1 Trigger Waves as Organizers of Mitotic Space-Time: A Roadmap for the Wave Paradigm of Cellular Regulation

Ronald Joseph Raiber
Raiber Clinic| Laboratory of Quantitative Cell Biology | Center for Submolecular Glinical Medicine: contact@raiber.clinic
Submitted to The New Science Monitor | Perspective / Roadmap Article

1.1 Abstract

Mitotic entry in large cells is coordinated not through the equilibrium establishment of concentration thresholds but through travelling trigger waves of Cdk1 activation propagating at velocities on the order of 60 micrometers per minute. This discovery, made by Chang and Ferrell in 2013, raised fundamental questions that remain unanswered. Is the Cdk1 wave singular, or does it represent merely the most visible component of a hierarchy of coupled waves involving multiple kinases and phosphatases? Does the kinetic structure of the wave carry information read by downstream cellular systems beyond the amplitude signal? Can disruption of the wave mechanism contribute to mitotic catastrophes in polyploid giant cancer cells? Does a feedback coupling exist between the biochemical Cdk1 wave and mechanical waves of cortical reorganization? The present article formulates an integrated research programme addressing these four questions. We propose concrete mathematical models based on coupled reaction-diffusion equations, detailed experimental designs including multiplex FRET imaging of several kinases simultaneously, and quantitative predictions that allow each hypothesis to be falsified. We introduce the concept of the mitotic age gradient as a measurable consequence of velocity differences between hierarchical waves, the concept of the cell-cycle wave code as an informational extension of the equilibrium model of signalling, and the concept of mechano-chemical frequency entrainment as a new type of coherent structure in cellular biophysics. The programme unifies systems biology, nonlinear physics, and oncology within a single formal framework, which we propose to call the wave paradigm of cellular regulation.

1.2 1. Introduction: From Thresholds to Waves

For three decades following the pioneering work of Novak and Tyson, mitotic entry was described primarily as a bistable switch. The cell accumulates cyclin B, the Cdk1-cyclin B complex reaches a threshold, and the system makes a sharp transition from the interphase to the mitotic state through a double-positive feedback mechanism in which Cdk1 activates its phosphatase Cdc25 and simultaneously inhibits its kinase Wee1. Mathematically this is described by an S-shaped curve of steady states with two stable branches and one unstable intermediate. The Hill coefficients measured by Trunnell and colleagues in 2011 and by Kim and Ferrell in 2007 confirmed the ultrasensitivity of this switch, with an effective cooperativity coefficient of approximately 11 for the Cdc25 system and approximately 3.5 for the Wee1 system.

Bistability as such, however, describes only the homogeneous state. A real cell has spatial extent, and the question of how the mitotic transition is coordinated in space remained open until the key experiment of Chang and Ferrell in 2013. Using cell-free extracts of Xenopus laevis oocytes in Teflon microfluidic channels with a fluorescent sensor of Cdk1 activity, they showed that Cdk1 activation propagates as a travelling front at a velocity of approximately 60 micrometers per minute. Critically, this velocity was independent of channel length, which excludes simple diffusion, for which the characteristic time scales as the square of distance, and points instead to a trigger wave of reaction and diffusion. Mathematically, the velocity of such a wave is determined by the relation , where denotes the diffusion coefficient of the signalling protein and the characteristic reaction constant, and is independent of system size.

This discovery, confirmed and extended by subsequent work of Cheng and Ferrell in 2018 on apoptotic waves, Vergassola, Deneke and Di Talia in 2018 on Drosophila embryos, and Nolet and colleagues in 2020 on Xenopus embryos in vivo, placed cell biology before the necessity of revising its foundational understanding of signalling. If coordination is achieved not through equilibrium concentrations but through the kinetics of travelling waves, then the conceptual apparatus of equilibrium biochemistry proves insufficient, and cell biology must borrow concepts from nonlinear physics: reaction-diffusion equations, Fisher-KPP theory, bistable fronts, phase synchronisation. The present article formulates four unsolved problems arising from the discovery of Chang and Ferrell and proposes an integrated research programme to address them. We restrict ourselves to the formulation of hypotheses, the description of theoretical models and experimental designs, and the discussion of expected results, leaving experimental realisation to future work.

