until arriving at a parameter set that optimized period robustness to parameter perturbation, locally in parameter space. Finally, one parameter bifurcation plots were acquired for each of the finalized parameter values. Figures A.1 - A.11 show the bifurcation diagrams and period dependencies of each parameter Phase Response PRC analysis was conducted on the new model (PRCs were acquired upon pulsing every pulsable parameter). Each PRC simulated for this model starts at CT0, defined as the point where F m reaches its maximum, as was done with Models 1-4. The PRC shapes for parameters in Model 5 corresponding to those in the simple (and more complex) models are all similar. In particular, pulsing the transcription of frq (multiplicative or additive) led to the same PRC shapes produced by pulsing the corresponding parameters in any of the models discussed in this research (see Figures 3.8, A.13 and A.14). Additionally, pulsing the degradation of WC-1 created the correct PRC pattern (Figure 3.9) as was seen in the complex model. The consistencies of PRC shapes across models of varying complexity suggests that the PRC shape is highly dependent on the dynamics of the system, and less dependent on the parameter set selected for the system. Acquiring the incorrect PRC shape when pulsing frq synthesis is not only consistent among the models presented here, but is also consistent among many of the models found in literature. This finding was initially unsettling, as this parameter is the general consensus for the location of light input in the Neurospora clock. While frq induction 29

43 Figure 3.8: PRCs of Model 5 pulsing frq transcription. Both PRCs are in response to a 5 minute square wave pulse of the transcription of frq. The multiplicative pulse (left) added 20 to k 3, and the additive pulse (right) added 0.3 to k 0. undoubtedly plays a major role in Neurospora light response, it is likely that other known reactions to light, such as WC-1 inactivation, are also functionally important. The fact that we consistently acquire the incorrect PRC shape when pulsing frq transcription, and we can acquire the correct PRC shape when pulsing WC-1 degradation, leads us to believe that accurately simulating response to light requires a pulse which affects multiple parameters simultaneously. 30

44 Figure 3.9: PRC of Model 5 pulsing WC-1 degradation. This PRC is in response to a one hour square wave pulse of the rate of degradation of the transcription factor WC-1. The pulse adds 0.02 to the parameter k 2. 31

45 Chapter 4 Experimental Phase Response Curves 4.1 Motivation We have been unable to produce the correct PRC shape by only pulsing frq induction with each simple Neurospora circadian model. This has led us to hypothesize that frq induction is not a sufficient mechanism to properly phase shift the clock. In order to test this hypothesis, we have conducted an experiment which measures Neurospora phase shifts in response to a pulse of frq induction. We have also acquired our own experimental light PRC to confirm the shape of the type 1 PRC. 4.2 Results Phase Shifting by Light As discussed previously, Neurospora s light PRC has been well documented [4, 14, 28]. Although, until recently, bioluminescence data for frq promoter activity was not readily accessible, which meant all PRCs were measured by conidia band formation. In [12], Neurospora s light PRC is acquired for the first time using bioluminescence data of frq promoter activity. In that study, a light PRC was acquired when growing Neurospora in petri dishes, yielding a Type 0 PRC from a 15 minute light pulse with low intensity 32

46 (approximately 3 lux). It was mentioned that large phase shifts were not easily acquired when growing Neurospora in race tubes, and we have confirmed this claim in our study. We have acquired a light PRC using bioluminescence data in race tubes with long and high intensity light pulses (30 minutes, 6666 lux), provided in Figure 4.1. The phase shifts are all in the range [-3,3], even though the light pulses administered are very high in intensity. Regardless, the PRC still has the correct shape (dead zone in the circadian day followed by delay and then advance regions in the circadian night). Although it is still unclear why Neurospora is less sensitive to light in race tubes, this result continues to highlight the importance of the circadian clock s consistent and precise phase response to light. This is why we have placed so much emphasis on acquiring the correct PRC shape when simulating the circadian clock with mathematical models. Figure 4.1: The phase response curve of Neurospora crassa in response to light. The x-axis represents the time (in circadian hours) of the applied pulse, and the y-axis represents the resulting phase shift in hours (positive values for advances, negative for delays). Each point represents the average phase shift resulting from 3 repeats, accompanied by standard deviation bars. The light pulse was 30 minutes in length, and approximately 6666 lux in intensity. The red dashed line represents the expected shape of the light PRC Pulsing frq Transcription We also acquired a Neurospora PRC in response to pulses of frq induction. To promote frq activation, we made use of a Neurospora strain, pqa-frq, which includes an additional 33

