28.13: Regulation of the Cell Cycle by Protein Kinases

Search Fundamentals of Biochemistry

Learning Goals 

(Learning goals written by Claude, Sonnet 4.6, Anthropic)

Cell Cycle Regulation by CDK-Cyclin Complexes and Their Inhibitors

• Describe the sequential activation of CDK-cyclin complexes through the mammalian cell cycle—CDK4/6:cyclin D (early G1) → CDK2:cyclin E (late G1/S transition) → CDK2:cyclin A (S phase) → CDK1:cyclin A (G2) → CDK1:cyclin B (mitosis)—and explain how the oscillatory expression of cyclins drives the unidirectionality of the cycle.
• Explain how CDK4/6:cyclin D initiates and CDK2:cyclin E completes the phosphorylation and inactivation of the retinoblastoma protein (pRb), releasing E2F transcription factors to drive expression of genes required for S phase entry, and describe how the CKI proteins p21^Cip1 and p27^Kip1 (sequestered by CDK4/6) and the INK4 family (which allosterically inhibits CDK4/6) provide negative regulation of this commitment step.
• Explain the dual regulation of CDK1:cyclin B at the G2/M transition: describe how CAK (CDK7:cyclin H:MAT1) activates the kinase by phosphorylating Thr14 in the activation loop, while WEE1 and Myt1 impose inhibitory phosphorylations at Tyr15 and Thr14, and how CDC25C phosphatase removes these inhibitory phosphates to trigger mitotic entry.

Structural Basis of CDK Activation and Substrate Recognition

• Describe the conserved structural features of CDK2 shared with all protein kinases—N- and C-terminal lobes flanking the ATP binding cleft, the C-helix containing a conserved Glu, the glycine-rich GXGXXG loop, and the activation loop (DFG…APE) containing Thr160—and explain how cyclin A binding repositions the C-helix to enable the Glu–Lys catalytic interaction and how subsequent Thr160 phosphorylation by CAK repositions the activation loop to fully open the substrate binding site.
• Explain how CDK-cyclin complexes achieve substrate specificity through two cooperative mechanisms: a local phosphorylation motif (S/T)PX(K/R) recognized at the active site, and a distal recruitment motif (RXL) on the substrate that docks to a hydrophobic patch (MRAIL sequence) on the cyclin surface, using the CDK2:cyclin A:CDC6 peptide complex as a concrete structural example.
• Describe the transcriptional roles of CDKs beyond direct cell cycle regulation—including CDK7's phosphorylation of RNA Pol II CTD heptad repeats (YSPTSPS) at Ser5 and Ser7 to promote transcriptional initiation, and CDK9's phosphorylation at Ser2 to promote elongation—and explain why dysregulation of these transcriptional CDKs contributes to cancer.

Mathematical Modeling of CDK1 Oscillations

• Explain conceptually why a simple one-component model with negative feedback (Model 1) produces a stable steady state rather than oscillations in CDK1 activity, why adding a second species (APC, Model 2) produces damped oscillations, and why a three-component model incorporating PLK1 as an intermediary between CDK1 and APC (Model 3) is the minimal network required to generate sustained oscillations—and explain why higher Hill coefficients (representing greater cooperativity or ultrasensitivity) are required to produce oscillatory behavior in each successive model.
• Explain how ubiquitin-mediated proteolysis of cyclin B by the APC/C-Cdc20 E3 ubiquitin ligase complex (which recognizes the D-box motif in cyclin B) drives cells out of mitosis and resets CDK1 activity, and describe how the mutual activation between CDK1:cyclin B and CDC25C and the mutual inhibition between CDK1:cyclin B and WEE1 create a bistable switch that generates the sharp, switch-like transition into and out of mitosis observed in the VCell models.

This section integrates referenced materials, with significant modifications and additions.

Aleem and Arceci. Targeting cell cycle regulators in hematologic malignancies. Article in Frontiers in Cell and Developmental Biology 2015. DOI: 10.3389/fcell.2015.00016. Creative Commons Attribution 4.0 International

Introduction

For a cell to undergo successful division, it has to perform four key tasks in a highly ordered fashion. First, a preparatory synthetic phase (G1) increases cell size in anticipation of DNA replication (S phase). Cells then proceed through the G2 Phase to prepare to segregate duplicated DNA equally (M phase) and finally divide into two equal daughter cells. From G1, a cell can exit the cell cycle and enter quiescence (G0), undergo differentiation, or re-enter the cell cycle to proliferate in response to mitogenic signals.

The core molecular machinery controlling the mammalian cell cycle consists of a family of serine/threonine protein kinases called cyclin-dependent kinases (CDKs). These catalytic subunits are, in most cases, activated by association with cyclin regulatory subunits. The activity of CDK/cyclin complexes is further regulated by CDK inhibitors (CKIs), phosphorylation and dephosphorylation, ubiquitin-mediated degradation, transcriptional regulation, substrate recognition, and subcellular localization. The family of CDKs/cyclins/CKIs contains more than 30 members. They are implicated in essential cellular functions such as transcription, DNA damage repair, epigenetic regulation, metabolism, proteolytic degradation, stem cell self-renewal, neuronal functions, and spermatogenesis. Figure \(\PageIndex{1}\) shows the cell cycle and the involvement of CDKs/cyclins at key points.

