Astronomy

Why do PMS stars on the Hayashi track remain at a constant temperature while they contract?

Why do PMS stars on the Hayashi track remain at a constant temperature while they contract?


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Why do pre-main-sequence stars on the Hayashi track remain at a constant temperature while they contract?

I've read the Wikipedia article, so no need to repeat the derivation.

What I took away from that is that the extra heat generated by contraction is given off almost exactly because the whole star is convective and the atmosphere is very capable of radiating that heat away if the opacity is mainly caused by hydrogen ions, which is the case in the cooler of the pre-main-sequence stars.

Is that correct?


The surface temperature of fully convective stars is set by their opacity mechanism, often emission when the H minus ion is created. The rate the star emits light is not set by the contraction rate, it is the other way around-- the star loses heat at a rate set by its structure, and that's what sets the contraction rate. (As said, the rate of gravitational energy release is always double the rate the star loses heat to starlight.) The reason the surface temperature always has to be something like 3000 K is pretty complicated, but it seems to be due to the fact that this temperature is a kind of peak in the surface's ability to emit light. Convection sets the temperature structure, but there are a range of possible solutions with different internal energy, so the history of heat loss picks out whichever solution has the right internal energy. But just why the surface is always about the same T is still the trickiest part to understand, I am hoping to get a better handle on that eventually. Ignore anything you see in introductory textbooks, they never get it right.


Talk:Star formation

  • Article requests : Current sites of star formation (possibly Gould Belt, Taurus, Ophiuchus, Orion, etc) frequency/duration of star formation (possibly Star formation efficiency). The free falling picture discussed should be suplemented by the standard model (subcritical cores collapsing after ambipolar diffusion makes them supercritcal) and supersonic turbulent model. Discussing initial mass functions and star formation rate, the effect of the magnetic field and turbulence are probably the central issues in star formation, but they aren't even mentioned in the article.

Plain Language Summary

Clouds influence the Earth's energy budget by reflecting sunlight and intercepting terrestrial radiation. The extent to which clouds modify these flows of energy is highly sensitive to the vertical distribution of cloud fraction, and changes in cloud fraction are a dominant source of uncertainty in future climate projections. Here we show that the prominent high cloud-fraction peak in simulations of the tropical troposphere is primarily produced by long cloud lifetimes, which result from the fact that very little condensed water can evaporate into cold air. Our results provide a revised interpretation for the extensive anvil clouds found in the deeply convecting tropics and highlight the importance of developing new cloud-fraction schemes for use in climate models that explicitly depend on cloud condensate lifetime.


Contents

The term "red dwarf" when used to refer to a star does not have a strict definition. One of the earliest uses of the term was in 1915, used simply to contrast "red" dwarf stars from hotter "blue" dwarf stars. [4] It became established use, although the definition remained vague. [5] In terms of which spectral types qualify as red dwarfs, different researchers picked different limits, for example K8–M5 [6] or "later than K5". [7] Dwarf M star, abbreviated dM, was also used, but sometimes it also included stars of spectral type K. [8]

In modern usage, the definition of a red dwarf still varies. When explicitly defined, it typically includes late K- and early to mid-M-class stars, [9] but in many cases it is restricted just to M-class stars. [10] [11] In some cases all K stars are included as red dwarfs, [12] and occasionally even earlier stars. [13]

The most recent surveys place the coolest true main-sequence stars into spectral types L2 or L3. At the same time, many objects cooler than about M6 or M7 are brown dwarfs, insufficiently massive to sustain hydrogen-1 fusion. [14] This gives a significant overlap in spectral types for red and brown dwarfs. Objects in that spectral range can be difficult to categorize.

Red dwarfs are very-low-mass stars. [15] As a result, they have relatively low pressures, a low fusion rate, and hence, a low temperature. The energy generated is the product of nuclear fusion of hydrogen into helium by way of the proton–proton (PP) chain mechanism. Hence, these stars emit relatively little light, sometimes as little as 1 ⁄ 10,000 that of the Sun, although this would still imply a power output on the order of 10 22 watts (10 trillion gigawatts). Even the largest red dwarfs (for example HD 179930, HIP 12961 and Lacaille 8760) have only about 10% of the Sun's luminosity. [16] In general, red dwarfs less than 0.35 M transport energy from the core to the surface by convection. Convection occurs because of opacity of the interior, which has a high density compared to the temperature. As a result, energy transfer by radiation is decreased, and instead convection is the main form of energy transport to the surface of the star. Above this mass, a red dwarf will have a region around its core where convection does not occur. [17]