1.3 2. Problem One: The Hierarchy of Mitotic Waves

1.3.1 2.1. Statement of the Problem

The discovery of Chang and Ferrell concerned a single wave, namely the activation of Cdk1. The real sequence of mitotic events, however, involves the activation of multiple kinases and phosphatases in a defined order: Cdk1 activates Plk1 through phosphorylation of Bora, Plk1 modulates the APC/C, Aurora A regulates centrosomal functions, Aurora B within the chromosomal passenger complex controls correction of kinetochore attachment errors, and mitotic exit is determined by reactivation of the phosphatases PP1 and PP2A-B55, which dephosphorylate mitotic substrates. Each of these components possesses its own activation kinetics, its own molecular weight determining the diffusion coefficient, and consequently its own characteristic propagation velocity. The central question of this section is whether all these components propagate as independent waves and, if so, what the consequences of their different velocities are for the spatial organisation of mitosis.

1.3.2 2.2. Theoretical Model

We propose to describe the spatiotemporal dynamics of mitotic kinases and phosphatases by a system of coupled reaction-diffusion equations:

[ = D_i ^2 C_i + f_i(C_1, , C_N), i = 1, , N]

For each component the reaction function is chosen from the class of regularised Hill kinetics with double positive feedback. For the active concentration of Cdk1, denoted , the basic function takes the form.

[ f(C, B) = k_s + k_a,+(C)(B – C) – k_i,-(C),C ]

where denotes the total concentration of the Cdk1-cyclin B complex available for activation, and the sigmoidal functions and incorporate cooperativity through Hill exponents:

[_+(C) = _0 + (1 – 0), -(C) = 1 – (1 – _1)]

with parameters , , and from experimental data. Cross-regulatory coupling terms are specified as Hill-type activating or inhibiting functions of the corresponding upstream component. Stationary travelling fronts are sought in the form , reducing the system to a set of ordinary differential equations with boundary conditions at infinity. The velocity is the eigenvalue of this boundary-value problem. Numerical continuation of fronts with respect to control parameters is performed by the pseudo-arclength method in AUTO-07p or BifurcationKit.jl, yielding complete bifurcation diagrams and critical surfaces in parameter space.

1.3.3 2.3. Quantitative Predictions

Substituting published diffusion coefficients for each protein, measured by fluorescence correlation spectroscopy in cytoplasm, together with characteristic kinetic constants from the literature, we obtain the following velocity estimates. The Cdk1 wave (complex approximately 90 kDa, µm²/s) should propagate at approximately 60 micrometers per minute, consistent with the Chang and Ferrell experiment. The Plk1 wave (approximately 66 kDa, µm²/s, but slower kinetics) is expected at approximately 40 micrometers per minute. The Aurora B wave within the chromosomal passenger complex (approximately 200 kDa, µm²/s) should slow to approximately 30 micrometers per minute. The PP2A-B55 phosphatase wave (approximately 150 kDa, slow dephosphorylation kinetics) is expected at approximately 20 micrometers per minute. The key qualitative prediction is that the fronts of different waves diverge linearly in time. The distance between fronts and grows as . In a large cell this creates a spatial gradient of mitotic progression, which we term the mitotic age gradient. At any given moment there exists a region of the cell that has already entered mitosis by the Cdk1 criterion but has not yet activated the APC/C, while an adjacent region may be in a different phase state. The magnitude of this gradient is directly proportional to cell size, which immediately creates a connection to the pathology of large cells discussed in Section 3.

