Can the Network Modify a Phase Over a Wide Range, Without Al-

Altering the Period?

Sensitivity information is local information; it is only valid for an infinitesimal perturba- tion from the point at which it was calculated. Thus, it is important to also examine whether local sensitivity information extends for larger parameter variations, such as those potentially undertaken by an organism either undergoing evolution or temporarily altering network performance to adapt to a change in environment. On a methodological level,

Table 3.11: Angle αi,j between peak-to-peak sensitivity and the period sensitivity vector,

where sin(αi,j) =

 ∂βi,j ∂p  T / ∂β ∂p

, from the peak of each species i to the peak of each other species j in the extended model. (n) indicates nuclear location, (c) indicates cytosolic location. Species P/C(c) P/C(n) mB(n) B(c) B(n) B*(n) RE(n) Per2/Cry mRNA(n) 27.7 3.4 4.2 12.7 15.6 9.3 7.5 PER2/CRY(c) 14.4 14.7 18.7 19.8 10.7 8.4 PER2/CRY(n) 64.8 38.8 31.5 13.8 10.9 Bmal1 mRNA(n) 37.7 32.2 14.6 11.0 BMAL1 (c) 46.9 18.2 14.8 BMAL1 (n) 17.5 19.6 BMAL1*(n) 7.5

it is also of interest to evaluate which of the two measures, α or L, computed in Section 3.3.10 corresponds better the ability of the network to modify the phase without altering the period.

Three relative phases in the basic model were selected for comparison. The relative phase β1 with the largest angle α = 78.4◦ (but one of the shortest relative lengths L = 0.026)

found in the basic model was the timing between the peak in nuclear PER2/CRY and Bmal1 mRNA. The relative phase β2 with the longest relative length (L = 0.271) and a smaller

angle (α = 27◦) was the time between the peaks of PER2/CRY complex in the cytosol and of BMAL1* activated transcriptional activator in the nucleus. Third, the relative phase β3

with the smallest angle (α = 4.2◦) and the shortest relative length (L = 0.024) was between the peaks of Per2/Cry mRNA and nuclear PER2/CRY.

The question was, how far can the network take the process of adapting a phase with- out altering the period? To this end, a very rudimentary optimization procedure was implemented in which a small, constant size step in the direction of the period-neutral relative phase sensitivity was taken in parameter space. The step size was kept constant at ∆p = µ∂β_∂p

T. The step length µ was selected based on L, if L was larger, the step

size µ was chosen smaller, and vice versa, to avoid going too far from the nominal point in one step. At the new parameterization, the new period and phase were determined and recorded. Then all sensitivities were recomputed and a new, period-neutral direction was found. This procedure was repeated until the period changed over 1% away from the

original value or no oscillation was detected at the new parameterization. At this point, the algorithm terminated. The resulting trajectories are represented in Figure 3-11, left column. The parameters at the end points were compared to the original parameters in Figure 3-11, right column.

The network has no trouble adapting the relative phase β1 by as much as 93% (corre-

sponding to a change by 1.3 hours) without disturbing the period by more than 1%. The relative phase β2 could be modified by 13.3%, which corresponds to an absolute change of

2.2 hours. Interestingly, the same two parameters were largely responsible for this change as the change in β1, even though the relative phase under study is in a different part of the

network.

The relative phase β3 could be modified by only 3.5% or 0.3 hours, neither a significant

relative nor absolute amount. Parameter #23 was the main effector of this small change. If the first and third relative phases are compared, it appears that the angle α is important in measuring the ability of the network to adapt the phase without changing the period. Both relative phase sensitivities have almost the same length, yet one is signigficantly more flexible than the other.

All numerical experiments were repeated with half the stepsize (µ = 0.005 or µ = 0.0005, respectively. The results were only slightly different in that the first relative phase β1 (large

α, small L) was modified from 1.43 h to 2.66 h instead of 2.75 h, β2 (large L, medium α)

was modified from 17.02 h to 19.40 h instead of 19.28 h, β3 (small α, small L) was modified

by the exact same amount. This leads to the conclusion that the step sizes that were chosen originally lead to a reasonable representation of what the network can do.

The relative phase with the wider angle and shorter length could be modified by a larger relative amount (92% vs. 13%) but a smaller absolute time (1.3 h vs. 2.3 h) than the second phase with a medium angle and a longer length. Both phases can be changed to a significant degree. Interestingly, only two parameters are used to regulate all relative phases, as is seen in Figure 3-11, right hand sides, respectively. These two parameters, #11 and #23, are the rates of nuclear export of PER2/CRY and deactivation of BMAL1*. The eleventh parameter ranks eighths in scaled period sensitivities, but neither parameter’s period sensitivity is very large in comparison to the top ranked. Both parameters represent processes in the network where posttranslational modification has been shown to play a role. PER2 phosphorylation by casein kinase I is shown to control nuclear trafficking and

0 10 20 30 40 50 60 70 80 90 100 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 % change in β % change in T 0 5 10 15 20 25 −1400 −1200 −1000 −800 −600 −400 −200 0 200

Relative change in parameters

Parameter index % change in parameter 0 2 4 6 8 10 12 14 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 % change in β % change in T 0 5 10 15 20 25 −900 −800 −700 −600 −500 −400 −300 −200 −100 0 100

Relative change in parameters

Parameter index % change in parameter 0 0.5 1 1.5 2 2.5 3 3.5 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 % change in β % change in T 0 5 10 15 20 25 −200 0 200 400 600 800 1000 1200

Relative change in parameters

Parameter index

% change in parameter

Figure 3-11: Trajectory of change of period and phase (left) and parameter changes (right) from a simple optimization aimed to increase three different phases β as much as possible without disturbing T more than 1%. Top: The phase β1 = 1.43h, with the largest angle

α = 78.4◦ _{and short length L = 0.026. The step size chosen was constant at µ = 0.01.}

Middle: The relative phase β2 = 17.02h, with the largest length L = 0.271 and intermediate

size angle α = 27.0◦. The step size chosen was constant at µ = 0.001. Bottom: The relative phase β3 = 7.46h, with the smallest angle α = 4.2◦ and shortest length L = 0.024. The

degradation [103]. The (de)activation step of BMAL1 is the binding of CLOCK, which is phosphorylated rhythmically as well [88]. The role of these two parameters is discussed further in Section 3.3.12.