47 frq promoter that is inducible by quinic acid (QA). This technique was also used in [18], although phase shifts were only determined for pulses applied in the range of CT To our knowledge, this is the only complete Neurospora PRC in response to frq induction. Due to complications with bioluminescence strains, phase was determined in this experiment by the position/time of the center of the conidia banding. Multiple runs were performed on points in the range of CT1-CT7 to investigate the possibility of a dead zone. The shape of the PRC in response to frq induction is unclear. The results are somewhat inconsistent in the range of CT1-CT7, and it is unclear whether or not this pulse results in a type 1 or a type 0 PRC. Additional data points are required to determine the shape of this PRC, and different pulse strengths may be required to reduce the phase shifts to develop a type 1 PRC. Additional experimentation is a future endeavor for model validation and hypothesis testing. Figure 4.2: The phase response curve of Neurospora crassa in reaction to frq induction. The x-axis represents the time (in circadian hours) of the applied pulse, and the y-axis represents the resulting phase shift in hours (positive values for advances, negative for delays). Each point represents the average phase shift resulting from 6 repeats, accompanied by standard deviation bars. The pulses were applied by adding 20 µl of 1M QA to conidia suspensions in 2mL of 0.1% LCM. 34

48 4.3 Materials & Methods Strains and Media In the light pulse experiment, we performed bioluminescence assays as described in [5] using the frq-luc strain [11]. In the frq pulsing experiment, pqa-frq was used. For both experiments, standard race tubes with standard race tube media were used with 0.1% glucose [7, 17]. In the light pulse experiment, the media contained 1% 1M luciferin Experimental Methods For the light pulse experiment, the frq-luc strain was initially inoculated in a minimal slant and grown in a 30 C incubator for 3 days. Conidia suspensions were then prepared by vortexing 1 ml of water in the minimal slant. 5 µl of the suspension were inoculated in each race tube, after which the tubes were transferred to the camera incubator chamber at 25 C where the lights were turned on. After at least 12 hours of LL (at approximately 6666 lux), sets of race tubes were transfered to the 25 C DD chamber (defined as CT12) at 4 hour intervals. When the race tubes were in DD for the appropriate amount of time, the race tubes to be pulsed were moved back into the LL camera incubator at 6666 lux with fluorescent lights while the controls remained in DD. After 30 minutes the lights were turned off in the camera incubator where the controls were returned and the camera was initiated. The camera captured images for the first 10 minutes of every hour and the data was analyzed using the software ImageJ. For the frq pulsing experiment, pqa-frq strains were initially inoculated in a minimal slant and grown in a 30 C incubator for 3 days. A conidia suspension was prepared by vortexing 1 ml of water in the minimal slant. 5 µl of the suspension were inoculated in sterile 10 ml culture tubes, filled with 2mL of media each. The tubes were moved to a 25 C LL incubator and shaken at 220 rpm. After at least 12 hours in LL the tubes were moved to a 25 C DD incubator where they continued to be shaken at 220 rpm. 35

49 When it was time for the pulse, 20 µl of 1 M QA were added to each tube to be pulsed. The pulse lasted for 2 hours, after which there was a one hour washout stage where all samples (including controls) were moved to sterile 10 ml culture tubes, each with 2 ml of fresh media. After the washout stage all samples were patted dry and inoculated in race tubes. Every 24 hours after inoculation the edge of the growth was marked. The data was analyzed using the software ChronosX. 36

50 Chapter 5 Phase Response Optimization 5.1 Problem Statement Consider the following system of ODEs and initial conditions: ẋ = f(x, p), (5.1.1) x(0) = x 0, where x(t) = (x 1 (t),..., x l (t)) is an array of real-valued variables and p = (p 1,..., p n ) is an array of real-valued parameters. Assume that the solution to (5.1.1) is a limit cycle solution with period T that is asymptotically orbtially stable with asymptotic phase [29]. We will refer to T as the free running period. Applying a square pulse to a parameter p i of system (5.1.1) yields: ẋ = f(x, p 1,..., p i (1 + h i ),..., p n ), (5.1.2) x(0) = x 0, where the pulse h i is a real-valued function and is defined by h i (t, a i, s i ) = a i H(t)H(s i t), where t is time, a i is the amplitude of the pulse, s i is the duration of the pulse, and H is the Heavyside step function. For simplicity, we will always set s i = 1. Notice that if a i = 0, (5.1.2) reduces to (5.1.1). Our general goal is to find the pulse(s) capable of producing a phase response curve (PRC) that is as close as possible to some reference PRC (denoted by P RC 0 ). In this context, the phase response curve is defined as the phase difference between (5.1.1) and 37

