### #10 The Doppler effect during development

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The development of somite is one of the most important hallmarks in vertebrate embryonic development. Somites are developed from presomatic mesoderm(PSM). Previous researches demonstrate the existance of a period Ts. One somite would grow from PSM every Ts. Because of such periodicity, the developmental biologists hypothesized that there are some genetic oscillation in cells and the period of somite growth is determined by the oscillation period of genetic circuits. After numerous researches, the oscillating genes were discovered. However, does that fully determine the period of somite formation?

During the somite formation of zebrafish, PSM would contract and become smaller from cranial side to caudal side. Only PSM cells have genetic oscillation. The genetic oscillation is produced in the caudal end of PSM by gene her1, and its oscillation could spread from caudal to cranial through the Notch signaling pathway [1]. (The detail is not that important here.) The authors of suggested reading use genetic engineering to make the oscillation of her1 gene directly visible under microscope, and they found that the oscillation period of her1 at the anterior(cranial) Ta and the posterior (caudal) Tp was not the same. Besides, it is Ta that determines the period of somite formation.

How could we explain such difference? The author suggested a physical model (we'll discuss later) and believed that the difference might come from Doppler effect and dynamic wavelength effect. Doppler effect states that the difference in period comes from the relative speed, while the dynamic wavelength effect tells that the difference in period comes from the changes of wavelength with time. They used wavelet transformation to convert the oscillation of her1 gene into the oscillation of phase, and they calculated the expected difference in period from his model. The expected period difference is 8.4% while the experimentally observed difference is 8.8%, so the Doppler effect and dynamic wavelength effect could fairly explain the period of somite formation. This result forces us to think above the clock-and-wavefront theory when it comes to the somite formation.

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Now let's define an coordinate x for PSM. x=0 at the caudal end of PSM while x = a(t) at the cranial end of PSM. (Noted that PSM becomes smaller as time pass, so da/dt<0.) The phase of oscillation at the cranial end and caudal end could be expressed as
So the angular frequency at the cranial and caudal end could then expressed as
No let's consider the phase difference between the cranial and caudal end. Define phase profile ψ as

Then the phase of cranial end could then be written

Differentiate with respect to t and we will get:
The term (∂ψ/∂x*da/dt) represents the contribution from Doppler effect while the term (∂ψ/∂t) means the changes in phase profile with time, or the dynamic wavelength effect. ψ and da/dt could be experimentally measured.

Next we will talk about the clock-and-wavefront theory. In clock-and-wavefront theory, the oscillation of PSM is homogeneous. So Φ(x, t) = Ωt and ψ(x,t)=0. However, this definitely can't explain the oscillation observed in zebrafish.

There is another theory called scaling wave pattern theory, which proposes that the wavelength becomes shorter as PSM contracts. Therefore
According to this hypothesis, we found that
And there is no period change because the effect of Doppler effect and the dyanmic wavelength effect cancel out each other. The somite formation of mice follows scaling wave pattern.

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