Effect of Model Selection on Prediction of Periodic Behavior in Gene Regulatory Networks

Abstract

One of the current challenges for cell biology is understanding of the system level cellular behavior from the knowledge of a network of the individual subcellular agents. We address a question of how the model selection affects the predicted dynamic behavior of a gene network. In particular, for a fixed network structure, we compare protein-only models with models in which each transcriptional activation is represented both by mRNA and protein concentrations. We compare linear behavior near equilibria for both cyclic feedback systems and a general system. We show that, in general, explicit inclusion of the mRNA in the model weakens the stability of equilibria. We also study numerically dynamics of a particular gene network and show significant differences in global dynamics between the two types of models.

Appendix

1.1 5.1 Proof of Theorem 3.3

We start by comparing the roots of the characteristic polynomial of (9)

$$ \operatorname{det}(A-D - \lambda I) =: \prod _{i=1}^n (\lambda- z_i) $$

(17)

with the characteristic polynomial of (11)

$$ \det Q := \det \left [ \begin{array}{@{}c@{\quad}c@{}} -D-\lambda I & C \\ A & -B-\lambda I \end{array} \right ] . $$

(18)

If we multiply the n+l column of Q by $\frac{d_{l}+\lambda}{c_{l}}$ and add it to the lth column of Q for l=1,…,n then the determinant in (18) does not change and we get

$$ \det Q = \det \left [ \begin{array}{@{}c@{\quad}c@{}} 0 & C \\ A-(B+\lambda I)C^{-1}(D+\lambda I) & -B-\lambda I \end{array} \right ] . $$

(19)

Now we multiply the l row of the above matrix by $\frac{b_{l}+\lambda }{c_{l}}$ and add it to the n+lth row for l=1,…,n. After this operation,

$$\det Q = \det \left [ \begin{array}{@{}c@{\quad}c@{}} 0 & C \\ A-(B+\lambda I)C^{-1}(D+\lambda I) & 0 \end{array} \right ] . $$

By exchanging the first n and second n rows, we finally get

(20)

Up to this point, our argument is general and allows arbitrary diagonal matrices B,C, and D. To make further progress, we assume that each of the diagonal matrices is constant. That is, we assume D=dI, B=bI, and C=cI, where I is the identity matrix. We denote the eigenvalues of matrix A by μ ₁,…,μ _n . Then the eigenvalues z ₁,…,z _n of the smaller problem (17) are related to the μ _i by

It follows from the Eq. (20) that the eigenvalues of Q satisfy det(A−uI)=0 where the constant

$$u = \frac{db + (d+b)\lambda+ \lambda^2}{c}. $$

Therefore, the eigenvalues come in pairs $\lambda_{i}^{\pm}$, where for each i they are related to eigenvalues μ ₁,…,μ _n of A by

$$\mu_i = \frac{db + (d+b)\lambda_i + \lambda_i^2}{c} $$

or

$$\lambda_i^2 + \lambda_i(d+b) + db - \mu_ic=0. $$

The solutions are

$$\lambda_i^\pm= \frac{1}{2} \Bigl( -(d+b) \pm \sqrt{ (d-b)^2 + 4 \mu_i c}\Bigr) . $$

We note that the sets U and V described in the Theorem 3.3, can be obtained from the following two sets:

• a. W ₁:={μ∈ℂ∣the real part of z=μ−d is negative},

• b. $W_{2} := \{ \mu\in\mathbb{C} \mid \mbox{the}\ \mbox{real}\ \mbox{part}\ \mbox{of}\ \mbox{at}\ \mbox{least}\ \mbox{one}\ \mbox{of}\ \lambda^{\pm}=\frac{1}{2}(-(d+b) + \sqrt{ (d-b)^{2} + 4 \mu c}) \mbox{ is positive} \}$.

Let W:=W ₁∩W ₂ and $Z:= \operatorname{Int}(\bar{W_{1}} \cap\bar{W_{2}})$, where the bar denotes the complement of a set in the complex plane, and $\operatorname{Int}$ is the interior of a set. It follows that the set Z is the set of those μ where the real part of z=μ−d is positive and the real parts of both λ ^± are negative. With these definitions, the sets

$$U= W - d,\quad\mbox{and}\quad V=Z-d $$

are translations of the sets W and Z, respectively.

The set W ₁ has a very simple shape—it is a half-plane in a complex plane of the form ℜe(μ)<d. We now investigate the set W ₂. We first observe that

$$\Re e\biggl( \frac{1}{2}\Bigl(-(d+b) \pm\sqrt{ (d-b)^2 + 4 \mu c}\Bigr)\biggr)>0 $$

is equivalent to

$$ d+b < \pm\Re e\Bigl( \sqrt{ (d-b)^2 + 4 \mu c} \Bigr). $$

(21)

We write

$$ \mu:=re^{i\phi} $$

(22)

and set

$$ u e^{i\theta} :=(d-b)^2 + 4c \mu=(d-b)^2 + 4cr e^{i\phi}. $$

(23)

Then the square root on the right-hand side (21) has two solutions

$$w_1= \sqrt{u}e^{i \theta/2} \quad\mbox{and} \quad w_2 = \sqrt {u} e^{i ( \theta/2 + \pi)} . $$

Taking the real part on the right-hand side of the Eq. (21), we see that the inequality is equivalent to

$$ d+b < \sqrt{u} \cos\theta/2 \quad\mbox{or} \quad d+b < \sqrt{u} \cos(\theta/2 + \pi) . $$

(24)

If we define the angle θ in (23) to be −π≤θ≤π, then for θ/2 we have −π/2≤θ/2≤π/2. This implies that cosθ/2>0 and cos(θ/2+π)<0 and the second inequality in (24) is never satisfied.