51 (5.1.2), dependent on the time (phase) of the applied pulse, the parameter being pulsed (p i ), and the amplitude of the pulsing function (a i ). Thus, it will be sufficient to denote each PRC as P RC i (t, a i ). As experimental PRC data is sparse, we generally have a limited number of data points corresponding to P RC 0. Thus, we will be comparing simulated PRCs with experimental PRCs point-wise, denoting experimental data points as (t 1, P RC 0 (t 1 )),..., (t N, P RC 0 (t N )). For each parameter p i, we wish to find the value of the control variable a i which minimizes the sum of squared differences (SSD) between P RC i and P RC 0 : N m i := (P RC i (t k, a i ) P RC 0 (t k )) 2. (5.1.3) k=1 Then, we can define A i and M i such that M i := min a i 0 (m i) = N (P RC i (t k, A i ) P RC 0 (t k )) 2. (5.1.4) k=1 The optimal PRC, allowing only one parameter to be pulsed, is defined by pulsing the parameter p k with pulse amplitude A k where. M k = min i=1,...,n (M i). (5.1.5) We can extend this idea by allowing a pair of parameters to be pulsed (p i, p j ) and determining the optimal pulse amplitudes (A i, A j ) where we redefine each variable accordingly: m i,j := M i,j := N (P RC i,j (t k, a i, a j ) P RC 0 (t)) 2, (5.1.6) k=1 N (P RC i,j (t k, A i, A j ) P RC 0 (t)) 2, (5.1.7) k=1 38

52 where P RC i,j (t, a i, a j ) is defined as the PRC acquired from pulsing two parameters p i and p j with pulse amplitudes a i and a j (still asserting that pulse lengths s i = s j = 1). In this case, the optimal PRC, allowing a pair of parameters to be pulsed, is defined by pulsing two parameters p k and p l with pulse amplitudes A k and A l where M k,l = min (M i,j). (5.1.8) i,j=1,...,n In particular, the goal is to compute each of these values for Model 5 which simulates time-dependent concentrations of transcripts and proteins in a circadian oscillator: d[w ] = k 1 k 2 [W ] + k 7 [W :F p ] k 8 [W ][F p ], dt (5.1.9) d[f m ] [W ] 7 = k 3 dt K 7 + [W ] k 4[F 7 m ], (5.1.10) d[f p ] = k 5 [F m ] k 6 [F p ] + k 7 [W :F p ] k 8 [W ][F p ], dt (5.1.11) d[w :F p ] = k 8 [W ][F p ] k 7 [W :F p ] k 9 [W :F p ], dt (5.1.12) with the following parameter values: k 1 = , k 2 = , k 3 = , k 4 = , k 5 = , k 6 = , k 7 = , k 8 = , k 9 = , K = Here, W represents a transcription factor, WC 1, which promotes synthesis of frq mrna, denoted by F m, which is translated into FRQ protein. The complex W :F p is formed between WC-1 and FRQ, inhibiting transcription and closing a negative feedback loop. One of the main difficulties in this project is to determine the phase response curve of an ODE system in a time-efficient manner. P RC i (t, 1) can easily be approximated point-wise by a direct method. For instance, P RC i (t 0, 1) is the phase difference between solutions of (5.1.1) and (5.1.2) where a i = 1 and the pulse is applied to a parameter p i over the interval [t 0, t 0 + 1]. As we will see later, the phase can be defined as t (mod T ) for the solution of (5.1.1). As the pulse applied to (5.1.2) will likely drive the solution away from the limit cycle, P RC i (t 0, 1) can be approximated by comparing phases of the 39

53 two solutions after allowing the solution to (5.1.2) to become sufficiently close to the limit cycle. As this method of approximation is computationally expensive, an alternative method is preferable. 5.2 Approximating the PRC with the IPRC One method of approximating the PRC is through the use of the infinitesimal PRC (IPRC). The following section discusses the IPRC as well as a numerically feasible approximation of the PRC via the IPRC. Note that in this section, the following notation is used: x Φ is used to denote the gradient of Φ with respect to the variable x, D x f is used to denote the Jacobian of f with respect to the variable x, and D p f is used to denote the Jacobian of f with respect to parameter set p Defining the IPRC Recall the general system of ODEs ẋ = f(x, p), (5.2.1) whose phase space includes the limit cycle (5.1.1). We will denote the solution to (5.1.1) as x γ (t), and the closed curve traced by the trajectory as γ. We will define the phase of a solution of (5.2.1) as a differentiable map Φ : A [0, T ) such that d Φ(x dt γ(t))=1, where A is in the region of attraction of γ. In particular, we will define Φ(x γ (t)) t (mod T ). Furthermore, we will define the phase of a solution apart from the limit cycle to match the phase of x γ belonging to the same isochron. For instance, if y(t) solves (5.2.1) and y(t 0 ) A \ γ, we say Φ(y(t 0 )) = φ if lim m y(t 0 + mt ) = x γ (φ). Taking the time derivative of Φ(x γ (t)), we quickly acquire the relationship x Φ(x γ (t)) f(x γ (t)) = 1. 40

