Supplemental Figures A Nonlinear Dynamical Theory of Cell Injury

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1 upplementary Figures, A Nonlinear ynamical Theory of Cell Injury - upplemental Figures A Nonlinear ynamical Theory of Cell Injury onald J. egracia, Zhi-Feng Huang, ui Huang A. f ( X ) = a Θ n n Θ + X n B..0 f ( X ) n = n = 2 n = 4 n = Θ = X X upplemental Figure. Characteristics of the Hill function. (A) Equation of the Hill function. a is an amplitude parameter that can be used as a normalizing factor. The ratio Θ n /(Θ n + X n ) ranges from 0 to, where n is the Hill coefficient. (B) Plots of a Hill function at various values of n. At low n, the function approximates a linear relationship. As n increases, the function assumes its characteristic sigmoidal form. At large n, the Hill function approximates a square wave, clearly illustrating an on/off effect with a threshold Θ. At Θ = X, f(x) = 0.5a; the threshold Θ then indicates a 50% inhibition effect. Conceptually, the use of the Hill function in the nonlinear dynamic model of this paper provides a threshold value of (i.e., Θ ) above which total damage will effectively inhibit total stress responses. And vice versa, there will be a threshold value of (i.e., Θ ) above which stress responses will be able to effectively inhibit damage. A key notion of Eq. (4) (main text) is that Θ and Θ change with injury magnitude, I. At low values of injury, the stress responses will have high Θ and be effective in inhibiting damage. As I increases so does Θ, making it progressively easier for damage to inhibit stress responses. In addition, Θ decreases with increasing I, making stress responses progressively less effective as injury magnitudes increase.

2 upplementary Files, A Nonlinear ynamical Theory of Cell Injury - 2 upplemental Figure 2. Eq. (4) in the main text is a normalized form (called rescaled in the physics literature) such that at each I, the maximum values of and are or 00%. We seek to modify Eq. (4) to output absolute values (referred to as unscaled ) of and. We therefore make one additional modeling assumption over that of Eq. (4): the v and v parameters are also defined as functions of I along with the Θ and Θ parameters [Eq. (2)], giving the following OE system: where the v and Θ parameters are the I-dependant functions: n d ' Θ' = v ' ' n n k dt Θ ' + ' n d ' Θ' = v' k' n n dt Θ ' + ' () v' = v Ie Θ ' = c I e + λ3i 2 + λi v' = v Ie Θ ' = c I e λ4i 2 λ2i (2) The transformation between the normalized/rescaled form Eq. (4) and the unscaled form Eq. () is: Ie Ie + λ3i λ4i ' = ( ), ' = ( ) which implies that λ = λ + λ 4 and λ = λ 2 + λ 3 in Eq. (4), and v, v, c, c, k, k, n, and n are unchanged from those in Eq. (4). Figure 2 shows the output of unscaled Eq. () (panels A, C) as compared to scaled Eq.(4) (panels B, ). (panels A and B) and * (panels C and ) shown in red. (panels A and B) and * (panels C and ) shown in green. There is no qualitative difference in the dynamics of Eq. (4) vs. Eq. () when λ 3 = λ 4 = 0. (3)

3 upplemental Figure 3. A brief primer on graphically solving an ordinary differential equation (OE). (A) Because in general nonlinear OEs cannot be solved exactly by integration, a graphical method invented by the French mathematician Henri Poincaré is now widely used to study OEs. This method is in fact relatively simple conceptually, as illustrated here. In short, a point in the plane, along with the OE parameters, serves as input to the OE (step ). The OE expression is then solved with the input numbers, and the OE outputs two magnitudes (d/dt and d/dt; the numerical values are listed as v and v 2 ). As the ratios d/dt and d/dt indicate, these are slopes and are vector quantities in the (, ) plane. The resultant of v and v 2 (labeled v 2 ) is then calculated (step 2) and the vector v 2 plotted at the point that originally serves as the input to the OE (step 3). One repeats this calculation, looping through the points in the plane, and plotting the A. Graphical construction of a vector field 0 0 (, ) Iλ n d ( cie ) = v I n n k = v λ dt ( cie ) + Iλ n d ( cie ) = v k I = v λ n n 2 dt ( c Ie ) + (2.) [ ij, ij, I, c, λ, c, λ, v, v, k, k, n, n ] = [0.2, 0.2, 8, 0., 0., 2.5, 0.9,,,,, 4, 4 ] vector output of the OE at each point to generate a vector field (also called a slope field). (B) A solution to the OE is any expression that satisfies the differential equation. For linear OEs, the exact solutions can be obtained by integration, but not for most nonlinear OEs. Therefore the vector field provides a graphical means to see all solutions to the OE. The solutions are known as trajectories. A trajectory is simply a flow path along the vector field. There are several well-known methods for numerically approximating trajectories on the vector field (e.g. the Runge-Kutta method). A specific trajectory is determined by the initial conditions, or the starting point of the trajectory. Illustrated are several trajectories from different initial conditions, where the initial conditions are depicted by stars. In general, the space of all trajectories is called a state space. ince our OE is two dimensional, the state space is referred to as the phase plane. In this example, it can be seen by visual inspection that every trajectory ends at the same point on the phase plane. This point is indicated by the yellow circle with red outline. ince all trajectories are attracted to this point, it is called an attractor point and is designated (*, *). More specifically, the attractor is the fixed point, or equilibrium point. Here the system no longer changes in time; the system is fixed at the attractor state. Because there is only one attractor on this example of phase plane, this phase plane is called monostable. A specific trajectory is mathematically equivalent to a matching pair of and time course. The trajectory itself is a curve, where each point on the curve indicates a pair (, ) which are the values of and that occur at the same point on a time course. This is illustrated in the figure (B) by showing three (, ) pairs (in red, green, and blue), and their corresponding locations on the time course. Thus, a trajectory is a more general way to view a time course. Finally, the fixed points or equilibrium points of the phase plane are, in a sense, the steady solutions to the OE because all trajectories in a specific phase plane go to these fixed points. The parameter values used for this example are listed in the figure. (.) v2 = v +v2 B. Phase Planes, trajectories, initial conditions and time courses [I, c, λ, c, λ, v, v, k, k, n, n ] = [3.5, 0., 0., 35, 0.9,,,,, 4, 4 ] Monostable phase plane with attractor (*, *) at (0.06, ) ( 0, 0 ) = (0, 0) and (3.) time, t 8