Figure \(\PageIndex{1}\): Cyclin-dependent kinases (CDKs) and their cyclin regulatory subunits. CDK-cyclin complexes with direct functions in regulating the cell cycle. Aleem and Arceci. Frontiers in Cell and Developmental Biology 2015. DOI: 10.3389/fcell.2015.00016. Creative Commons Attribution 4.0 International

CDK3/cyclin C drives cell cycle entry from G0. CDK4/6/cyclin D complexes initiate phosphorylation of the retinoblastoma protein (pRb), and they sequester p21^Cip1 and p27^kip1 (not shown), which are both inhibitors of CDK2, thus promoting the activation of CDK2/cyclin E complex. In late G1, the CDK2/cyclin E complex completes phosphorylation and inactivation of pRb, which releases the E2F transcription factors, and the G1/S transition takes place. DNA replication takes place in the S phase. CDK2/cyclin A complex regulates progression through the S phase, and CDK1/cyclin A complex through the G2 phase in preparation for mitosis (M). Mitosis is initiated by CDK1/cyclin B complex (which will be modeled at this section's end). The activity of CDK1/cyclin B is tightly regulated by activating phosphorylation by the CDK-activating kinase CAK (a heterodimer of cyclin H-CDK7-MAT1) and inhibitory phosphorylations by Wee1 and Myt1 on Tyr15 and Thr14 (not shown). Some specific CDK4/CDK6 pharmacological inhibitors are also shown

CDKs with Direct Functions in Cell Cycle Regulation

The classical CDKs that directly regulate the mammalian cell cycle in complexes with cyclin subunits include CDK3, CDK4, CDK6, CDK2, and CDK1. CDK3 promotes cell cycle entry from quiescence in association with cyclin C. CDK8 has also been suggested to play a role in cell cycle entry from G0 and in the G1/S transition. In its simplest model, the mammalian cell cycle proceeds as follows:

• In early G1, CDK4/CDK6 in complex with cyclin D receive mitogenic signals that activate cell cycle entry, as shown in Figure \(\PageIndex{1}\). Key signaling events include the initiation of retinoblastoma protein (pRb) phosphorylation and the sequestration of p21^Cip1 and p27^kip1, both CDK2 inhibitors, thereby promoting activation of the CDK2/cyclin E complex. In late G1, CDK2 in complex with cyclin E completes the phosphorylation and inactivation of pRb, releasing the E2F transcription factors. E2F promotes transcription of cyclin E, which is necessary for the G1/S transition.
• Progression through the S phase is mediated by CDK2/cyclin A complex.
• CDK1/cyclin B complexes then initiate mitosis. We will model the regulation of CDK1 later in this section. CDK1/cyclin A complexes contribute to the preparation for mitosis in the G2 phase. The activity of CDK1/cyclin B is tightly regulated by activating phosphorylation by the CDK-activating kinase (CAK) (a heterodimer of cyclin H and CDK7) and inhibitory phosphorylations by WEE-1 and Myt1 on Tyr15 and Thr14. Mitosis starts after WEE-1 is degraded and CDC25C phosphatase releases the inhibitory phosphorylation on CDK1/cyclin B.

The cyclins are also expressed in a coordinated fashion throughout the cell cycle. Figure \(\PageIndex{2}\) shows the cyclin expression cycle. The timing of expression is consistent with the explanations above.

Figure \(\PageIndex{2}\): Cyclin expression cycle. https://commons.wikimedia.org/wiki/F...Expression.svg

Figure \(\PageIndex{3}\) shows a graph showing multiple progressions through the cell cycle.

Figure \(\PageIndex{3}\): Sustained oscillations of the CDK network in mammalian cells. Gérard and Goldbeter, Front. Physiol., 02 November 2012
Sec. Systems Biology Archive. https://doi.org/10.3389/fphys.2012.00413. Creative Commons Attribution License

The time evolution of cyclin D/Cdk4–6 (in black), cyclin E/Cdk2 (in blue), cyclin A/Cdk2 (in green), and cyclin B/Cdk1 (in red) is shown in the presence of a suprathreshold level of growth factor. Cyclin D/Cdk4–6 is the total active form of the kinases, which is composed of cyclin D/Cdk4–6 and also the complex formed by cyclin D/Cdk4–6 and p21/p27.

Note again the oscillatory nature of the cyclin B/Cdk1 complex, which will be explored at the end of this section.

CDKs with Transcriptional and Other Functions

In addition to their direct role in the mitotic cell cycle regulation, some classical CDK/cyclin complexes have essential functions in meiosis, such as CDK2, in transcription, and/or DNA repair. Other CDKs act by activating the classical CDKs, such as CDK7/cyclin H (CAK) and the related CDK20, also known as cell cycle-related kinase (CCRK). Some CDKs primarily influence transcription by phosphorylating the carboxy-terminal domain (CTD) of ribonucleic acid (RNA) polymerase II (RNA pol II). This phosphorylation also serves as a platform for RNA processing and chromatin regulation.