54 In fact, this relationship holds true for all solutions of (5.2.1) in A as a consequence of defining the phase via isochrons. To prove this, let us assume that x Φ(y(0)) f(y(0)) 1 for some solution y(t) of (5.2.1), y(t) A \ γ for all t. Then there exists some δ > 0 and some ɛ > 0 such that Φ(y(δ)) Φ(y(0)) 1 δ > ɛ. On the contrary, because x γ has asymptotic phase, lim t Φ(y(t)) = 1, so there is some N N such that Φ(y(δ+NT )) Φ(y(NT )) 1 δ < ɛ. But, by our definition of phase, Φ(y(δ+N T )) = Φ(y(δ)), and Φ(y(N T )) = Φ(y(0)), which yields a contradiction. Hence, we have that x Φ(x(t)) f(x(t)) = 1 if x solves (5.2.1), x A. (5.2.2) We will now define the infinitesimal PRC (IPRC), Q(t), as the change in phase with respect to a change in a state variable from the limit cycle, or Q(t) x Φ(x γ (t)). It will be shown that Q(t) can be approximated by the numerically attainable solution to the adjoint equation: Ġ(t) = [D x f(x γ (t))] T G(t). (5.2.3) To show this, we will compare the phases of two solutions of (5.2.1), x(t) = x γ (t) and y(t) = x γ (t) + ɛδx(t), where 0 < ɛ << 1, δx(t) = O(1) (element-wise t), and y(0) / γ. Notice that because y(0) / γ uniqueness implies that y(t) / γ t and δx(t) 0 t. We can examine the evolution of δx(t) over time: d dt (δx) = d dt ( ) y(t) xγ (t) = 1 ɛ ɛ (ẏ(t) ẋ γ(t)) = 1 ɛ (f(y(t)) f(x γ(t)) = 1 ɛ (f(x γ(t) + ɛδx(t)) f(x γ (t)) = D x f (x γ (t)) δx(t) + O(ɛ), which, neglecting O(ɛ) terms, gives us the useful approximation d dt (δx) D xf (x γ (t)) δx(t). (5.2.4) 41

55 Also, we can consider the difference in phase between the two solutions, φ := Φ(y) Φ(x): φ = Φ(y) Φ(x) = Φ (x γ (t) + ɛδx) Φ (x γ (t)) = x Φ(x γ (t)) ɛδx + O(ɛ 2 ), which, again neglecting O(ɛ 2 ) terms, gives us another useful approximation φ ɛq(t) δx. (5.2.5) We must now recognize that the phase difference between the two solutions is constant with respect to time, which is a direct consequence of (5.2.2). Using this fact, along with approximations (5.2.4) and (5.2.5), we can justify the following: 0 = d dt [ φ] d [ɛq(t) δx] (from (5.2.5)) dt = ɛ Q(t) δx + ɛq(t) d dt (δx) ɛ Q(t) δx + ɛq(t) D x f (x γ (t)) δx(t) (from (5.2.4)) = ɛ Q(t) δx + ɛ[df(x γ (t))] T Q(t) δx(t) [ ] = ɛ Q(t) + [Df(x γ (t))] T Q(t) δx(t). As δx 0, this implies that Q(t) is an approximate solution to (5.2.3). This proof can also be found in [29] PRC Approximation We can use the notion of the IPRC to determine an approximation to a PRC resulting from perturbing a fixed set of parameters, as in (5.1.2), by considering the following: ẋ = f(x, p + ɛ p δp), x(t 0, 0) = x 0, (5.2.6) 42

56 where x = x(t 0, t), ẋ = t x, and δp = δp(t 0, t). The support of δp in terms of t is an interval of length 0 < σ << 1 (supp(δp) = [t 0, t 0 + σ]) and 0 < ɛ p << 1 such that x(t 0 + σ) A. We will assume the parametric perturbation is small enough such that x(t 0, t) = x γ (t) + ɛ x δx(t 0, t), where 0 < ɛ x << 1 and δx(t 0, t) = O(1) element-wise for all t. Notice that δx(t 0, t) = 0 for t [0, t 0 ]. Defining the phase difference between the pulsed and limit cycle solutions as φ(t 0, t) := Φ(x(t 0, t)) Φ(x γ (t)), we can define the PRC resulting from a pulse applied at φ = t 0 (V (t 0 )) as the total change in φ, which is only accumulated during the pulse. In other words, V (t 0 ) = t0 +σ t 0 s φ(t 0, s)ds. (5.2.7) Hence, in order to approximate the PRC (V ), we wish to approximate t φ(t 0, t). First, we will need to approximate φ(t 0, t): φ(t 0, t) = Φ(x(t 0, t)) Φ(x γ (t)) = Φ(x γ (t) + ɛ x δx(t 0, t)) Φ(x γ (t)) = x Φ(x γ (t)) ɛ x δx(t 0, t) + O(ɛ 2 x), giving us the important approximation: φ(t 0, t) x Φ(x γ (t)) ɛ x δx(t 0, t). (5.2.8) Next, we will need to approximate t δx(t 0, t): t δx(t 0, t) = 1 ɛ x t [x(t 0, t) x γ (t)] = 1 ɛ x [f(x γ (t) + δx(t 0, t), p + ɛ p δp(t 0, t)) f(x γ (t), p)] 1 ɛ x [D x f(x γ (t), p)ɛ x δx(t 0, t) + D p f (x γ (t), p) ɛ p δp(t 0, t)], 43