Therefore, the region Y is bounded by the curve

$$ \sqrt{u} > \frac{b+d}{\cos(\theta/2)}, \quad\mbox{where } {-}\frac {\pi}{2} \leq\theta/2 \leq\frac{\pi}{2} . $$

(25)

We now express this inequality in terms of u and θ, rather than $\sqrt{u}$ and θ/2. Since cosθ/2≥0 in the range of possible θ/2, the inequality (25) is equivalent to $\cos^{2} (\theta/2) > \frac{(b+d)^{2}}{u}$. We multiply by 2, subtract 1 from both sides and use the double angle formula to get

$$\cos\theta> 2\frac{(b+d)^2}{u} -1 , $$

which yields

$$ u > 2\frac{(b+d)^2}{1+ \cos\theta}, \quad-\pi< \theta< \pi. $$

(26)

This formula has a clear geometric interpretation. As angle θ sweeps the range from −π to π, the critical modulus

$$u^*(\theta) := 2\frac{(b+d)^2}{1+ \cos\theta} $$

varies from (b+d)² at θ=0 (the positive real axis) to ∞ in the limit θ→±π (the negative real axis). The function u ^∗(θ) in polar coordinates has apex at (0,(b+d)²) and is symmetric around the real axis since u ^∗(θ)=u ^∗(−θ). The region Y={ue ^iθ∈C∣u>u ^∗(θ)} is the set of all complex numbers whose modulus is larger than u ^∗(θ).

From definition (23), we see that the set W ₂ is a shifted and scaled version of the set Y. Indeed, the affine transformation X:=Y−(d−b)² shifts the region Y to the left, but does not change the shape of Y. The coordinate of the apex of X on positive real axis will be

$$(b+d)^2 - (d-b)^2 = 4bd >0 . $$

Finally, a complex number Re ^iϕ∈X if, and only if, 4cre ^iϕ∈W ₂. This means that $r=\frac{R}{4c}$ and the set W ₂ can be obtained from the set X by dividing the radius along each ray by 4c. The intersection of the set W ₂ with the positive real axis is then

$$ \biggl\{ \mu\biggm{|} \mu>\frac{bd}{c} \biggr\}. $$

(27)

Now we turn to the sets W=W ₁∩W ₂ and $Z= \operatorname{Int}(\bar {W_{1}} \cap\bar{W_{2}})$, where $\operatorname{Int}$ denotes the interior of a set. Since W ₁ is given by μ<d we see from (27) that if b≥c, the intersection of W with the real axis is empty, while if b<c this intersection is an interval $(\frac{b}{c}d, d)$. The set W is an unbounded subset of the set W ₁ (see Fig. 1). The set U is just a translation of the set W; if b≥c then U does not intersect the real axis, while if b<c, then this intersection is an open interval $((\frac{b}{c} - 1)d, 0)$.

The interior of the intersection $\bar{W}_{1} \cap\bar{W}_{2} = \emptyset$ when b≤c, while when b>c it is a bounded set around the positive real axis, intersecting this axis in the interval $(d,\frac{b}{c}d)$. The set V is just a translation of this intersection and intersects the positive real axis in the interval $(0,(\frac{b}{c} - 1)d)$.

1.2 5.2 Computer Simulations

Numerical analysis was performed with custom software written in Python and C, using AUTO (http://indy.cs.concordia.ca/auto/) for dynamical continuation, and the CVODE module of Sundials (https://computation.llnl.gov/casc/sundials/main.html) for evaluation of particular time-series solutions. Source code is publicly available at https://github.com/gic888/msu_rna_dynamics), and is licensed under the GNU Public license (http://www.gnu.org/copyleft/gpl.html). Separate, but equivalent, specifications of both models (with and without explicit RNA) were implemented in C for use by Sundials, and FORTRAN for use by AUTO. Evaluation protocols and visualizations were written in Python.

We report results from two sorts of numerical investigations. The boundaries of the region of stable oscillations, in both of Figs. 2 and 3, were computed by AUTO. The amplitudes of oscillation, as encoded by the colors in Fig. 3 were computed from time-series evaluations of the system.

To compute the oscillation amplitude values, we sampled the reported range of the parameters α ₂ and ϵ with a 100 by 100 grid of parameter values. For each point in this grid, we numerically integrated the model, using CVODE’s BDF/Newton integrator, for 600 time units. We then discarded the first half of this time window, to remove the initial transients, and measured the peak-to-peak amplitude of the remaining time-series.

During the scans reported here, we used only a single initial condition (all state variables equal to 2). As a result, it is not surprising that in the interior of the fold, we sometimes do not detect oscillations even where a stable oscillatory state should exist according to the dynamics. It appears that, as α ₂ increases and ϵ decreases, the oscillatory state becomes less and less numerically accessible. In additional scans using many sets of initial conditions, distributed in the vicinity of previously located oscillations, we were able to detect oscillations in more, but still not all, of this region. These explorations were computationally expensive, prone to numerical errors, and did not affect our conclusions about the overall system dynamics, and are therefore not reported here.