Because low-mass red dwarfs are fully convective, helium does not accumulate at the core, and compared to larger stars such as the Sun, they can burn a larger proportion of their hydrogen before leaving the main sequence. As a result, red dwarfs have estimated lifespans far longer than the present age of the universe, and stars less than 0.8 M have not had time to leave the main sequence. The lower the mass of a red dwarf, the longer the lifespan. It is believed that the lifespan of these stars exceeds the expected 10-billion-year lifespan of our Sun by the third or fourth power of the ratio of the solar mass to their masses thus, a 0.1 M red dwarf may continue burning for 10 trillion years. [15] [19] As the proportion of hydrogen in a red dwarf is consumed, the rate of fusion declines and the core starts to contract. The gravitational energy released by this size reduction is converted into heat, which is carried throughout the star by convection. [20]

Typical characteristics of M dwarfs [21]
Stellar
class
Mass
( M )
Radius
( R )
Luminosity
( L )
Teff
(K)
M0V 60% 62% 7.2% 3,800
M1V 49% 49% 3.5% 3,600
M2V 44% 44% 2.3% 3,400
M3V 36% 39% 1.5% 3,250
M4V 20% 26% 0.55% 3,100
M5V 14% 20% 0.22% 2,800
M6V 10% 15% 0.09% 2,600
M7V 9% 12% 0.05% 2,500
M8V 8% 11% 0.03% 2,400
M9V 7.5% 8% 0.015% 2,300

According to computer simulations, the minimum mass a red dwarf must have to eventually evolve into a red giant is 0.25 M less massive objects, as they age, would increase their surface temperatures and luminosities becoming blue dwarfs and finally white dwarfs. [18]

The less massive the star, the longer this evolutionary process takes. It has been calculated that a 0.16 M red dwarf (approximately the mass of the nearby Barnard's Star) would stay on the main sequence for 2.5 trillion years, followed by five billion years as a blue dwarf, during which the star would have one third of the Sun's luminosity ( L ) and a surface temperature of 6,500–8,500 kelvins. [18]

The fact that red dwarfs and other low-mass stars still remain on the main sequence when more massive stars have moved off the main sequence allows the age of star clusters to be estimated by finding the mass at which the stars move off the main sequence. This provides a lower limit to the age of the Universe and also allows formation timescales to be placed upon the structures within the Milky Way, such as the Galactic halo and Galactic disk.

All observed red dwarfs contain "metals", which in astronomy are elements heavier than hydrogen and helium. The Big Bang model predicts that the first generation of stars should have only hydrogen, helium, and trace amounts of lithium, and hence would be of low metallicity. With their extreme lifespans, any red dwarfs that were a part of that first generation (population III stars) should still exist today. Low-metallicity red dwarfs, however, are rare. The accepted model for the chemical evolution of the universe anticipates such a scarcity of metal-poor dwarf stars because only giant stars are thought to have formed in the metal-poor environment of the early universe. As giant stars end their short lives in supernova explosions, they spew out the heavier elements needed to form smaller stars. Therefore, dwarfs became more common as the universe aged and became enriched in metals. While the basic scarcity of ancient metal-poor red dwarfs is expected, observations have detected even fewer than predicted. The sheer difficulty of detecting objects as dim as red dwarfs was thought to account for this discrepancy, but improved detection methods have only confirmed the discrepancy. [22]

The boundary between the least massive red dwarfs and the most massive brown dwarfs depends strongly on the metallicity. At solar metallicity the boundary occurs at about 0.07 M , while at zero metallicity the boundary is around 0.09 M . At solar metallicity, the least massive red dwarfs theoretically have temperatures around 1,700 K , while measurements of red dwarfs in the solar neighbourhood suggest the coolest stars have temperatures of about 2,075 K and spectral classes of about L2. Theory predicts that the coolest red dwarfs at zero metallicity would have temperatures of about 3,600 K . The least massive red dwarfs have radii of about 0.09 R , while both more massive red dwarfs and less massive brown dwarfs are larger. [14] [23]

The spectral standards for M type stars have changed slightly over the years, but settled down somewhat since the early 1990s. Part of this is due to the fact that even the nearest red dwarfs are fairly faint, and their colors do not register well on photographic emulsions used in the early to mid 20th century. The study of mid- to late-M dwarfs has significantly advanced only in the past few decades, primarily due to development of new astrographic and spectroscopic techniques, dispensing with photographic plates and progressing to charged-couple devices (CCDs) and infrared-sensitive arrays.