57 giving us another necessary approximation: t δx(t 0, t) 1 ɛ x [D x f(x γ (t), p)ɛ x δx(t 0, t) + D p f (x γ (t), p) ɛ p δp(t 0, t)]. (5.2.9) With the previous two approximations, and the fact that Q(t) = x Φ(x γ (t)) is an approximate solution to the adjoint equation, we can now approximate d dt φ(t 0, t): t [ φ(t 0, t)] t [ xφ(x γ (t)) ɛ x δx(t 0, t)] (from (5.2.8)) = t [ xφ(x γ (t))] ɛ x δx(t 0, t) + x Φ(x γ (t)) ɛ x t [δx(t 0, t)] Q(t) ɛ x δx(t 0, t)+ [ ] 1 Q(t) ɛ x [D x f(x γ (t), p)ɛ x δx(t 0, t) + D p f (x γ (t), p) ɛ p δp(t 0, t)] ɛ x (from (5.2.9)) [ = Q(t) + [D x f(x γ (t))] Q(t)] T ɛ x δp(t 0, t) + Q(t) [D p f (x γ (t), p) ɛ p δp(t 0, t)] Q(t) [D p f (x γ (t), p) ɛ p δp(t 0, t)]. This last approximation is due to the fact that Q(t) is an approximate solution to the adjoint equation. This phase reduction technique was originally developed in [21]. Finally, we have enough information to approximate the PRC (V (φ)) resulting from a pulse that starts at phase φ and last for length of time σ: V (φ) φ+σ φ Q(s) [D p f(x γ (s), p)ɛ p δp(φ, s)] ds. (5.2.10) Hence, a good approximation of a phase response curve can be computed by solving for Q, x γ, and ɛ x δx, and then substituting the functions into (5.2.10). This can be accomplished by first solving (5.2.1) to approximate x γ. Q can then be approximated by solving (5.2.3) backwards in time, substituting the approximation of x γ. Because Q and x γ are functions independent of φ, they only need to be solved for once. As ɛ x δx is a function of φ, it 44

58 must be approximated for each φ. This can easily be accomplished by solving (5.2.6) on the interval [φ, φ + σ], and comparing the solution to x γ. 5.3 Optimization As we have now developed a numerically feasible formulation of P RC i (t, a i ), namely through the function V, the optimization problem pulsing one parameter p i, explicitly expressed in (5.1.2), becomes an optimization of the following quantity in terms of the amplitude of the pulse a i : where V i (t k, a i ) = N (V i (t k, a i ) P RC 0 (t k )) 2, (5.3.1) k=1 tk +1 t k Q i (s) δf δp i (x γ (s))h i (s)ds. Recalling that h i (t, a i, s i ) = a i H(t)H(s i t), we can define g i (s) = H(t)H(s i t) and find that tk +1 V i (t k, a i ) = a i t k Q i (s) δf δp i (x γ (s))g i (s)ds. What s more, if we define α(t) = t+1 Q t i (s) δf δp i (x γ (s))g i (s)ds the problem now becomes an optimization of the following function: We can take the derivative F (a i ) = N (a i α(t k ) P RC 0 (t k )) 2. (5.3.2) k=1 N F (a i ) = 2 (a i α(t k ) P RC 0 (t k )) α(t k ), (5.3.3) k=1 and asserting that F = 0, we can directly solve for the optimal value of a i, A i : A i = N k=1 P RC 0(t k )α(t k ) N, (5.3.4) k=1 α2 (t k ) where the second derivative test confirms that this value results in a local minimum as F = 2 N k=1 α2 (t k ) > 0. 45