CDKs with important transcriptional roles include the CDK7/cyclin H/MAT1 complex, a component of the basal transcription factor TFIIH, which facilitates transcriptional initiation. CDK8/cyclin C, in addition to its role in transcription, is involved in the Wnt/β-catenin pathway and in inhibiting lipogenesis. Cyclin C can recruit CDK8 or CDK19 to the CDK8 module of the Mediator complex, which can function as a positive or negative transcription regulator by RNA pol II. CDK3/cyclin C also plays a role in NHEJ-mediated DNA damage repair. While CDK9 in complex with cyclin T forms the phospho-transcription elongation factor b (p-TEFb) and promotes transcriptional elongation, CDK9 also functions in the DNA damage response when complexed with cyclin K. CDK10/cyclin M phosphorylates the Ets2 transcription factor and positively controls its degradation by the proteasome. Ets2 plays a key role in cancer and development. CDK11/cyclin L controls the RNA pol II mediator complex assembly. CDK12 and CDK13, in complex with cyclin K, control RNA pol II transcription, and CDK12/cyclin K controls the DNA damage response.

Structure of CDKs–cyclins

Open Biology. Wood and Jane A. Endicott (2018) https://doi.org/10.1098/rsob.180112. Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/,

The structures of inactive cyclin-free kinases are very similar but vary at the N-terminal and C-terminal ends. Figure \(\PageIndex{4}\) shows an interactive iCn3D model of the prototypical active human cyclin-dependent kinase 2 with a bound ATP (1HCK).

Figure \(\PageIndex{4}\): Human cyclin-dependent kinase 2 with a bound ATP analog (1HCK) (Copyright; author via source). Click the image for a popup or use this external link: https://structure.ncbi.nlm.nih.gov/i...3Sf81pwLKQZx9A

The model shows that CDK2 has structural features shown in all the kinases we have studied previously:

• a smaller N-terminal lobe (light cyan) and larger C-terminal lobe (light magenta) in between which ATP binds (along with Mg²⁺).
• the C-helix (residues 45 – 55, purple) contains a conserved Glu. It forms an interaction with and helps position a key lysine in the active site, which facilitates ATP binding and transition state stabilization;
• hinge (residues 80 – 84, yellow),
• activation loop (residues 145– 172, red), which contains T160 (sticks, CPK colors, labeled) that becomes phosphorylated on activation by yet another kinase called CDK-activation kinase (CAK). When T160 is phosphorylated, the kinase binds to cyclin A. The loop starts and ends with the conserved residues DFG and APE, respectively.
• Not highlighted in the model is a conserved conformationally flexible glycine-rich region (residue 12-16) with the motif GXGXXG

In the inactive form, the N-terminal end of the activation loop has a short alpha helix that prevents the C-helix from adopting the correct position for catalysis. Activation requires movement of the C-helix, allowing the Glu in the C-helix to position the active site Lys.

CDK2–cyclin A activation

Cyclin A's binding to CDK2 activates it through the repositioning of the C-helix and the activation loop. When CDK2 is phosphorylated and bound to cyclin A, there is a large shift in the C-helix, allowing the interaction of the C-helix Glu with the active site Lys.

First, let's look at the structure of cyclins. Each cyclin has a unique sequence and structural features that allow it to interact with specific CDKs and associated proteins. However, they all have a conserved "cyclin box" structure containing about 100 amino acids.

Figure \(\PageIndex{5}\) shows an interactive iCn3D model of bovine cyclin A (1VIN).

Figure \(\PageIndex{5}\): Bovine cyclin A (1VIN) (Copyright; author via source). Click the image for a popup or use this external link: https://structure.ncbi.nlm.nih.gov/i...6j7kLcPzpRTAA9

Cyclin A has two linked cyclin box folds, each containing around 100 amino acids and five helices. They interact with the more disordered regions of unphosphorylated CDK2, resulting in low activity levels. Cyclin binding causes a large movement of the C-helix, enabling the Glu-Lys interaction. Phosphorylation of T160 leads to the repositioning of the activation loop.

Figure \(\PageIndex{6}\) shows an interactive iCn3D model of Phosphorylated cyclin-dependent kinase 2 bound to cyclin A (1JST)

Figure \(\PageIndex{6}\): Phosphorylated cyclin-dependent kinase 2 bound to cyclin A (1JST) (Copyright; author via source). Click the image for a popup or use this external link: https://structure.ncbi.nlm.nih.gov/i...cP6YsNcgUtGHW8

CDK2 is shown in cyan and cyclin A in gray. Here are some structural features represented in the model.

• the C-helix (residues 45 – 55, purple),
• activation loop (residues 145– 172, red), which contains pT160 (sticks, CPK colors, labeled)
• the catalytic "triad" Lys³³, Glu⁵¹, and Asp¹⁴⁵

Figure \(\PageIndex{7}\) shows an animation of structural changes in just CDK2 when "apo"-CDK2 (without bound cyclin A, pdbID 1HCK) binds cyclin A (1JST).