4 upplementary Files, A Nonlinear ynamical Theory of Cell Injury - 4 upplemental Figure 4: A bistable phase plane. Prior to the widespread availability of computers, it was generally unappreciated how complex OE solutions were: the focus on linear OEs gave a misleadingly simple impression of OE behavior. However, when considering the larger class of nonlinear OEs, one cannot, in general, know a priori the complicated flow patterns of the vector field. An element of surprise can occur, called emergence, when the vector field is plotted and reveals unanticipated features in the state space. Our nonlinear model embodied in Eq. (4) is relatively simple, yet under some parameter conditions, a vector field more complex than the monostable type can emerge, as shown here. When one visually inspects the vector field generated by the above set of parameters, and traces out trajectories, it is seen that there are two possible fixed point attractors to which the trajectories can converge. One of these is in the upper left (green) at (0.03, ), and the other in the lower right (red) at (0.99, 0.06) of the - unit plane. This type of phase plane is called bistable because there are two possible stable fixed points. In addition there is a third point at (0.57, 0.38) (blue) to which the trajectories appear to be drawn toward; but as they get closer to this point, they are repelled from it and towards one of the attractors. This point is thus called a repeller. When there is more than one fixed point, the attractors and repeller fixed points always appear together (trogatz, 994). The repeller point divides the phase plane into those trajectories that approach the attractor at (0.03, ) and those that approach (0.99, 0.06). The yellow dotted line indicates this division and is called the separatrix. Thus, all trajectories to the left of the separatrix form the basin of attraction for the attractor at (0.03, ). Trajectories to the right of the separatrix form the basin of attraction for the attractor at (0.99, 0.06). The emergence of bistable phase planes in the solutions to Eq. (4) constitutes one of the most important features of our nonlinear cell injury model. Because our theory associates attractors of * > * with survival, and those of * > * with death, bistable phase planes represent clear-cut avenues for therapeutic treatments based on the injury dynamics. As discussed in the article text, an injured system within the basin of attraction of the death attractor can be diverted, at least in principle, via some manipulation of the system, to the survival attractor. Initial conditions for the two plotted trajectories are marked by stars and their coordinates listed on the plot. The input parameters for this phase plane are also listed in the figure.

5 upplemental Figure 5: Often an OE system contains various parameters and it is of interest to study the behavior of the OE system as the parameters change. Parameters are not variables. A simple example explains this distinction. Condier the familiar equation of a line: y=mx+b. For this equation, x is the variable, and m and b are parameters. ifferent lines are obained by changing the m and b parameters, but all of them are lines and all are functions of the variable, x. An analogous logic holds for the nonlinear dynamic model of cell injury. Its parameters give different injury systems analogous to how the m and b parameters generate different lines. It is of interest to understand how the OE system behaves when the parameters change; this study is called variously bifurcation analysis, qualitative analysis or fixed point analysis and is illustrated here. Again, the basic notion is relatively simple. In general, the fixed points change as a parameter changes. (A) This is illustrated by showing several vector fields resulting from changing the parameter I (leaving all others constant). As can be seen, the fixed points are different amongst the vector fields. In particular, the number of fixed points changes, in this instance, from one to three. When the number of fixed points changes, a bifurcation occurs. In this example, the system bifurcates twice: it goes from one to three fixed points, and then goes from three to one fixed point. The presence of bifurcations is an example of emergence in nonlinear dynamics (see trogatz, 994, for further information). The numerical values of the fixed points (*, *) are listed below each respective vector field. (B) To characterize a system as the parameters change, one constructs a bifurcation diagram by simply plotting the fixed points of the OE vs. the parameter of interest. Because our system is two dimensional, every fixed point is a (*, *) pair. The system bifurcates at the same parameter values for * and * so only one needs to be studied. Here we plot * vs. I. When the system bifurcates, all three values of *, corresponding to each fixed point, are plotted at the given value of I (arrows). The repellers are here colored blue to distinguish them from the attractors. In bistable regions of OE solutions, there are two possible outcomes. In our model one outcome is interpreted as a survival phenotype (green), the other as a death phenotype (red). The * value from each fixed point is similarly color-coded in the bifurcation diagram of (B). The type of bifurcation diagram shown, which results from varying only the single parameter, I, is the simplest way to analyze Eq. (4). Because this OE has 2 variables and parameters, a full bifurcation analysis would vary all 3 quantities within the model constraints (e.g. c > c and λ > 0 and λ > 0) resulting in 3-dimensional bifurcation diagrams. Parameter values are listed on the figure.