59 We can again extend this idea to pulsing two parameters at once. Keeping the same notation as before we wish to minimize the following: N (V ij (t k, a i, a j ) P RC 0 (t k )) 2, (5.3.5) k=1 where V ij (t k, a i, a j ) = a i α(t k ) + a j β(t k ), α(t k )= t k +1 t k Q i (s) δf δp i (x γ (s))g i (s)ds and β(t k )= t k +1 t k Q j (s) δf δp j (x γ (s))g j (s)ds. We can define the function we want to minimize in terms of the control parameters a i and a j : F (a i, a j ) = N (a i α(t k ) + a j β(t k ) P RC 0 (t k )) 2, (5.3.6) k=1 where we can calculate each of the partials necessary for optimization: F ai (a i, a j ) = 2(a i Z 1 + a j Z 3 Z 4 ), (5.3.7) F aj (a i, a j ) = 2(a i Z 3 + a j Z 2 Z 5 ), (5.3.8) F ai a i (a i, a j ) = 2Z 1, (5.3.9) F aj a j (a i, a j ) = 2Z 2, (5.3.10) F ai a j (a i, a j ) = 2Z 3, (5.3.11) where we have defined the following quantities: Z 1 = N k=0 α2 (t k ), Z 2 = N k=0 β2 (t k ), Z 3 = N k=0 α(t k)β(t k ), Z 4 = N k=0 α(t k)p RC 0 (t k ), and Z 5 = N k=0 β(t k)p RC 0 (t k ). If we set the first partials to 0 we can solve for the optimal values of a i and a j, (A i and A j ): A i = Z 4 Z 1 Z 3 Z 1 A j, (5.3.12) A j = Z 1Z 5 Z 3 Z 4. (5.3.13) Z 1 Z 2 Z3 2 Clearly, F ai a j (A i, A i ) > 0. We must now consider the discriminant, D = F ai a i F aj a j F 2 a i a j evaluated at (A i, A j ): 46

60 N D(A i, A j ) = 4 α 2 (t k ) = 4 k=0 N i,j=1,i j N = 2 i,j=1,i j N = 2 i,j=1,i j ( N N β 2 (t k ) 4 α(t k )β(t k ) k=0 k=0 α 2 (t i )β 2 (t j ) α(t i )α(t j )β(t j )β(t i ) ) 2 α 2 (t i )β 2 (t j ) 2α(t i )α(t j )β(t j )β(t i ) + α 2 (t j )β 2 (t i ) (α(t i )β(t j ) α(t j )β(t i )) 2 > 0, which confirms that (A i, A j ) does indeed result in a local minimum. In fact, we can extend this method even further to account for pulsing an arbitrary number of parameters p l1,..., p lm. Continuing to adopt similar notation as before we wish to minimize N (V l (t k, a l1,..., a lm ) P RC 0 (t k )) 2, (5.3.14) k=1 where V l (t k, a l1,..., a lm ) = a l1 α 1 (t k ) + + a lm α M (t k ), and α j (t k )= t k +1 t k Q i (s) δf δp lj (x γ (s))g lj (s)ds. minimize in terms of the control parameters a l1,..., a lm : Now, we can define the function F we wish to F (a l1,..., a lm ) = N (a l1 α 1 (t k ) + + a lm α M (t k ) P RC 0 (t k )) 2, (5.3.15) k=1 or written differently, F (B) = XB Y 2 2 = Y T Y 2Y T XB + B T X T XB, (5.3.16) α 1 (t 1 ) α 2 (t 1 ) α M (t 1 ) α 1 (t 2 ) α 2 (t 2 ) α M (t 2 ) where X =., B =..... α 1 (t N ) α 2 (t N ) α M (t N ) 47 a l1 a l2. a lm P RC 0 (t 1 ) P RC 0 (t 2 ), and Y =.. P RC 0 (t N )

61 Again, we can differentiate F : F (B) = 2X T Y + 2X T XB, (5.3.17) and optimize by setting F = 0: F (B) = 0 2X T XB = 2X T Y B = (X T X) 1 X T Y. (5.3.18) It is clear that this is a minimum as F is a convex function. To make things more complicated, we would like the pulse amplitudes to be positive. That is, we would like to enforce the condition that B 0, element-wise. To this extent, we initially attempted to minimize F using a non-negative least squares method on Model 5, which will be shown in the next section. It will quickly become clear that this method can often produce inaccurate results. This can be attributed to the fact that the amplitudes produced from the non-negative least squares method are too large for the PRCs created from the iprc method to be accurate Applying the iprc Method In a first attempt to accurately optimize the PRC for Model 5, we only allowed the pulse to include those parameters known to be affected by light. That is, we allowed for the pulse to have a positive effect on the parameters associated with frq transcription (k 0 and k 3 ), wc-1 transcription (k 1 ), and WC-1 degradation (k 2 ). Using the iprc method outlined in the previous section, we found the amounts by which to increase each of these parameters (pulse amplitudes), resulting in the optimal PRC. Surprisingly, it was found that the optimal PRC approximation only included pulsing k 2. The time-dependent WC- 1 profile responding to this optimal pulse is provided in Figure 5.1 and the PRC is in Figure 5.2. There could be a number of reasons the pulse of the optimal PRC only includes k 2. It is likely that the model is simply not complex enough to produce accurate phase shifts 48