The revised Yerkes Atlas system (Johnson & Morgan, 1953) [24] listed only two M type spectral standard stars: HD 147379 (M0V) and HD 95735/Lalande 21185 (M2V). While HD 147379 was not considered a standard by expert classifiers in later compendia of standards, Lalande 21185 is still a primary standard for M2V. Robert Garrison [25] does not list any "anchor" standards among the red dwarfs, but Lalande 21185 has survived as a M2V standard through many compendia. [24] [26] [27] The review on MK classification by Morgan & Keenan (1973) did not contain red dwarf standards. In the mid-1970s, red dwarf standard stars were published by Keenan & McNeil (1976) [28] and Boeshaar (1976), [29] but unfortunately there was little agreement among the standards. As later cooler stars were identified through the 1980s, it was clear that an overhaul of the red dwarf standards was needed. Building primarily upon the Boeshaar standards, a group at Steward Observatory (Kirkpatrick, Henry, & McCarthy, 1991) [27] filled in the spectral sequence from K5V to M9V. It is these M type dwarf standard stars which have largely survived as the main standards to the modern day. There have been negligible changes in the red dwarf spectral sequence since 1991. Additional red dwarf standards were compiled by Henry et al. (2002), [30] and D. Kirkpatrick has recently reviewed the classification of red dwarfs and standard stars in Gray & Corbally's 2009 monograph. [31] The M dwarf primary spectral standards are: GJ 270 (M0V), GJ 229A (M1V), Lalande 21185 (M2V), Gliese 581 (M3V), Gliese 402 (M4V), GJ 51 (M5V), Wolf 359 (M6V), van Biesbroeck 8 (M7V), VB 10 (M8V), LHS 2924 (M9V).

Many red dwarfs are orbited by exoplanets, but large Jupiter-sized planets are comparatively rare. Doppler surveys of a wide variety of stars indicate about 1 in 6 stars with twice the mass of the Sun are orbited by one or more of Jupiter-sized planets, versus 1 in 16 for Sun-like stars and the frequency of close-in giant planets (Jupiter size or larger) orbiting red dwarfs is only 1 in 40. [32] On the other hand, microlensing surveys indicate that long-orbital-period Neptune-mass planets are found around one in three red dwarfs. [33] Observations with HARPS further indicate 40% of red dwarfs have a "super-Earth" class planet orbiting in the habitable zone where liquid water can exist on the surface. [34] Computer simulations of the formation of planets around low-mass stars predict that Earth-sized planets are most abundant, but more than 90% of the simulated planets are at least 10% water by mass, suggesting that many Earth-sized planets orbiting red dwarf stars are covered in deep oceans. [35]

At least four and possibly up to six exoplanets were discovered orbiting within the Gliese 581 planetary system between 2005 and 2010. One planet has about the mass of Neptune, or 16 Earth masses ( M ). It orbits just 6 million kilometers (0.04 AU) from its star, and is estimated to have a surface temperature of 150°C, despite the dimness of its star. In 2006, an even smaller exoplanet (only 5.5 M ) was found orbiting the red dwarf OGLE-2005-BLG-390L it lies 390 million km (2.6 AU) from the star and its surface temperature is −220°C (53 K).

In 2007, a new, potentially habitable exoplanet, Gliese 581c , was found, orbiting Gliese 581. The minimum mass estimated by its discoverers (a team led by Stephane Udry) is 5.36 M . The discoverers estimate its radius to be 1.5 times that of Earth ( R ). Since then Gliese 581d, which is also potentially habitable, was discovered.

Gliese 581c and d are within the habitable zone of the host star, and are two of the most likely candidates for habitability of any exoplanets discovered so far. [36] Gliese 581g, detected September 2010, [37] has a near-circular orbit in the middle of the star's habitable zone. However, the planet's existence is contested. [38]

On 23 February 2017 NASA announced the discovery of seven Earth-sized planets orbiting the red dwarf star TRAPPIST-1 approximately 39 light-years away in the constellation Aquarius. The planets were discovered through the transit method, meaning we have mass and radius information for all of them. TRAPPIST-1e, f, and g appear to be within the habitable zone and may have liquid water on the surface. [39]

