Carryover of a saddle-node bifurcation after transforming a parameter into a variable

In this article, we introduce and study the carryover of a saddle-node bifurcation, a concept that describes how a saddle-node bifurcation of a dynamical system is carried over into an extended dynamical system obtained by transforming one of the parameters of the original system into a variable. We show that additional transversality and singularity conditions are needed to guarantee the carryover of a saddle-node bifurcation and provide a graphical methodology with a two-parameter bifurcation diagram to verify that such conditions are met. The results are applied to a gene activation model when the parameter describing the signal for activation is transformed into a variable, and to a cell cycle regulatory model when the parameter describing the cell mass is transformed into a variable. In both cases, we show that a saddle-node bifurcation carryover takes place.


Introduction
Consider a dynamical systemż = f (z; µ 1 , µ 2 ), where z ∈ R n and one of the parameters µ 1 , µ 2 ∈ R drives a saddle-node bifurcation as it crosses a critical value. Now, transforming one of the parameters, for example, µ 1 , into a variable, we obtaiṅ z = f (z, µ 1 ; µ 2 ), where (z, µ 1 ) ∈ R n+1 and g(µ 1 ; µ 2 ) is the vector field of the new variable µ 1 . We are interested in the conditions under which a saddle-node bifurcation in the extended system occurs as µ 2 varies. If the bifurcation point in the extended system, (2), is inherited (in a sense that will be clarified later) from the bifurcation point in the original system, (1), then we say that the bifurcation point in the extended system is the carryover of the bifurcation in the original system. We are choosing the word carryover to express the fact that the bifurcation can also be transferred to a new parameter that was not originally the bifurcation parameter, but other choices include inheritance, persistence, extension and continuation.
The main objective of this work is to find conditions that guarantee the carryover of a saddle-node bifurcation. Moreover, we would like to unfold the relationship, if any, between the saddle-node bifurcation in the original system and the saddle-node bifurcation in the extended system. Our work is motivated by the study of saddle-node bifurcations in the mathematical models of biological systems. These saddle-node bifurcations often are associated with biologically meaningful properties such as biological switches and hysteresis. When adding more complexity to the model of a biological system exhibiting a saddle-node bifurcation, it might be important to preserve the existence of the bifurcation. In particular, if a set of differential equations describing a biological system (original system) exhibits a saddlenode bifurcation driven by a parameter, it might be relevant to add more complexity to the system by transforming the bifurcation parameter into a dynamical variable and obtain a system with one additional dimension (extended system). Note that in the extended system, a saddle-node bifurcation might still be present but with respect to another parameter. If this is the case, we call this property the carryover of a saddle-node bifurcation.
To illustrate the applicability of our work, we introduce the following two examples, which will be fully analyzed in Section 5. First, consider the dimensionless model for the activation of gene x by a biochemical signal substance s given byẋ 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 where r > 0 is the degradation rate, and s ≥ 0 (Strogatz 1994;Lewis et al. 1977). This model is characterized by an irreversible switch-like activation (critical transition) of gene x as s increases from zero above a threshold s * : This switch is driven by a saddle-node bifurcation at s * (see Section 5 for more details). The biochemical signal needs to be sufficiently high to increase the activity of the gene from low to high. Suppose that we include the dynamics of the substance s by adding a differential equation for s in the original system (3). We would be interested in finding the conditions that guarantee the carryover of the saddle-node bifurcation for the extended system with respect to the parameter r. In other words, we would like to know if the activation or inactivation of the gene x could be driven now by the degradation rate r.
As a second example, we consider the regulatory network for the cell cycle involving the mutual antagonism of Cyclin B and APC (Anaphase Promoting Complex) that regulates the beginning and the end of the cell cycle. The beginning of the cell cycle, during the G1 phase, is characterized by low levels of Cyclin B caused by the action of active APC. Cell growth promotes APC inactivation, which in turn allows for Cyclin B synthesis in a critical transition, known as the start of the cell cycle. During the S-G2-M phase, Cyclin B activity is high which promotes activity of Cdc20, an activating subunit of APC. At the end of the S-G2-M phase, active APC lowers the level of Cyclin B, thus closing the cycle and defining the finish of the cell cycle. In this regulatory network, the cell mass acts as a parameter that drives the critical transition towards the beginning of the cell cycle through a saddle-node bifurcation in which the activity of Cyclin B increases quickly as the cell mass increases. The system describing the cell cycle is modelled by the following system of differential equations dY dt = k 1 − (k 2p + k 2pp P )Y, dP dt = (k 3p + k 3pp A)(1 − P ) J 3 + (1 − P ) − k 4 m Y P J 4 + P , dA dt = k 5p + k 5pp (mY /J 5 ) n 1 + (mY /J 5 ) n − k 6 A, where Y is the concentration of Cyclin B, P is the concentration of active APC (given by subunit Cdh1), A is the concentration of active of Cdc20, m is the cell mass, k parameters are production and decay rates, and J parameters are saturation constants, and n is the Hill coefficient (Segel and Edelstein-Keshet 2013;. This basic model for the cell cycle accounts for the G1 (or start) checkpoint, identified with an equilibrium where the concentration of Cyclin B is low, and the irreversible transition to S-G2-M phase, driven by a saddle-node bifurcation where the low level equilibrium of Cyclin B is lost (see Section 5 for more details).