62 0.60 WC-1 Conc. (a.u.) No Pulse Pulse at CT17 Pulse at CT Time (h) Figure 5.1: WC-1 Profile from an Optimal Pulse. Time-dependent WC-1 profiles respond to the pulse resulting from iprc optimization. The pulse increases k 2 by and is applied at CT17 (red line), causing a phase delay, and CT21 (blue line), causing a phase advance. The pulsed profiles are compared to the unpulsed solution (black dashes). with known responses to light. Model 5 omits elements such as compartmentalization, phosphorylation, and the light response element, vvd. It s possible that these details play an important role in the light response mechanism. It could also be that there are more reactions to light we have yet to discover. For instance, it is unknown whether or not FRQ protein concentrations have a direct response to light. It is possible that a model as simple as Model 5 is incapable of producing the correct PRC. A slightly less obvious reason for PRC inaccuracy is the use of the iprc as an approximation of the PRC. Comparing the direct PRC to the iprc using the optimal pulse, we see in Figure 5.2 that the iprc is an excellent approximation to the PRC in this case. However, the iprc is not always a good approximation of phase response. To demonstrate this, we used the iprc optimization method on Model 5 under the assumption that every parameter excluding the hill coefficient could be involved in the pulse. The results are provided in Figure 5.3. Clearly, the iprc method fails to produce accurate results. Although the iprc resulting from the iprc optimization technique accurately simulates the experimental data, the iprc does not match the direct PRC, 49

63 Phase Shift (h) Direct PRC iprc CT (h) Figure 5.2: An Optimal iprc. The simulated PRCs above are the result of applying the iprc optimization method on Model 5 against experimental PRC data from [14] including k 0, k 1, k 2 and k 3 in the pulse. The pulse acquired from this method increases k 2 by for five minutes. The phase response curve is approximated using the direct method (blue dots) and iprc method (red dashes). which is an accurate approximation of phase response. This is likely due to the fact that the pulse includes large increases of k 3 ( ), k 9 (241.31), and K (136.98). The iprc loses accuracy as pulses become large, driving the oscillating system away from the stable limit cycle, where the infinitesimal phase response curves most accurately approximate phase response. However, it is possible that there exists a pulse which produces a much more accurate PRC. The iprc method might fail to find this pulse due to the inaccuracy of the iprc with pulses of larger strength as demonstrated above. This led us to explore other options of PRC optimization that avoid using the iprc A Geometric Approach to PRC Optimization The iprc optimization method has led us to conclude that WC-1 degradation may play a much more important role in Neurospora phase response than previously understood. We have decided to test this theory on the more complex csp-1 model [5] discussed in Section 3.2. By switching to a more complex model, we can simulate light response in 50

64 10 Phase Shift (h) Direct PRC iprc Experimental Data CT (h) Figure 5.3: An Inaccurate iprc. The simulated iprc above (green dashes) is the result of applying the iprc optimization method to Model 5 to match the WT experimental data from [14] (blue dots). The iprc inaccurately predicts phase shifting, which is demonstrated by applying the same pulse with the direct method to simulate phase response (red line). The pulse increases k 1 by 3.232, k 3 by , k 5 by 2.98, k 7 by 14.72, k 8 by 18.03, k 9 by and K by for five minutes. greater detail. For instance, the csp-1 model can explicitly simulate the upregulation of frq and wc-1 mrna, whereas frq is the only mrna concentration explicitly modeled in Model 5. The known responses to light capable of being reproduced with the csp-1 model include frq and wc-1 upregulation, as well as WC-1 protein degradation. Experimental data suggest that light-initiated frq upregulation is consistently large, independent of the circadian time of the pulse [3]. With the csp-1 model, however, pulsing the linear rate of transcription, k 1, produces almost no frq induction during the subjective day. For this reason, we only include a constant upregulation of frq in the pulse through the parameter q Mf. We also exclude the rate of cytoplasmic degradation (k 11 ) due to the fact that most WC-1 protein resides in the nucleus of the cell. Including both linear and constant rates of csp-1 upregulation (k 7 and q Mw, respectively) in our pulse, we attempt to simulate light response in the csp-1 model with these four parameters (q Mf, k 7, q Mw and k 12 ). 51