Figure \(\PageIndex{7}\): Structural changes CDK2 when it binds cyclin A

Gray represents the CDK2 structure in the absence of cyclin A. The structure of just CDK2 in the presence of cyclin A is shown in cyan. Large shifts in the C-helix (purple) and activation loop (red) occur on binding cyclin A.

Cyclin partners of CDK1 and CDK2

CDK1 is the closest member of the CDK family to CDK2, for which structures of the cyclin-free and authentic cyclin-bound forms can also be compared.

Depending on cyclin availability and concentration, CDK2 can bind cyclin A, B (if CDK1 expression is knocked down), and E (see Figure \(\PageIndex{1}\). The binding interface between CDK2 and the cyclins is quite large compared to that between CDK1 and the cyclins. Three large aromatic side chains (Y170, Y177, and Y258) are conserved in the binding interface. The corresponding amino acids in cyclin E are smaller (N112, I119, and L202).

The binding interface between CDK1 and the cyclins is smaller so it appears that it might preferentially interact with cyclins A and B to gain binding affinity through the more robust interactions with the aromatic groups in the interface in the CDK2: Cyclin A and CDK2:cyclin B complexes.

A comparison of the CDK1–cyclin B and CDK2–cyclin A/B/E structures also highlights the potential for these closely related CDKs to be differentially regulated by reversible phosphorylation. The antagonistic activities of Wee1/Myt1 kinases and Cdc25 phosphatases regulate the phosphorylation status of the CDK glycine-rich loop (defined by the GXGXXG motif, residues 11–16 in CDK2). The structure of CDK2–cyclin A phosphorylated on Y15 illustrates how phosphorylation promotes a glycine loop structure that antagonizes both peptide substrate binding and the ATP conformation required for catalysis. The flexibility of the glycine-rich loop is compatible with a model in which the phosphorylated Y15 side chain is solvent-exposed and accessible to both kinases and phosphatases. CDK1 is also regulated by active-site phosphorylation, and the conserved structure in this region suggests that the mechanism of inhibition is conserved.

CDK substrate recognition

Local and distal sequence motifs must be used to confer specificity to the binding of specific cyclins and other proteins to specific CDKs. One interesting example is examining the structure of a phospho-CDK2 Cyclin A in complex with a peptide substrate derived from the protein CDC6. Figure \(\PageIndex{8}\) shows an interactive iCn3D model of Phospho-CDK2:Cyclin A complex with a peptide containing both the substrate and recruitment sites of CDC6 (2CCI)

Figure \(\PageIndex{8}\): Phospho-CDK2 Cyclin A in complex with a peptide containing both the substrate and recruitment sites of CDC6 (2CCI) (Copyright; author via source). Click the image for a popup or use this external link: https://structure.ncbi.nlm.nih.gov/i...Ge8kdSBjxkaNd8

The color coding is the same as the models above:

• CDK2 is shown in cyan and cyclin A in gray.
• the C-helix (residues 45 – 55, purple),
• activation loop (residues 145– 172, red), which contains pT160 (sticks, CPK colors, labeled)
• the catalytic "triad" Lys³³, Glu⁵¹, and Asp¹⁴⁵

The 30mer peptide (numbers 67-96) shown in gold is a substrate for phosphorylation by the CDK2:cyclin A complex. It derives from an actual biological substrate, the cell division control protein 6 homolog, called CDC6-related protein. It is involved in a checkpoint control of the cell cycle that "checks" that DNA replication is completed before mitosis. It is discontinuous in the model since part of the bound peptide is intrinsically disordered and not observed in the crystal structure.

• The 1st fragment of the CDC6 peptide (67-73) contains the binding motif sequence S/T)PX(K/R) (the CDC6 substrate has the sequence 70Ser-Pro-Arg-Lys). Ser 70 is the target amino acid for phosphorylation by CDK2: cyclin A.
• The second fragment seen in the model (amino acids 85-96) contains another binding motif, RXL (in this peptide, RRL), which recruits cyclin A. This motif binds to the sequence MRAIL (210-214) in cyclin A.

Interestingly, this second binding site of cyclin A for its target protein is so far from the active site of CDK2:cyclin A. These interactions help determine CDK specificity and the binding partners of cyclins.

CDKs 7, 9, 12, and 13 phosphorylate the RNA polymerase carboxy-terminal domain (CTD). The CTD sequence is unusual, comprising 52 heptad repeats in humans, with the consensus sequence YSPTSPS. Extracted from cells, CTD residues S2 and S5 are the most abundantly phosphorylated serine residues, while S7 is phosphorylated to a lesser extent. CDK7 has been shown to predominantly phosphorylate Ser5 and Ser7; CDK9 phosphorylates all three serines; and CDK12 and CDK13 predominantly phosphorylate S2.

CDKs form complexes with target protein substrates and other proteins, which can serve as scaffolding anchors that bind both the CDK and the cyclin. Figure \(\PageIndex{9}\) shows an interactive iCn3D model of human CDK-activating kinase (CAK), a complex composed of cyclin-dependent kinase (CDK) 7, cyclin H, and the scaffolding protein MAT1 (6xbz).