1.3.4 2.4. Proposed Experimental Approach

Testing the hierarchical wave hypothesis requires simultaneous visualisation of multiple kinase activities in real time with micrometer spatial resolution. We propose a multiplex set of four FRET sensors with widely separated spectral characteristics. The Cdk1 sensor is built on the Eevee-CDK construct with the fluorophore pair mTurquoise2 (excitation 434 nm, emission 474 nm, quantum yield 0.93, fluorescence lifetime 4.0 ns) and sYFP2 (emission 527 nm), with a Förster radius of 5.8 nm. The Plk1 sensor is designed de novo on the phosphopeptide substrate FoxM1 (sequence GRSSPIMTPYPS) with a POLO-box intramolecular binding domain and the pair mOrange2 (excitation 549 nm, emission 565 nm, lifetime 2.8 ns) and mCherry (emission 610 nm), Förster radius 6.0 nm. The Aurora B sensor uses the infrared pair iRFP670 (excitation 643 nm, emission 670 nm, lifetime 0.75 ns) and iRFP720 (emission 720 nm), Förster radius 5.5 nm, which frees the visible spectral range for other sensors and reduces autofluorescence in Xenopus embryo imaging. The PP2A-B55 sensor operates in FLIM mode on the pair mNeonGreen (excitation 506 nm, emission 517 nm, lifetime 3.0 ns in the free state and 2.1 ns in the FRET-active conformation) and mScarlet-I (emission 594 nm), Förster radius 6.2 nm, allowing quantitative separation from overlapping intensity sensors through donor lifetime measurement by TCSPC. Imaging is performed on a confocal microscope with a hyperspectral detector in lambda-stack mode from 460 to 750 nm in steps of 8.9 nm, with subsequent spectral decomposition by non-negative matrix factorisation. The FLIM channel for PP2A-B55 is implemented through two-photon excitation at 950 nm with TCSPC detection and two-exponential decay fitting. The experimental platform consists of Xenopus cell-free extracts in PEG-silane-passivated PDMS microfluidic channels, with wave initiation through a local thermal impulse. Control experiments include pharmacological inhibition of individual components (BI 2536 for Plk1, ZM447439 for Aurora B, okadaic acid for PP2A) with the predicted disappearance of the corresponding wave without affecting upstream components, as well as a size-independence test for wave velocities in droplets of variable volume.

1.4 3. Problem Two: Wave Pathology of Polyploid Giant Cancer Cells

1.4.1 3.1. Statement of the Problem

Aneuploidy is present in approximately 90 percent of solid tumours. Polyploid giant cancer cells (PGCC), described in the work of Niu and Liu in 2017 and Mirzayans in 2018, play a key role in resistance to chemotherapy and disease relapse, with volumes 10 to 100 times greater than normal. Accepted mechanisms of aneuploidy, namely defects in the spindle assembly checkpoint, cohesion disruptions, and multipolar spindles, do not fully account for the high frequency of mitotic catastrophes in these cells. We propose the hypothesis that an increase in cell size leads to functional dysregulation of the wave mechanism for mitotic coordination, making an independent contribution to PGCC mitotic catastrophes.

1.4.2 3.2. Biophysical Argument

The logic of the hypothesis is elementary. The time for a wave to traverse the cell scales linearly with radius: . The degradation time of cyclin B is determined by the enzymatic kinetics of the APC/C and does not depend on cell size: . When the cell is enlarged by a factor , wave traversal time grows as , while degradation time remains unchanged. When exceeds , a situation arises in which mitotic exit begins in one part of the cell before the entry wave has reached another part. The cell simultaneously occupies two mutually incompatible states, which constitutes the definition of mitotic catastrophe. The critical enlargement factor at typical values µm/s, µm, and s⁻¹ gives , which coincides with the observed volume threshold for PGCC. This is a non-trivial quantitative coincidence requiring experimental verification.

1.4.3 3.3. Proposed Experiments

Testing the hypothesis requires direct measurement of Cdk1 wave parameters in isogenic cell pairs of different sizes: diploid HCT116 and their tetraploid derivatives, RPE1 and PGCC induced from them by inhibition of cytokinesis with dihydrocytochalasin B, and clinically relevant lines MDA-MB-231 after docetaxel treatment and HEY after cisplatin. The measured wave parameters include front velocity, width, amplitude, and coherence, defined as the reciprocal of the standard deviation of front arrival time at different peripheral points of the cell. The hypothesis predicts a systematic decrease in front coherence with increasing cell size and a positive correlation between loss of coherence and frequency of mitotic abnormalities. The decisive causal test is a rescue experiment in which the effective diffusion coefficient in PGCC is artificially increased through optogenetic activation of intracellular motors. If this restores front coherence and reduces the frequency of mitotic catastrophes, the causal role of the wave mechanism will be established. A pharmacological rescue variant through Wee1 inhibition with adavosertib should increase wave velocity and shift the critical cell size upward, which is in principle testable.

1.4.4 3.4. Therapeutic Implications

If the hypothesis is confirmed, it would open a qualitatively new class of therapeutic targets. Unlike molecular targets such as receptors, enzymes, and immune checkpoints, the wave target attacks the spatiotemporal coordination of cellular regulation and is theoretically not subject to point-mutation-based resistance, since it acts at the level of signal propagation physics rather than at the level of individual molecules. This creates the prospect of a strategy complementary to existing approaches and particularly valuable for eliminating the PGCC fraction responsible for clinical relapses.

Trigger Waves as Organizers of Mitotic Space-Time

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