65 Figure 5.4: Optimizing the PRC of the csp-1 model. The plots above represent the error between PRC simulations of the csp-1 model and the WT PRC measured in [14]. Green isosurfaces represent error values 7% above the absolute minimum error. The minimum error, approximately , results from pulse amplitudes of 2.7, 15.20, and 0 to q Mf, k 12, q Mw and k 7, respectively. In order to capture the most accurate phase response curve with the csp-1 model, we determine the relative amounts by which each parameter is increased (we will refer to these values as pulse amplitudes). Under the assumption that exactly four parameters are involved in the pulse, this becomes a four-dimensional global optimization problem. We approached this problem by solving for an error function on a four-dimensional grid of nonnegative amplitude values. We can visualize the errors for each amplitude combination using a three dimensional animation. Results from this animation are provided in Figure 5.4, where isosurfaces encapsulate the location of the optimal amplitude combinations between q Mf, k 12 and q Mw for three separate values of k 7 amplitudes. Interestingly, the optimal pulse for the csp-1 model does not involve k 7. This suggests 52

66 Figure 5.5: Variable Profiles from an Optimal Pulse. Time-dependent frq mrna, it wc-1 mrna and WC-1 profiles respond to the pulse resulting from PRC optimization. The pulse increases q Mf by 2.7, q Mw by and k 12 by 15.2 and is applied at CT18 (red line), causing a phase delay, and CT23 (blue line), causing a phase advance. The pulsed profiles are compared to the unpulsed solution (black dashes). that, like frq, wc-1 upregulation is consistent, independent of the circadian time of the light pulse. The time dependent profiles of frq mrna, wc-1 mrna and WC-1 protein are provided in Figure 5.5 and 5.6 shows the optimal PRC of the csp-1 model compared to the wild type (WT) PRC from [14]. Although the simulated PRC captures the general shape of the correct PRC (dead zone, delay, advance), the advances are much larger than those measured experimentally, and the dead zone has a positive slope, inconsistent with the data Simulated Annealing Our next attempt at accurate simulation of the Neurospora PRC made use of the complex model developed in [32]. This model includes a detailed set of light dependent interactions, including upregulation of frq, wc-1 and vvd genes, as well as negative feedback induced by VVD binding to WCC, the transcription factor of each of frq, wc-1 and vvd. This model is capable of simulating phodoadaptation, a phenomenon that allows the clock to adjust to long-term light exposure such as moonlight, and respond to multiple light inputs of different levels of intensity. The wiring diagram for the vvd model is presented in Figure 5.7. The parameter values used for this model are provided in Tables A.7 and A.8. Although the vvd model accurately simulates photoadaptation, it does not produce an 53

67 Phase Shift (h) Experimental Data csp-1 Model Time of Pulse (CT) Figure 5.6: Optimal PRC of the csp-1 model. The optimal PRC of the csp-1 model (green squares) is plotted against the Heintzen et al WT PRC (blue circles). The optimal PRC results from a pulse which increases q Mf by 2.7, k 12 by and q Mw by accurate phase response curve with the original parameter set. Figure 5.8 demonstrates this by comparing the phase response curve of the model with the Heintzen et al WT PRC. The light response compartment of the vvd model is complex. The added complexity allows for more capability of the model, with the trade-off of adding uncertainty with an increased number of parameters. In fact, none of the parameters which define the light response in the vvd model (upregulation of transcription factors, transcription/translation of vvd, etc.) are directly supported by experimental measurements. We assume the default parameter set in the vvd model is the source of inaccuracy of the model s phase response. The vvd model simulates light response through a single parameter, k 26, which represents the rate at which active WCC becomes light activated. The light activated form of WCC then quickly degrades while initiating a series of responses by upregulating frq, wc-1 and vvd. In order to investigate how each of these responses affects the vvd model, we simulated phase response to pulses which only increase these 54

68 Figure 5.7: vvd Model Wiring Diagram [32]. The wiring diagram of the [32] model (left) is presented with explanations of symbols (right). Each reaction is numbered, corresponding to the model parameter numbers. This figure was originally presented in [32]. rates of transcription, or the rate of active WCC degradation. Pulsing vvd alone will not affect the model because VVD is only assumed to interact with light activated WCC. Figure 5.9 presents the different effects caused by pulsing these components. We adopted the computational search method Simulated Annealing (SA) [20] to randomly search parameter space of the vvd model for a parameter set which allowed for the optimal response to light pulses. The SA method will theoretically find a global optimal control set given enough computation time. The method works by making large jumps through parameter space often, at first, before reducing the jumps size and regularity as a parameter called temperature decreases exponentially. We constructed an error function that compares model simulations to four sets of experimental data. Two data sets come from an experiment in [14], where a PRC 55