Modern evidence suggests that planets in red dwarf systems are extremely unlikely to be habitable. In spite of their great numbers and long lifespans, there are several factors which may make life difficult on planets around a red dwarf. First, planets in the habitable zone of a red dwarf would be so close to the parent star that they would likely be tidally locked. This would mean that one side would be in perpetual daylight and the other in eternal night. This could create enormous temperature variations from one side of the planet to the other. Such conditions would appear to make it difficult for forms of life similar to those on Earth to evolve. And it appears there is a great problem with the atmosphere of such tidally locked planets: the perpetual night zone would be cold enough to freeze the main gases of their atmospheres, leaving the daylight zone bare and dry. On the other hand though, a theory proposes that either a thick atmosphere or planetary ocean could potentially circulate heat around such a planet. [40]

Variability in stellar energy output may also have negative impacts on the development of life. Red dwarfs are often flare stars, which can emit gigantic flares, doubling their brightness in minutes. This variability makes it difficult for life to develop and persist near a red dwarf. [41] While it may be possible for a planet orbiting close to a red dwarf to keep its atmosphere even if the star flares, more-recent research suggests that these stars may be the source of constant high-energy flares and very large magnetic fields, diminishing the possibility of life as we know it. [42] [43]


Why do PMS stars on the Hayashi track remain at a constant temperature while they contract? - Astronomy

Astrophysics is a branch of astronomy that applies the theories and techniques of physics to the study of celestial objects. For the large and massive objects in our universe, the most important force is gravity, and the base of our present theoretical understanding is Einstein's theory of gravitation and Einstein's theory of general relativity. Other ideas of physics are also used in astrophysics, such as the use of emission spectrum from stars to determine their composition, or doppler shifts in the emission spectrum to determine the speed of stars with respect to the Earth.

Cosmology is the study of the organization and structure of the universe and its evolution. Cosmology studies the universe as a whole and seeks to find answers to questions such as how the universe was formed, why does it look the way it does, and what will happen to it in the future. It looks for answers to what seems to be impossible questions, such as "Has the universe always existed, or did it have a beginning in time?" "If it had a beginning, what came before that?" "Is it finite or infinite?" "If it is finite, what is out there beyond it?"

Stars and Galaxies

1x10^13km) In these units, the size of our solar system is about 0.001ly, and the closest star to us, other than the Sun, is Proxima Centauri, which is 4.3ly away.

On a clear and moonless night, thousands of stars of varying degrees of brightness can be seen from Earth, as well as the elongated cloudy stripe known as the Milky Way. The elongated cloudy stripe is actually composed of countless individual stars that are part of our Galaxy. Our Galaxy is a spiral-armed galaxy that is flat like a disc, and has a central bulging nucleus as depicted on page 2. Our Galaxy is about 100,000ly in diameter and 2000ly thick, with the central bulge about 6000ly thick. Our Galaxy contains about 100 billion stars and our Sun is just one of them. Our Sun is a little more than half way out from the galactic center and orbits the galactic center once every 200 million years.

Our Milky Way Galaxy

Shown above are two views of our Milky Way Galaxy, one from above and one from the side.

Using powerful telescopes such as the Hubble space telescope, which is in orbit about the Earth, billions and billions of other galaxies can be seen beyond our Milky Way Galaxy. The nearest one is the Andromeda galaxy, which is over 2 million light years away. When we look up and see the Andromeda galaxy, what we see is the light that left the Andromeda galaxy over 2 million years ago. What does the Andromeda galaxy look like today? We won't know for over 2 million years, since light leaving the Andromeda galaxy will not arrive on Earth for over 2 million years. In this way, astronomers are able to look back into the past. The farther away a star or galaxy is, the 'older' the light is that reaches the Earth. The orbiting Hubble space telescope can see more clearly galaxies farther away than other telescopes because it is above the Earth's atmosphere and its images are free from distortions that are caused by the Earth's atmosphere. The Hubble space telescope has seen galaxies as far away as 14 billion light years, which means it has looked back into the past 14 billion years.

Stellar Evolution

As nuclear fusion begins in the core of the star, the star continues its gravitational collapse. What keeps a star from continuing to collapse due to gravity at this point in its evolution? The answer is radiation pressure. The process of nuclear fusion in the core, in which hydrogen is 'burned' to become helium, releases large amount of energy in the form of electromagnetic radiation. As this radiation moves out from the core of the star, it exerts an outward pressure on the particles of the star. When enough nuclear fusion takes place to create a radiation pressure that balances the gravitational collapse, the protostar becomes a star and its size becomes stable. Our Sun and other stars with similar masses take about 30 million years to stabilize and become a star. They have enough hydrogen to remain stable through hydrogen 'burning' for about 10 billion years, assuming our theories about stellar evolution are correct. Although most stars are billions of years old, there is evidence (and some beautiful pictures from the Hubble space telescope, http://hubble.stsci.edu) that stars are being born right this moment.