Figure \(\PageIndex{9}\): Human CDK-activating kinase (CAK), a complex composed of cyclin-dependent kinase (CDK) 7, cyclin H, and MAT1 (6xbz)(Copyright; author via source). Click the image for a popup or use this external link: https://structure.ncbi.nlm.nih.gov/i...3bdn39ARgLijB7

The gray protein is CDK7, the cyan is Cyclin H, and the orange is MAT1. The purple again represents the C-helix of the CDK, and the red is the activation loop. The catalytic triad side chains in the CDK's active site are shown in CPK-colored sticks. Also shown is phospho-Ser in the activation loop.

The CDK-activating kinase (CAK) shown above phosphorylates the target S/T in the activation loop (the T-loop) of CDKs, thereby activating the kinase. In addition, it regulates the initiation of transcription by phosphorylating the YSPTSPS repeats in the C-terminus of RNA polymerase II subunit RPB1. There are 15 consecutive repeats in the sequence, and others are dispersed in the C-terminal domain.

We already mentioned the RXL motif found in cyclin-binding proteins, which recruits them to cyclins (for example, through its interaction with MRAIL (210-214) in cyclin A). Likewise, short motifs in cyclins bind to proteins that increase or decrease CDK activity.

Several cyclin-encoded protein-binding sites or short peptide motifs have been structurally characterized. A well-characterized example is the recycling of the cyclin RXL recruitment site, which is exploited to either enhance or inhibit CDK activity. Alternatively, short motifs encoded within the cyclin sequence can be used to dock cyclins to substrates, thereby enhancing CDK activity and localizing them to CDK regulators, frequently resulting in a loss of CDK activity. Members of the p27KIP1/p21CIP1 cyclin-dependent kinase inhibitor (CKI) family share an RXL motif with RXL-containing substrates and compete with them for CDK–cyclin association. The INK (inhibitors of CDK) family of CKIs selectively inhibits CDK4 or CDK6 and, through an allosteric mechanism, disfavors CDK–cyclin binding. Their tandem ankyrin repeat structures, exemplified by Cy CDK6–p19INK4d, bind near the CDK hinge, on the interface opposite the surface remodeled upon cyclin association.

Modeling the Cell Cycle - Oscillations

Lastly, we will focus on mathematical models that show how the oscillatory behavior of CDK1/Cyclin B arises (remember that CDKs become active on binding to a cyclin). When CDK1 is activated, the cell is driven into mitosis. It is driven out of mitosis by activation of the anaphase-promoting complex (APC), which contains APC-Cdc20, an E3 ubiquitin ligase. Yeast Cdc20 is an activator protein that regulates the ubiquitin ligase activity of APC by binding at the right time in the cell cycle to B cyclins that contain a D box motif. This recruits the B cyclin:CDK1 to APC, which ubiquitinates the cyclin B, leading to its degradation by the proteasome.

Figure \(\PageIndex{10}\) shows the levels of both cyclin B/CDK1 (red) and cyclin A/CDK2 (green) with time. Now, in your mind, imagine another curve on the graph showing similar oscillations of activated APC, only time-shifted a bit so that the active APC trails that of active cyclin B/CDK1. When cyclin B/CDK1 is at its active peak, active APC is already beginning its rise.

Figure \(\PageIndex{10}\): The switch to sustained oscillations of cyclin A/Cdk2 (in green) and cyclin B/Cdk1 (in red) is shown following the overexpression of AP1. Gérard and Goldbeter, ibid.

The dissociation constant, K_D, for CDK1 and cyclin B is about 28 nM, representing high-affinity binding. Cyclin B's ubiquitination and degradation allow the release of CDK1 and the inhibition of its activity. The oscillatory behavior in activity occurs only on the overexpression of yet another protein, AP1.

Let's explore in detail a model proposed by Ferrell et al. that accounts for the 25-min oscillatory behavior of CDK1 in Xenopus (frog) eggs. The eggs are very large and, perhaps because of their size, face different constraints in their cell cycle. For example, cells can enter mitosis before DNA synthesis is complete. The players that regulate their activity are shown below in Figure \(\PageIndex{11}\).

Figure \(\PageIndex{11}\): Proteins that help regulate the activity of CDK1 in Xenopus eggs.

• Cyclin: binds to and activates CDKs; active CDK1 drives cells into mitosis.
• APC: anaphase-promoting complex; active APC drives the cell out of mitosis with its E3 ubiquitin ligase activity, modifying cyclin and targeting it for proteolysis;
• Wee1: nuclear Ser/Thr kinase
• Cdc25: cell cycle division phosphatase that activates cyclin-CDK1.

Our goal in this discussion is to model the actual oscillatory behavior of CDK1 and show you how models are built and tested. When modeling enzyme inhibition data, it is important to fit the data to multiple models (reversible competitive, uncompetitive, mixed, and noncompetitive) to determine which model best fits. Typically, one starts with the simplest possible model and then advances to more complex models until the model that best fits the data is found. Other examples, which are more relevant here, involve fitting binding and kinetic data with equations that yield hyperbolic (for saturation binding and simple Michaelis-Menten kinetics) and sigmoidal (for cooperative binding and allosteric enzymes) fits.