This model can be extended by including more elements of the cell cycle, such as other Cyclin proteins and checkpoints (Tyson et al. 2002(Tyson et al. , 2003, or by considering the mass, m, as a variable (Contreras et al. 2019). In this latter case, we would have an extended system where the beginning of the cell cycle occurs dynamically as regulated by the G1 checkpoint, described by the saddlenode bifurcation with respect to the parameter m in system (4). This leads to the question of whether or not there exists another parameter that drives the G1 checkpoint in the extended system through a saddle-node bifurcation. In other words, our interest would be to find the conditions that guarantee the carryover of the saddle-node bifurcation for the extended system with respect to a new parameter.
Note that in both examples, we transform the parameter that drives a saddle-node bifurcation into a variable, and aim to find the conditions for the extended system that guarantee the carryover of the saddle-node bifurcation so that the particular biological characteristics of the original system are preserved in the extended system.
In this paper, we provide sufficient conditions to guarantee the carryover of the saddle-node bifurcation going from the original system (1) to the extended system (2). For this purpose, we review the basic concepts of a saddle-node bifurcation in Section 2, state the main results for the carryover of a saddle-node bifurcation for the one-dimensional case in Section 3, and extend the results to the n-dimensional case in Section 4. We provide a practical graphical approach that can be used to guarantee the carryover of a saddle-node bifurcation. In Section 5, we will apply the results to the two examples introduced above. Finally, in Section 6, we discuss the limitations of our results and future work.

Mathematical Background
Saddle-node bifurcations in R n are characterized by three conditions: singularity, nondegeneracy, and transversality conditions. They guarantee the creation (or destruction) of two equilibria as one parameter crosses the bifurcation value. This is summarized in the following results taken from Meiss (2007, Ch. 8).
Equations (5) and (7) are the singularity and nondegeneracy conditions, respectively. They are necessary conditions for the function f 1 to be zero up to the zero-and first-order approximations about the bifurcation point, but nonzero in the second-order approximation. The function y = η(x; µ) allows us to reduce the dynamics in a neighborhood of the bifurcation point to onedimension, i.e.,ẋ = f 1 (x, η(x; µ); µ).
The extremal value function m(µ) determines a single condition on the parameters, m(µ) = 0, along which two equilibria are created (or destroyed). Having one condition on the parameters means that the bifurcation that takes place has codimension-one. In order to be a saddle-node bifurcation, the equilibria need to be created as some combination of the parameters crosses the bifurcation point. This can be guaranteed with a simple condition.
Corollary 1 If µ 1 is a single parameter such that then a saddle-node bifurcation takes place when µ 1 crosses zero.
Equation (8) is known as the transversality condition because, geometrically, m(µ) = 0 is crossed transversally as µ 1 crosses zero. Note that µ 1 is an arbitrary parameter, and that the transversality condition can hold for several parameters at the same time. We only consider saddle-node bifurcations that take place as a single parameter crosses the bifurcation point.
In the context of this paper, we only consider two parameters for the sake of simplicity, i.e., µ = (µ 1 , µ 2 ) ∈ R 2 . We show that as long as a saddle-node bifurcation takes place in system (1) for at least one of the parameters, the extended system (2) also has a saddle-node bifurcation for the other parameter (i.e., the extended variable does not need to be a bifurcation parameter in the original system) under some singularity and transversality conditions. The Implicit Function Theorem is an essential tool in the study of saddlenode bifurcations in general. For instance, the function y = η(x; µ) in Theorem 1 is consequence of this theorem. The following form of the Implicit Function Theorem is taken from Meiss (2007, Ch. 8 Theorem 2 (implicit function) Let U be an open set in R n × R k and F ∈ C r (U, R n ) with r ≥ 1. Suppose there is a point (x 0 , µ 0 ) ∈ U such that F (x 0 ; µ 0 ) = c and D x F (x 0 ; µ 0 ) is a nonsingular matrix. Then there are open sets V ⊂ R n and W ⊂ R k and a unique C r function ξ(µ) : W → V for which x 0 = ξ(µ 0 ) and F (ξ(µ); µ) = c.