When the supply of hydrogen in the core is sufficiently depleted, the core contracts and heats up. This results in an increase in hydrogen fusion, which is an increase in radiation pressure, and the star expands into a red giant. A star entering the red giant phase can be thought of as having two regions, a core and an outer shell. In the core, a buildup of helium from the hydrogen 'burning' has reduced the radiation pressure, so gravitational collapse gets ahead and contracts the core. But as the core collapses, it heats up and the process of hydrogen 'burning' goes faster, creating more radiation pressure. The core will become stable again, but more dense and at a temperature ten to a hundred times hotter. The outer shell doesn't have the helium from the hydrogen burning, but it does see the increased radiation pressure from the core, so it expands until radiation pressure and gravitational collapse are balanced. Our Sun and the planets of our solar system are about 5 billion years old. Our Sun will enter the red giant phase in about another 5 billion years. When it does, it should expand until its radius is about equal to the distance from the Sun to the Earth. In the core of a red giant the temperature is ten to a hundred times hotter than it is in a star. At this temperature, not only does hydrogen 'burning' go faster, but there is also enough kinetic energy for other types of nuclear fusion. Helium can be fused into beryllium, and if the star is massive enough, elements all the way up to iron can be made.

What is the fate of a red giant? That depends on the size of the star. Stars with a mass a little larger than our Sun or smaller will run out of stuff to 'burn' and cool, becoming white dwarfs. The white dwarf will continue radiating away energy until it eventually becomes a black dwarf, a dark, cold piece of ash. More massive stars can continue to contract, due to their larger gravitational forces, until the atoms are crushed and the atomic nuclei are mashed up against each other. Under these high pressures, the electrons combine with the protons to form neutrons and the result may be a neutron star. Neutron stars are thought to be so dense as to have diameters of only a few kilometers (and masses greater than our Sun. ) It is thought that one way in which a supernovae can be created happens when a neutron star undergoes its final collapse. When the atoms give way and it crushes to nuclear density, tremendous energy can be released in a short period of time. This cataclysmic event has enough energy to create virtually all of the elements of the period table.

Stars with masses much greater than our sun will continue to collapse even beyond the density of a neutron star, crushing not only the atoms, but the neutrons as well, pushing the quarks together. When this happens, a black hole is formed, an object whose gravitational attraction is so great that no matter or light can escape it. If our Sun had the same density as a black hole, it would be about the size of a golf ball. It is believed that there are black holes at the center of galaxies, and that our own Milky Way Galaxy has a black hole at its center with a mass estimated to be several million times that of our Sun.

The Expanding Universe

So we can calculate that 14 billion years ago the Earth and everything around it were in roughly the same spot, but surely the Earth isn't at the center of the universe? Heck, it's not even at the center of our Galaxy (which is a good thing since there is probably a massive black hole there). So an assumption must be made that there is nothing special about our little corner of the universe. This assumption is called the cosmological principle, and it states that the universe looks the same no matter which way we look or where we are. This assumption is only valid on a large scale, which for the universe is a size much bigger than our Galaxy. On a local scale (for Cosmology), this clearly isn't true our Galaxy is a slowly rotating flat disc appearing different when we look at it from the side or from the top. The Cosmological principle is only assumed for really big things. So what does that imply for the Earth and everything around it? In the 14 billion years it took for everything around us to expand to where it is today, the Earth moved from the center of the universe to where it is today. We've certainly come a long way from the ideas of Aristotle with the Earth at the center of the universe. A view of the universe that was held until only a few hundred years ago!

The Cosmic Microwave Background

The Standard Cosmological Model: The Early History of the Universe

In order to produce a universe as large as the one we observe today, both in size and mass, the Big Bang must have been huge. So huge in fact, that much of the very interesting features of the Big Bang happened very quickly, as the universe was rapidly expanding away from a single point and rapidly cooling. The two variables we will keep track of when we talk about the evolution of the universe are time and temperature. The standard cosmological model can only make predictions about the universe back until about 10^-43 seconds, when the temperature of the universe was about 1032K. We simply do not know enough about what might have happened before that because we have no good theory to predict the behavior of gravity at such a high temperature. It is believed that gravity must be quantized, but no good quantum mechanical theory for gravity has yet to be developed and experimentally verified. That is not to say that no theory exist, lots of theories exist, but none of them can be experimentally verified. Such is the nature of science.