6 upplementary Files, A Nonlinear ynamical Theory of Cell Injury - 6 upplemental Figure 6: The four types of injury courses are cross-sections of surfaces in 3 parameter space and interconvert via the surface topology. (A) Illustration of type A bistable injury course dynamics converting to monostable injury course dynamics. Yellow and orange dashed lines show characteristic 2 bifurcation diagram profiles for * and *, respectively, for type A bistable injury course dynamics. (B) Illustration of type B bistable injury course dynamics converting to monostable injury course dynamics. Yellow and orange dashed lines show characteristic 2 bifurcation diagram profiles for * and *, respectively, for type B bistable injury course dynamics. For 3 plots of *, attractors are indicated as red and repellers blue; for 3 plots of *, attractors are green and repellers are blue. Insets are rotations of surfaces. ecay time vs. I plots shown under 3 plots correspond to the

7 cross sections at c values indicated in the 3 panels; decay times are in red, decay times in green, and dashed magenta lines are for I = I X. Parameters (c, λ, λ, c ) used to generate plots in A and B were [0., 0., 0.9, 2.5 < c < 50] and [0.075, 0.0, 0.0, < c < 0.57], respectively. Figure 6 illustrates the bifurcation diagrams of * and * along families of injury courses resulting from varying c and I. Figure 6A shows how type A bistable injury courses transform into monostable ones. The repellers form a curved wall within the surface that disappears at a discreet value of c : the subsequent injury courses are then monostable. In the (t R, t ) vs. I plots below the 3 figures, at lower c the decay time plots are clearly of the type A bistable variety; however, with increasing c, they gradually morph into the monostable decay time plots with the characteristic spike around I X. Figure 6B shows a family of injury courses that transform from the type B bistable into the monostable dynamical pattern. A different 2 surface emerges with a repeller wall forming an almost conical shape. In the decay time plots, as c increases, the post-i X decay hump, characteristic of the type B bistable dynamical pattern, merges with and transforms into the spike characteristic of the monostable dynamical pattern. These examples illustrate that the dynamics of a specific injury course can be obtained from the cross-sections within the higher dimension parameter space of the solutions to Eq. (4). They also illustrate that the model can output many variations of the four basic injury course dynamic patterns. In fact, multi-dimensional bifurcation analysis suggests an intimate relationship between different injury systems and their dynamics. That is, overtly different forms of injury could be related to each other based on the dynamical patterns of their injury courses. At present, different injury modalities, for example, ischemia, trauma, shock, seizures, etc., are treated as completely different. While there is increasing recognition of common damage mechanisms (e.g. oxidative damage, induction of stress responses such as the heat shock response), our model shows there are natural relationships amongst the dynamics of injury courses.

8 upplementary Files, A Nonlinear ynamical Theory of Cell Injury - 8 upplemental Figure 7. Manipulation of initial conditions to cause a death outcome at I < I X. As discussed in the main text, altering initial conditions can cause the system to flip state in a bistable phase plane. (A) Plots of Θ and Θ vs. I; I X = 3.3. (B) Phase plane at I = 3. A number of survival (green) and death (red) trajectories are plotted and reveal that 0 = is the cutoff below which the system survives and above which it dies. Also indicated are the survival attractor (green), death attractor (red), and repeller (blue). (C) The injury course of the system. ashed magenta line is I = I X, dotted blue line is I = 3, which corresponds to the level of the phase plane in B. This is a type B bistable injury course. The parameter values used for this example are listed in the figure. In this bistable phase plane at I < I X, from initial conditions (0, 0) the system approaches the survival attractor (green circle) and survives. However, if 0 > (interpreted as the system being damaged to 6.9% of its maximum), the system will divert to the death attractor and die from what would be a sublethal insult when induced from the uninjured [e.g. ( 0, 0 ) = (0, 0)] state. This example is reminiscent of the worsened stroke outcome experienced by diabetic patients. It is well-known that diabetics suffer worse stroke outcome than non-diabetics. Research in this area seeks to elucidate specific diabetes-induced damage pathways that worsen stroke outcome. While it has been known that increased cardio-vascular disease in diabetics increases stroke risk, this does not fully explain the worsened outcome. Our model suggests that the worsened outcome is due to the underlying injury dynamics of the system, and that diabetes serves as a role of setting the initial conditions of the stroke injury such that 0 > 0.