One-dimensional case
In this section, we focus on the case where the variable z in the system (1) is one-dimensional, i.e., we consider the systeṁ where x ∈ R, µ 1 , µ 2 ∈ R, and f is a sufficiently smooth function on (x, µ 1 , µ 2 ). This system is extended by transforming one of the parameters, for example, µ 1 , into a variable to define the extended systeṁ where g(µ 1 ; µ 2 ) is the sufficiently smooth vector field of the new variable µ 1 . We want to find conditions for the carryover of a saddle-node bifurcation in the original system (9) to the extended system (10). Suppose, without loss of generality, that the saddle-node bifurcation for (9) occurs at the origin as one parameter, µ 1 , crosses zero. That is, f at (x, µ 1 , µ 2 ) = (0, 0, 0) satisfies the singularity conditions and the nondegeneracy and transversality conditions The left hand side of system (11) defines the function This matrix has full rank since  (11) and (12). The Implicit Function Theorem 2 applied to F guarantees the existence of an interval I and unique functions x = X (µ 2 ), for µ 2 ∈ I, such that X (0) = 0, M(0) = 0, and the singularity conditions (11) are satisfied in I. This defines a smooth onedimensional curve Γ that follows the bifurcation point (x, µ 1 , µ 2 ) = (0, 0, 0), and is parameterized by µ 2 , i.e., and satisfies (11). By continuity, we can start from (x, µ 1 , µ 2 ) = (0, 0, 0) and follow the points that satisfy the singularity conditions (11) and the nondegeneracy and transversality conditions (12) to extend Γ . If the transversality condition is violated at some point, D µ1 f = 0, but the same condition is satisfied for the other parameter, D µ2 f = 0, we apply similar arguments to parameterize Γ by µ 1 in that section. Hence, we can extend Γ from the bifurcation point (x, µ 1 , µ 2 ) = (0, 0, 0) beyond the interval I as long as the transversality condition is satisfied for at least one of the parameters (see Figure 1). The projection of the extended Γ onto the (µ 1 , µ 2 )-plane, given by π : (x, µ 1 , µ 2 ) → (µ 1 , µ 2 ), defines an implicit function h(µ 1 , µ 2 ) = 0, known as bifurcation boundary, commonly plotted numerically using continuation (for example, using PyDSTool, XPPAUT, or MatCont). Note that although Γ is a smooth curve, h(µ 1 , µ 2 ) = 0 is not necessarily smooth at every point. For more details on birfucation curves and two-parameter bifurcations see Kuznetsov (2004). If the extended system (10) has a saddle-node bifurcation that is the carryover of the saddle-node bifurcation of interest in the original system (9), then this bifurcation must take place on Γ as it is the set of points satisfying the conditions for a saddle-node bifurcation.
This proposition says that in order to find the saddle-node bifurcation points for the extended system, we plot the two-parameter bifurcation diagram of the original system, superimpose the nullclines of the new equation in the extended system, and look for transverse intersections between the saddle-node bifurcation curve and the nullclines. This is enough to verify the singularity and transversality conditions in the extended system.
Propositions 1 and 2 will be used in Section 5 in the analysis of the first biological application, namely the gene activation model (3).

n-dimensional case
In the previous section, we showed the carryover of a saddle-node bifurcation for the one-dimensional case. In this section, we show that this result also holds for the n-dimensional case. In short, this is true because the saddle-node bifurcations can be reduced to one-dimension around the bifurcation point.
Now, assume that f 1 satisfies the transversality condition of Corollary 1 for either µ 1 or µ 2 . Then, in a similar fashion as in the one-dimensional case, the Implicit Function Theorem 2 guarantees the existence of an interval I and unique functions X and M such x and µ 1 can be parameterized in terms of µ 2 (for example), i.e., x = X (µ 2 ), µ 1 = M(µ 2 ), where µ 2 ∈ I, and X (0) = 0, M(0) = 0.
As we might expect, Proposition 2 also applies to the n-dimensional case. The proof follows similar arguments to the one-dimensional case.

Gene activation
Consider the gene activation model (3), which is in the form of (1), with f (x; r, s) = s − rx + x 2 1 + x 2 .