From 10^-43 seconds to 10^-35 seconds, the standard cosmological model predicts a temperature drop from 1032K to 1027K. At these temperatures, there is enough kinetic energy for the quarks to freely exist with the leptons. The quarks are going so fast that they cannot be bound into things like neutrons and protons. We call this era the Grand Unified Era where quarks and leptons are whizzing about in a dense soup.

From 10^-35 seconds to 10^-6 seconds (one millionth of a second), the standard cosmological model predicts a temperature drop from 1027K to 1013K. At these temperatures, there is not enough kinetic energy for the quarks to exist freely and they bind together to form hadrons (hadrons is the name for the class of particles that contains neutron, protons, and things like them). We call this era the Hadron Era.

From 10^-6 seconds to 10 seconds the standard cosmological model predicts a temperature drop from 1013K to 1010K. At these temperatures, there is not enough kinetic energy to form hadrons, but there is still enough energy to form leptons, such as electrons, which are lighter than neutrons and protons. We call this era the Lepton Era.

From 10 seconds to about 1 million years, the standard cosmological model predicts a temperature drop from 1010K to 3000K. At these temperatures, there is not enough kinetic energy to form even leptons, and by now most matter had disappeared through particle-anti-particle annihilations. The universe is still quite hot however and is dominated by radiation. We call this era the Radiation Era. It is believed that towards the end of the Radiation era the temperature of the universe had become low enough for electrons to combine with nuclei to form stable atoms.

From 1 million years to 14 billion years (the present), the standard cosmological model predicts a temperature drop from 3000K to 3K. As the universe expands and cools, the density of radiation is dropping. During this era in the evolution of the universe, the radiation density has dropped low enough that it is below the density of matter in the universe. We call this era the Matter-dominated Era. It is believed to be in this era that stars and galaxies formed, probably from self-gravitation around mass concentrations.


Future Prospects

Prospects for the future study of extrasolar giant planets seem bright. Transit searches from ground and space will allow characterization of the properties of a fraction of the planets detected by Doppler spectroscopy. Astrometry from interferometers on the ground and in space will fill in the statistics of the occurrence of giant planets in orbits extending to 5 AU or so, for planets down to 10 Earth masses (49). Direct imaging searches implemented on 8-meter or larger telescopes and culminating in spaceborne imaging interferometers (41), will allow giant planets in orbits beyond 5 AU around nearby stars to be detected and atmospheric properties studied through spectroscopy. Microlensing surveys are capable of mapping the mass distribution of planets around stars in distant regions of the observable galaxy (but without detailed study of individual planets) (50). Space does not permit a more detailed analysis of the outcome of such future studies. However, the ability to search for and characterize giant planets by a variety of techniques certainly bodes well for a time, perhaps two decades hence, when we will thoroughly understand the frequency, nature, and dynamical effects on terrestrial planets of giant planets around other stars.


5. MIDDLE LEVEL MESOSCALE VORTICES

[30] One implication of the vertical structure of heating in an MCS (Figure 4) is that a vortex tends to form in middle levels at the base of the stratiform cloud (Figure 21). The formation of a mesoscale vortex in the stratiform region of an MCS was first noticed in the tropics [e.g., Houze, 1977 Gamache and Houze, 1982 ]. However, it is even more prominent in midlatitude MCSs. In a case study of synoptic and satellite data, Menard and Fritsch [1989] and Zhang and Fritsch [1988] in a modeling study pointed out that an MCC can develop a middle level mesoscale vortex in its mature and later stages. Cotton et al. [1989] found middle level positive relative vorticity in a composite analysis of MCCs. This feature is now called a “mesoscale convective vortex” or MCV. Bartels and Maddox [1991] compiled a satellite-based climatology of MCVs over the United States by identifying spiral-banded structures in the visible satellite images of the middle level cloud remains of old MCSs. Associating the observed MCVs with sounding data, they found that the MCVs were favored by weak flow, weak vertical shear, weak background relative vorticity, and strong gradients of humidity. They further concluded that the MCVs could be explained by the stretching term of the vorticity equation.

[31] Menard and Fritsch [1989] and Cotton et al. [1989] suggested that this middle level vortex could become inertially stable. Thus energy would be retained by the system that otherwise would propagate away in the large-scale environment's gravity wave response to the convective disturbance. The stable mesoscale vortex, according to this reasoning, would be supported by a secondary (vertical-radial) circulation and therefore have a built-in mechanism for supporting continued release of potential instability in air drawn into and upward through the system.