So, let's start with the simplest model that might lead to CDK1's oscillatory activity. In this section, the active form of CDK1 will be designated as CDK1*.

Model 1: One-step process with negative feedback - CDK1* inhibits its own activation (for example, by activating APC and hence ubiquitinylation of cyclin)

We saw in the Vcell models for the MAP kinase cascade that feedback phosphorylation of the first enzyme in the cascade (MAPKKK) by the last enzyme in the cascade, MAPKPP, leads to oscillations in enzyme activity. Could it explain the oscillations in the activity of CPK1 in Xenopus eggs? Let's make the following set of assumptions, which are all understandable from the material presented in previous chapters:

1. CDK1* is inhibited by APC* (active APC), and to make it simpler, APC* can be expressed as a simple function of CDK1*, so there is just one species that varies in the flux equation;
2. CDK1 activation occurs on rapid, high-affinity binding of cyclin, which is synthesized at a constant rate a1;
3. The rate of CDK1 activation to produce CDK1* is given by mass action = rate activation - rate inactivation;
4. APC is activated to APC "instantaneously" by CDK1*, so APC* is a very sensitive “cooperative” function of CDK1*, which can replace APC*. We will use the Hill equation for this "instantaneous" (or highly cooperative) effect, which yields a sharp, sensitive, cooperative rise in complex formation rather than a simple formation of a complex between CDK1* and APC. We explored this property of the Hill equation in Chapter 5.3.5, Mathematical Analysis of Cooperative Binding.

Here is the Hill expression commonly used to fit the fractional saturation of a species empirically

\begin{equation}
Y=\frac{L^n}{K_D^n+L^n}
\end{equation}

It's a bit different than the Hill equation we saw for modeling the cooperative binding of O₂ to hemoglobin (Chapter 5.3.5) since the K_D term is also raised to the power n (which is not in the actual Hill equation). However, we did see that for oxygen binding to hemoglobin,

\begin{equation}
\mathrm{K}_{\mathrm{D}}=\mathrm{P}_{50}^{\mathrm{n}}
\end{equation}

Hence, the empirical expression used in fitting Model 1 is completely in accord with the Hill treatment of cooperativity. Again, we use the Hill equation to model binding and kinetic data that are highly sensitive to conditions. It provides an additional parameter to aid in fitting the data and testing models.

Two different representations of a reaction diagram showing Model 1 are shown in Figure \(\PageIndex{12}\).

Figure \(\PageIndex{12}\): Two different representations of a reaction diagram for Model 1 - Activity of CDK1.

The representation on the right is from Vcell. The one on the left shows that CDK1 can inhibit itself.

Model 1 and its associated assumptions lead to the following differential equation that can be solved numerically in Vcell. All of the VCell outputs below were obtained from VCell models kindly provided by Leslie Loew.

\begin{equation}
v=\frac{d C D K 1^*}{d t}=a_1-b_1\left(C D K 1^*\right)\left(A P C^*\right)=a_1-b_1\left(C D K 1^*\right)\left(\frac{C D K 1^{* n 1}}{K_1^{n 1}+C D K 1^{* n 1}}\right)
\end{equation}

This rate equation has two terms (assumption 3). The first is the rate that CDK1* forms (a constant a1 defined by the rate of cyclin synthesis) and the rate at which it is degraded by APC*. The constant b1 in the second term can be interpreted as a second-order rate constant for the interaction between CDK1* and APC*, a process that inactivates CDK1*.

The [APC*] in the middle equation is replaced with the Hill equation for APC's effective fractional saturation concentration (see assumption 4 of Model 1 described above) in the right-hand side.

\begin{equation}
\left(A P C^*\right)=\left(\frac{C D K 1^{* n 1}}{K_1^{n 1}+C D K 1^{* n 1}}\right)
\end{equation}

We will define the activity of the system as the rate at which CDK1* forms.

\begin{equation}
\text { Activity }=\frac{d C D K 1^*}{d t}
\end{equation}

Now let's see if changing the Hill coefficient n1 can cause oscillations in CDK1*.

---

MODEL

CDK - Model 1

Initial Conditions

Select Load [model name] below

 Select Start to begin the simulation.

Interactive Element

Select Plot to change Y axis min/max, then Reset and Play  |  Select Slider to change which constants are displayed |  Select About  for software information.

Move the sliders to change the constants and see changes in the displayed graph in real-time.

Time course model made using Virtual Cell (Vcell), The Center for Cell Analysis & Modeling, at UConn Health.  Funded by NIH/NIGMS (R24 GM137787); Web simulation software (miniSidewinder) from Bartholomew Jardine and Herbert M. Sauro, University of Washington.  Funded by NIH/NIGMS (RO1-GM123032-04)

---

 

Note that all the graphs plateau quickly, at which point the CDK1* activity is constant. The graph (gray) of the curve with the highest value of the Hill coefficient (n1=24) is linear and then abruptly plateaus. The slope of the velocity curve over the entire linear part of the n1=24 graph curve is 0.1, which is the value set for the rate of activation of CDK1. Then, suddenly, at around 4 seconds, an "almost infinitely cooperative" shift to a constant formation rate occurs, arising from an abrupt increase in the inactivation rate of CDK1* by the APC complex. These graphs do not show oscillations.