For r < 1 2 , this model is characterized by the existence of a switch activation of the gene via a saddle-node bifurcation as s increases from zero (see Figures 5ab). With initial condition x(0) = 0, increasing s would drive a critical transition that brings the activity of gene x to a high value, thus activating the gene. However, if the value of s is decreased, the activity of the gene remains in the active mode. For r = 0.4 the bifurcation occurs at s * ≈ 0.0418 .   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 For this simple application, we can compute all conditions satisfying Theorem 1 and Corollary 1. From the singularity conditions, f = 0 and D x f = 0, we obtain a parametrization of r and s in terms of 0 < x ≤ 1, The nondegeneracy condition requires but the transversality conditions, D s f = 1 = and D r f = −x = 0, do not impose extra conditions. Thus, the bifurcation curve is given by and there is a saddle-node bifurcation at any (x, a, b) ∈ Γ except for x = √ 3 3 . In Figure 5c, we show the projection of Γ (bifurcation boundary) onto the rs-plane. Here we see that the bifurcation boundary separates the parameter space into two regions which differ in the number of equilibria, and that the switch activation of the gene via s corresponds to crossing the upper bifurcation boundary from below. Now consider transforming s into a variable with constant synthesis and linear degradation termsṡ = g(s; r) = a − bs.
where a, b > 0. Note that the signal s goes to a stable equilibrium s = c = a b . If c = 6 169 ≈ 0.0355 < s * ≈ 0.0418, starting from initial conditions x(0) = 0 and s(0) = 0, the gene will not become active since the trajectory always remains within the bistability region (see Figure 5c). Would it be possible to activate the gene by varying the inactivation rate r? Note that the point (x, s, r) = ( 1 5 , 6 169 , 125 338 ) belongs to Γ and satisfies the singularity conditions (15), g = 0 and D s g = −b = 0; and transversality condition (16), Thus, by Proposition 1, there is a saddle-node bifurcation at r * = 125 338 ≈ 0.3698 as r decreases from 0.4. In fact, this point was found by substituting x = 1 5 into (24).
We can also apply Proposition 2 to the two-parameter bifurcation Figure 5c, where we observe that the nullcline g = 0 crosses the bifurcation boundary transversally and the nullcline is not tangential to the s-axis at the intersection. In this figure, it is clear that the switch activation occurs in both the original and the extended systems as the corresponding bifurcation parameter crosses the upper bifurcation boundary from the bistability region into   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 the region with only one equilibrium. This can be verified from the bifurcation diagram of the extended system as shown in Figure 5d. Therefore, the saddlenode bifurcation associated with the switch activation of the gene driven by r in the extended system is the carryover of the saddle-node bifurcation associated with the switch gene activation of the gene driven by s in the original system.

Cell cycle start
Consider the cell cycle model (4) where z = (Y, P, A). This basic model is characterized by a saddle-node bifurcation driven by the cell mass, m, which is associated with the start of the cell cycle, as illustrated in Figure 6a. The lower stable equilibrium is associated with the G1 phase since the concentration of Y is low. If m remains lower than approximately m * ≈ 0.8, the cell remains in G1 checkpoint as the concentration of Y needs to increase for the progression of the cell cycle. As m increases beyond the bifurcation value m * , the lower equilibrium is lost, allowing for the concentration of Y to increase abruptly and the start of the cell cycle. It can be shown that this bifurcation is in fact a global saddle-node-loop bifurcation and the details of this can be found in Segel and Edelstein-Keshet (2013), but for the purpose of this paper we focus on the local saddle-node bifurcation.