[32] The middle level vortex forms in the stratiform region of an MCS at the level of maximum convergence (Figure 4). In midlatitudes the Coriolis force accentuates the development of MCVs. Fortune et al. [1992] suggested the MCV in midlatitude storms might have a baroclinic character analogous to a larger-scale frontal cyclone. However, model simulations carried out later have indicated a fluid dynamical explanation for MCV formation in a leading-line/trailing-stratiform MCS and a more specific role of the Coriolis force. Skamarock et al. [1994] showed that trailing “bookend” vortices form on each end of the squall line and that a midlatitude cyclonic MCV can develop from the bookend vortex favored by the Coriolis force (Figure 22). This type of development leads to a distortion of the trailing-stratiform precipitation region, where the stratiform region is biased toward the poleward end of the line (Figure 23b). The stratiform region behind the poleward end of the line is advected rearward by the cyclonic flow, while dry air is advected toward the central and equatorward ends of the line. It takes several hours for the Coriolis force to act and form the asymmetric structure. The echo structure in the earlier stages of the MCS tends to be symmetric, with the stratiform region more or less centered behind the convective line (Figure 23a).

[33] Parker and Johnson [2000] further examined the symmetric and asymmetric paradigms of MCS structure identified by Houze et al. [1990] . They used radar data over the central United States to track 88 MCSs and analyzed how the echo structure evolved in each case. They determined that the spatial arrangement of the stratiform precipitation relative to the convective line on radar was a function of the life cycle stage of the MCS and that several variations on the structural paradigms could occur (Figure 24). The most common life cycle scenario (trajectory 1 in Figure 24) sees an initial line of convective cells develop a stratiform region first in a symmetric juxtaposition with the line and then evolving into an asymmetric form. Evidently, the system became more asymmetric as the Coriolis force had longer to act. The second most common evolution (trajectory 2 in Figure 24) had the only stratiform precipitation forming on the northeast end of the convective line, as old cells weakened and new ones formed on the southwest end of the line. After time went by, it too took on an asymmetric form, with the stratiform precipitation on the northern end of the line being swirled around to the rear of the system. The third most common pattern of echo development (trajectory 3 in Figure 24) showed stratiform precipitation developing ahead of the convective line, a system behavior also seen by Houze and Rappaport [1984] also in a tropical case.

[34] While the Coriolis force accentuates MCV development in midlatitudes, especially in asymmetric squall lines, the development of a mesoscale middle level vortex in the stratiform region also occurs in squall systems at tropical latitudes [e.g., Gamache and Houze 1982 ]. The effect is not, however, strong enough to generate asymmetric squall line structures as seen in midlatitudes.

[35] Bosart and Sanders [1981] found that the long-lived, self-regenerating MCS that produced the famous Johnstown, Pennsylvania, flood was characterized by a middle level cyclonic vortex detectable by the synoptic sounding network. Knievel and Johnson [2002 , 2003] have used profiler data from a mesonetwork to describe an MCV. Their vorticity budget indicates that the middle level vortex is made up both of vorticity advected in from the environment and generated by the MCS perturbation itself. Bosart and Sanders [1981] postulated that within and near the MCS the vertical circulation of the evidently balanced or quasi-balanced circulation was responsible for the regeneration of the convection.

[36] Raymond and Jiang [1990] provided a theoretical framework for such a circulation associated with a heating anomaly of the type associated with an MCS. They suggested that an environment of weak middle level shear but stronger low level shear (as observed in MCV environments [ Bartels and Maddox, 1991 ]) could support a mesoscale rotational circulation in an MCS. The postulated circulation consisted of a warm core vortex characterized by a positive potential vorticity anomaly (i.e., MCV) in middle levels beneath a negative potential vorticity anomaly at upper levels. The idealized MCV overlays a cold pool, presumably formed by precipitation evaporation and melting associated with the MCS. Using a numerical model, Chen and Frank [1993] found MCV formation consistent with the theory of Raymond and Jiang [1990] . Their result is depicted in Figure 21. The middle level vortex forms in the stratiform region of the MCS. As the stratiform cloud develops, air in middle-to-upper levels saturates over the mesoscale breadth of the storm. The saturation causes the Rossby radius of deformation to become smaller since the buoyancy frequency is determined by the moist static stability rather than the dry static stability, and the stratiform cloud deck is made up of buoyant air from the upper portions of previously more active convective cells (as discussed by Houze [1997] ). The buoyancy of the middle-to-upper level cloud leads to a low-pressure perturbation at the base of the stratiform cloud, and the lowered Rossby radius allows a quasi-balanced cyclonic vortex to form there (Figure 21b).