Now let's see the graph with no feedback inhibition, much as we did with the MAPK cascade in Chapter 12.4. The easiest way to do that is to set b1 = 0, the "second"- order rate constant for the interaction between CDK1* and APC* in the model. The graph is shown in Figure \(\PageIndex{13}\).

Figure \(\PageIndex{13}\): Activity of CDK1* in Model 1 in the absence of feedback inhibition.

The activity of CPK1* continues to increase. When feedback inhibition is added, the curve "bends" to a plateau, but it does not start to decrease and shows no signs of oscillations. Time to move on to a more complex model!

Model 2: Two-species model with activation and inhibition-

This model is more complicated and shows two species (CDK1 and APC) both of which are activated and inhibited. We need two mass action differential equations, one for each. Figure \(\PageIndex{14}\):

Figure \(\PageIndex{14}\): Two different representations of a reaction diagram of Model 2

The equation for dCDK1*/dt is the same as in Model 1, as is repeated below.

\begin{equation}
v=\frac{d C D K 1^*}{d t}=a_1-b_1\left(C D K 1^*\right)\left(A P C^*\right)=a_1-b_1\left(C D K 1^*\right)\left(\frac{C D K 1^{* n 1}}{K_1^{n 1}+C D K 1^{* n 1}}\right)
\end{equation}

Likewise, the equation for dAPC*/dt consists of two terms, one for its activation and one for its inhibition.

\begin{equation}
v=\frac{d A P C^*}{d t}=k_f A P C-k_r A P C^*
\end{equation}

Assume that k_f, the rate constant of the activation of APC, is equal to a constant a₂ times a Hill function of CDK1*, and k_r, the rate constant for the inactivation of APC*, is simply b₂. Then the equation becomes

\begin{equation}
v=\frac{d A P C^*}{d t}=a_2\left(\frac{C D K 1^{* n 2}}{K_2^{n 2}+C D K 1^{* n 2}}\right) A P C-b_2 A P C^*
\end{equation}

Let's look at the output graphs for the following initial condition:

• CDK1* = 0 uM
• APC = 1 uM
• APC* = 0 uM

 

---

MODEL

CDK Model 2

 

See equations in text.

Select Load [model name] below

 Select Start to begin the simulation.

Interactive Element

Select Plot to change Y axis min/max, then Reset and Play  |  Select Slider to change which constants are displayed |  Select About  for software information.

Move the sliders to change the constants and see changes in the displayed graph in real-time.

Time course model made using Virtual Cell (Vcell), The Center for Cell Analysis & Modeling, at UConn Health.  Funded by NIH/NIGMS (R24 GM137787); Web simulation software (miniSidewinder) from Bartholomew Jardine and Herbert M. Sauro, University of Washington.  Funded by NIH/NIGMS (RO1-GM123032-04)

---

 

Figure \(\PageIndex{15}\) shows static graphs of just CDK1* activity vs time for n1 values of 1, 4, 8, and 24 for Model 2.

Figure \(\PageIndex{15}\): Graphs of just CDK1* activity vs time in Model 2 for n1 values of 1, 4, 8, and 24.

Wow! By simply adding an additional species to the model and a second differential equation for it, we see the first signs of oscillatory behavior in CDK1*'s activity. The output at higher n1 values is best described as damped oscillations. Now, let's try a final third model.

 

Model 3: Three-species model with activation and inhibition-

This model contains the enzyme Plk1 (Polo-like kinase 1, also called serine/threonine protein kinase 10-A), APC, and CDK1. These three species are all activated and inhibited. Assume that CDK1 activates Plk1 and that it also helps activate APC. Following the arrows in the left part of the figure below shows that it acts as an "intermediary" between CDK1 and APC. Two different representations of a reaction diagram of Model 3 are shown in Figure \(\PageIndex{16}\).

Figure \(\PageIndex{16}\): Two different representations of a reaction diagram of Model 3

We have three species, so we need three differential equations, as shown below.

\begin{equation}
v=\frac{d C D K 1^*}{d t}=a_1-b_1\left(C D K 1^*\right)\left(\frac{A P C^{* n 1}}{K_1^{n 1}+A P C^{* n 1}}\right)
\end{equation}

\begin{equation}
v=\frac{d P l k 1^*}{d t}=a_2\left(1-P l k 1^*\right)\left(\frac{C D K 1^{* n 2}}{K_2^{n 2}+C D K 1^{* n 2}}-b_2 C D K 1^*\right.
\end{equation}

and

\begin{equation}
v=\frac{d A P C^*}{d t}=a_3\left(1-A P C^*\right)\left(\frac{P l k 1^{* n 3}}{K_3^{n 3}+P l k 1^{* n 3}}-b_3 A P C^*\right.
\end{equation}

The equation for the activation of CDK1 is the same as in Models 1 and 2.