We now include the dynamics of the slow parameter m by transforming it into a variable following logistic growth where µ is the growth rate, and K is the carrying capacity. If K is larger than the bifurcation value (m * ≈ 0.8), then the start of the cell cycle occurs naturally as m increases dynamically beyond m * . We are interested in G1 checkpoint activation driven by k 5p as this parameter can be thought to be modulated by Polo-like kinase 1 (Plk1), a enzyme that plays a role in Cdc20 activation and has the potential to be used in cancer treatment through regulation of the cell cycle (Hansen et al. 2004;Liu et al. 2017). That is, we are interested in the carryover of the saddle-node bifurcation into the extended system with k 5p as the bifurcation parameter. Note that, as opposed to the previous application, computing the bifurcation curve Γ in this example is considerably more complicated. We therefore study the potential carryover of the saddle-node bifurcation graphically by applying Proposition 4. The two-parameter bifurcation diagram (using m and k 5p as bifurcation parameters) for the original system (4) is shown in Figure 6b. The start of the cell cycle is seen here as crossing the lower portion of the bifurcation boundary from the G1 region to the right. With the addition of equation (26), the nullcline g = 0 appears at m = K. If K = 2, for example, there is a saddlenode bifurcation for k 5p ≈ 0.025 as the intersection between the nullcline and the bifurcation boundary is transversal and not parallel to the m-axis, and   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 Fig. 6 a) Bifurcation diagram for the original system (4) with m as the bifurcation parameter. The arrow indicates the transition from G1 to S-G2-M at the start of the cell cycle when the cell mass increases and the stable equilibrium of low Y concentration disappears via a saddle-node bifurcation (labeled LP1). In the S-G2-M region, a limit cycle appears (globally, LP1 is a saddle-node-loop bifurcation), indicated in pink by maximum and minimum concentration of Y , in which the concentration of Y increases and decreases (the decrease in Y concentration defines the finish of the cell cycle). b) Two-parameter bifurcation diagram for the original system (4) with m and k 5p as bifurcation parameters. The horizontal arrow matches that in a indicating the start of the cell cycle. The red curve has two branches corresponding to the continuation of LP1 and LP2 in a. The nullcline m = K of (26) is indicated in blue for K = 2 and K = 3. For k = 2, the nullcline crosses the bifurcation curve transversally and Proposition 4 applies. For k = 3, the nullcline does not cross the bifurcation curve and thus G1 checkpoint activation is not possible if varying k 5p . c) Bifurcation diagram for extended system (4) and (26) with k 5p as the bifurcation parameter. The arrow indicates the G1 checkpoint activation, and matches the vertical arrow in b.

Discussion
The carryover of a bifurcation is a concept that contributes to the development and analysis of mathematical models. In this paper, we studied the carryover of a saddle-node bifurcation; we take a system where a saddle-node bifurcation is present, extend it by transforming a parameter into a variable (adding one extra equation), and find the conditions that guarantee that a saddle-node bifurcation is still present. Our focus was restricted to the case where there are two parameters of interest and the vector field of the new equation does not depend on the variables already present in the original system. We showed that additional singularity and transversality conditions for the extended system are sufficient for the carryover of a saddle-node bifurcation on top of the necessary singularity, non-degeneracy, and transversality conditions for the original system. We provided a graphical approach to verify such conditions using a numerical continuation software, such as AUTO, XPP, MatCont, and PyDSTool.
We provided a proof of our statement for the one-dimensional case in Proposition 1, which was later used to prove the n-dimensional case in Proposition 3. Our proofs require the Implicit Function Theorem 2 to follow the bifurcation curve, and the Saddle Node Bifurcation Theorem 1 to either verify the conditions for a saddle-node bifurcation in the one-dimensional case or to reduce the system to one dimension at the saddle-node bifurcation for the n-dimensional case. Equivalent graphical results for the one-dimensional and n-dimensional cases are given in Propositions 2 and 4, respectively. These graphical results provide a practical numerical tool to study all the points where a carryover is possible based on a two-parameter bifurcation diagram and the nullclines of the new vector field.
We illustrate our results with two applications in mathematical biology where the carryover of a saddle-node bifurcation is of interest. The first application is the one-dimensional gene activation model, in which we are able to show the carryover of a saddle-node bifurcation both theoretically and numerically. The second application is a model for the cell cycle progression, in which we are only able to show the carryover numerically. In practice, the study of bifurcations is generally carried out numerically; thus, our graphical approach becomes a powerful tool to study the carryover of a saddle-node bifurcation in applications.
The development of the carryover of a saddle-node bifurcation was motivated by the results in Contreras et al. (2019), where the authors found a saddle-node bifurcation driven by radiation in a larger version of the cell cycle model with radiation pathway. Furthermore, in Contreras (2020), the author showed that this saddle-node bifurcation is the carryover of the saddle-node bifurcation driven by the mass in the cell cycle model studied by . Other applications in which the carryover of a saddle-node bifurcation might take place include: variations of the cell cycle model , a model of inflammatory response (Reynolds et al. 2006), a model of   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 glucose and insulin levels in diabetes (Topp et al. 2000), and coupling-bursting models (De Vries and Sherman 2001).
Our result applies whether or not the parameter that is being transformed into a variable is the bifurcation parameter in the original system; we only need to start from a saddle-node bifurcation point and continue the bifurcation curve with respect to two parameters. Interestingly, a bifurcation for the extended system can take place at a point at which a bifurcation is not possible in the original system with respect to the same parameter because the transversality condition is not satisfied. Cases like this one are illustrated in this article with examples.