Relative Angular Phase Sensitivities

The scaled relative angular phase sensitivities ∂ ln γ_{∂ ln p} for two differently defined dawn-to- dusk relative phases were computed for the original and extended models using unlumped parameters (Figure 6-4). Dawn was defined as the point at which mCry1 reaches a minimum concentration and dusk was defined alternatively as “mCry1 dusk” or “mPer2 dusk” as described in Section 6.1.2. The results are similar between the two models; the results for the extended model will be discussed, as they provide more detail. When the peak of mCry1 is used as the marker for dusk, the two dominant sensitivities are degradation and export

0 5 10 15 20 25 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 rank sensitivity

unbinding of Cry from RE GBCC

unbinding of Cry from Per2 GBCC transcription

binding of Per2ppCCry1 to RE GBCC binding of Cry1 to RE GBCC

binding of Per2ppCCry1 to Per2 GBCC unbinding of Per2ppC from Cry1 (n)

mRNA export degradation of Per2ppC (n) primary phosphorylation (a) 0 5 10 15 20 25 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 rank sensitivity

unbinding of Cry from RE GBCC unbinding of Cry from Per2 GBCC

transcription basal unbinding of Per2 from Bmal1 DNA

binding of Per2 to Bmal1 DNA transcription 1

unbinding of Per2ppC from Cry1 (n) mRNA export transcription translation (b) 0 5 10 15 20 25 −2 −1 0 1 2 3 4 rank sensitivity

unbinding of Cry from ROR GBCC

unbinding of Cry from Per2 GBCC binding of Per2ppCCry1 to ROR GBCC

mRNA export

binding of Cry1 to ROR GBCC transcription

unbinding of Per2ppC from Cry1 (n)

primary phosphorylation mRNA degradation

binding of Per2ppCCry1 to Per2 GBCC

(c)

Figure 6-3: Scaled, rank-ordered relative amplitude sensitivities for the different mRNA concentrations, in the extended model; (a) mRE; (b) mBmal1; (c) mROR; black, Cry1- related parameter; blue, Cry2-related parameter; green, Per1-related parameter; red, Per2- related parameter; magenta, Bmal1-related parameter; cyan, Rev-Erbα-related parameter; yellow, ROR-related parameter.

of Cry1 mRNA. This is interesting, given that both PLCs that define the beginning and end of apparent daytime were formulated with reference to mCry1, but it is noteworthy that reactions involving mCry1 synthesis do not dominate. The degradation and export reactions dominate the relative angular phase between trough and peak. The reactions that follow in the ranking are reminiscent of the high-period sensitivities shown in Figure 6-1, and mostly located in the Per2 negative feedback loop. That is, reactions that strongly affect the period do not do so exclusively. They also affect phase relationships. It is important to note that period-changing reactions are not unalterable in the context of a 24-hour day. Rather, compensating changes in such reactions can lead to unaltered period yet changed

0 5 10 15 20 25 −0.05 0 0.05 0.1 0.15 mRNA degradation mRNA export mRNA degradation mRNA export degradation (n)

unbinding of Cry from Per2 GBCCRn nuclear/cytoplasmic volume

primary phosphorylationtotal CK1 concentration unbinding of Per2pC from Cry1 (n)

rank dln γ /dlnp (a) 0 5 10 15 20 25 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 mRNA degradation mRNA export mRNA degradation mRNA export

unbinding of Cry from Per2 GBCCRn

primary phosphorylation degradation (n) total CK1 concentration binding to kinases degradation of Per2ppC (n) rank dln γ /dlnp (b) 0 5 10 15 20 25 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

unbinding of Cry from Per2 GBCC

mRNA degradation

mRNA export

unbinding of Cry from Cry1 GBCC translation

binding of Per2ppCCry1 to Per2 GBCC unbinding of Cry from Per1 GBCC

degradation of Per2ppC (n) translation transcription rank dln γ /dlnp (c) 0 5 10 15 20 25 −0.2 −0.1 0 0.1 0.2 0.3 0.4

unbinding of Cry from Per2 GBCC

mRNA degradation

mRNA export translation

degradation of Per2ppC (n) unbinding of Per2ppC from Cry1 (n)

degradation (n)

transcription translation

unbinding of Cry from Cry1 GBCC

rank

dln

γ

/dlnp

(d)

Figure 6-4: Scaled relative angular phase sensitivities _{∂ ln p}∂ ln γ for the dawn-to-dusk phase, ranked by absolute magnitude. (a) Original model, mCry1 dusk; (b) Extended Model, mCry1 dusk; (c) Original model, mPer2 dusk; (d) Extended model, mPer2 dusk; Black, Cry1-related parameter; blue, Cry2-related parameter; green, Per1-related parameter; red, Per2-related parameter; magenta, Bmal1-related parameter; cyan, Rev-Erbα-related pa- rameter; yellow, ROR-related parameter.

phase relationships.

Conversely, when daytime was defined using a definition of dusk that involves the Per2 mRNA concentration (but the same, Cry1-based definition of dawn), the Per2 related reac- tions moved even farther to the top of the ranking and the Cry1 mRNA degradation remains the second most influential. A group of Cry1 related parameters populates ranks 7–11; these reactions are all important in determining the amount of Cry1 in circulation (transcription, translation, the binding of the inhibitory Cry-complexes from the BCC at the Cry1 binding site and degradation of nuclear and cytosolic CRY1). Interestingly, a number of reactions

were found to be of significant influence in the relative angular phase sensitivities that were already found previously to be important in setting the period and amplitudes of the mRNA oscillations. This finding will be discussed in more detail in Section 6.2.4.

The second definition of the relative phase not only leads to somewhat different sensitiv- ity rankings, but also to higher magnitudes of scaled sensitivity than found in the previous definition. This observation leads to the conclusion that if one is to understand how the clock could regulate the dawn-to-dusk phase, it is important to understand how both dawn and dusk are “tracked” on a molecular basis. This knowledge could be important for under- standing the molecular origin of seasonal affective disorder and possibly could point towards novel therapeutic intervention.