The equation for the activation of APC is similar to Model 2, with k_f modeled as a Hill function of Plk1star, which activates APC

The equation for the activation of Plk1 is similar to Models 2 and 3, with k_f modeled as a Hill function of CDK1star, which activates Plk1

Although you probably can't write these differential equations by yourself, hopefully, you can see that they make sense.

---

MODEL

CDK Model 3

 

Initial values

Select Load [model name] below

 Select Start to begin the simulation.

Interactive Element

Select Plot to change Y axis min/max, then Reset and Play  |  Select Slider to change which constants are displayed |  Select About  for software information.

Move the sliders to change the constants and see changes in the displayed graph in real-time.

Time course model made using Virtual Cell (Vcell), The Center for Cell Analysis & Modeling, at UConn Health.  Funded by NIH/NIGMS (R24 GM137787); Web simulation software (miniSidewinder) from Bartholomew Jardine and Herbert M. Sauro, University of Washington.  Funded by NIH/NIGMS (RO1-GM123032-04)

---

 

 

Figure \(\PageIndex{17}\) shows graphs of just CDK1* activity vs time for n1 values of 1, 4 and 8 for Model 3

Figure \(\PageIndex{17}\): Graphs of CDK1* activity vs time for n1 values of 1, 4 and 8

Finally, we observe oscillatory behavior in CDK1*activity, but only at higher values of the Hill coefficient (n1 = 4 and 8).

Other models can produce oscillations, but this one seems perhaps the most comprehensible to students who have studied mass-action equations, Hill binding, and kinetic equations. Of course, the three-component system described in Model 3 is embedded within a larger pathway of inputs and outputs, so other factors likely affect the oscillatory behavior.

Summary

(Summary written by Claude, Sonnet 4.6, Anthropic)

The eukaryotic cell cycle is driven by a molecular oscillator built from cyclin-dependent kinases (CDKs) and their cyclin regulatory partners. The four phases of the cycle—G1 (growth and preparation), S (DNA replication), G2 (preparation for division), and M (mitosis)—are governed by the sequential activation and inactivation of distinct CDK-cyclin complexes whose timing is coordinated by oscillatory cyclin expression, CDK inhibitor proteins (CKIs), and phosphorylation-dependent switches. A critical commitment point occurs at the G1/S transition, where CDK4/6:cyclin D initiates phosphorylation of the retinoblastoma tumor suppressor protein (pRb) and sequesters the CKI proteins p21^Cip1 and p27^Kip1; CDK2:cyclin E then completes pRb hyperphosphorylation, releasing E2F transcription factors that drive expression of S-phase genes. Cells that fail to complete this transition exit into quiescence (G0), differentiate, or await appropriate mitogenic signals.

CDKs share a conserved bilobal kinase domain whose activation requires two cooperative events: binding to a cyclin, which repositions the C-helix to enable the catalytic Glu–Lys interaction, and phosphorylation of a threonine in the activation loop (Thr160 in CDK2) by the CDK-activating kinase (CAK, a trimeric complex of CDK7, cyclin H, and MAT1). CAK itself serves dual roles—activating the cell cycle CDKs and phosphorylating the heptad repeat sequence YSPTSPS in the C-terminal domain of RNA polymerase II to initiate transcription. CDK1:cyclin B, the master kinase driving mitotic entry, is held inactive by WEE1/Myt1-mediated phosphorylation of Tyr15 and Thr14 until CDC25C phosphatase removes these marks in a positive feedback loop. Exit from mitosis is driven by the APC/C-Cdc20 E3 ubiquitin ligase, which recognizes a D-box degron in cyclin B, ubiquitinates it, and targets it for proteasomal degradation, thereby inactivating CDK1 and resetting the cycle.

CDKs achieve substrate selectivity through a two-site docking mechanism: a local consensus phosphorylation motif (S/T)PX(K/R) positions the target Ser or Thr at the active site, while a distal RXL recruitment motif on the substrate binds a conserved hydrophobic patch on the cyclin surface, as illustrated by the CDK2:cyclin A:CDC6 peptide complex structure. The INK4 family of CKIs (p16, p19, etc.) inhibits CDK4/6 through allosteric ankyrin-repeat domain interactions that disfavor cyclin binding, while the CIP/KIP family (p21, p27) uses RXL-containing sequences to compete with substrates for cyclin binding and occupies the CDK active site as pseudosubstrates.

Mathematical modeling of CDK1 oscillations reveals that the minimal network sufficient to generate sustained oscillatory behavior requires at least three interacting species with cooperative (high Hill coefficient) responses. A one-component negative feedback model (CDK1 inhibiting its own activation through APC) produces only stable steady states. Adding APC as an explicit second species introduces damped oscillations, and incorporating PLK1 as an intermediary activator of APC creates a three-node network capable of generating the ~25-minute oscillations observed in Xenopus egg extracts. These models illustrate a general principle that emergent oscillatory behavior in biochemical networks arises from combinations of positive and negative feedback loops with ultrasensitive (cooperative) switching—the same principle underlying MAPK cascade oscillations and Ca²⁺ spiking discussed in earlier chapters of this textbook.