[37] Fritsch et al. [1994] combined concepts from Rotunno et al. [1988] , Raymond and Jiang [1990] , and their own detailed mesoanalysis of a major MCS over the United States in a conceptual model (Figure 25). They found that the MCV that develops in the stratiform region of the MCS can grow upscale and become somewhat larger and much longer lived than the parent MCS. Such was evidently the type of mesovortex described by Bosart and Sanders [1981] . These results suggest that the MCV may achieve a state of near balanced flow. Davis and Weisman [1994] examined numerically the potential vorticity development associated with the formation of an MCV in an asymmetric squall line. They found a pattern of balanced ascent on the warmer side of the MCS and sinking on the colder side, consistent with Raymond and Jiang's [1990] theory. However, they found that while a warm core vortex of the type discussed by Raymond and Jiang [1990] and Fritsch et al. [1994] may achieve a state of quasi-balance, it must go through an unbalanced convective phase while part of the convection is evolving into a stratiform region. Consistent with this notion, Fritsch et al. [1994] found that a sequence of MCSs occurred in the long-lived mesovortex. Thus, in some extreme cases, an MCS can foster the development of a longer-lived vortex that can, in turn, support development of new MCSs within the long-lived vortex. Fritsch et al. [1994] further investigated the mechanism by which the MCV may promote the formation of new convection extending the overall life of the MCS. They found, in the case they studied, that the subsequent MCSs tended to break out in the center of the vortex (i.e., not at the edge of the low level cold pool). They hypothesized that some sort of temporal or spatial nonhomogeneity of the low level cold pool (such as might be promoted by mesoscale banding of the precipitation) allows the warm boundary layer to penetrate horizontally toward the center of the region occupied by the middle level vortex.

[38] Another aspect of the MCV was brought out in a modeling study by Zhang [1992] . He described the low as being a cold core low rather than a warm core. The cold core evidently developed from evaporative cooling and/or cooling by melting below the base of the stratiform cloud. Jorgensen and Smull [1993] , analyzing airborne Doppler radar data, showed that the cyclonic “bookend” vortex (of the type in Figure 22) in a midlatitude MCS consisted of two intertwined flows: a rising warm flow on the north side and a cold sinking flow on the south side. Evidently, the middle level vortex is not always easily classifiable as being purely warm or cold core. Some studies have focused on the cold branch of the circulation [e.g., Zhang, 1992 ]. The modeling study of Chen and Frank [1993] emphasized the saturated warm branch of the vortex. We note, however, that Figure 21, taken from the Chen and Frank [1993] article, is a two-dimensional cross section through a highly three-dimensional storm, and the rear inflow in Figure 21b appears in the schematic sketch to be detached from the vortex. Three-dimensional analysis of the model results, however, shows that the subsiding unsaturated rear inflow depicted in Figure 21b is actually circulating cyclonically around the vortex center and is intertwined with the warm saturated air circulating around the vortex center (S. S. Chen, personal communication, 2004).


Results

The compound eye epithelium is established during embryogenesis as an internal disc of cells called the eye imaginal disc (Wolff and Ready, 1993). During the larval phase of the life cycle, the disc grows in size by asynchronous cell division. During the final 50 hr of the larval phase, a morphogenetic furrow (MF) moves across the eye disc from posterior to anterior (Figure 1A,B). All cells arrest in G1 phase within five cell diameters anterior to the furrow, and then as the furrow passes through them, periodic clusters of cells express the proneural gene atonal (Jarman et al., 1994). Atonal expression is subsequently restricted to one cell per cluster, which becomes the R8 photoreceptor. Each R8 cell then secretes an EGFR ligand that activates the receptor in neighboring cells and causes them to transit from multipotent progenitor to differentiated states (Figure 1C) (Freeman, 1996). Transitions occur in a sequence of symmetric pairs of multipotent progenitor cells that differentiate into R2/R5, R3/R4, and R1/R6 photoreceptors (Figure 1C) (Wolff and Ready, 1993). Thereafter, a single progenitor transits to a R7 photoreceptor fate followed by two pairs of cells, C1/C2 and C3/C4, that differentiate into cone cells. These cone cells are non-neuronal and form the simple lens that overlies each cluster of eight photoreceptors. The furrow induces the nearly simultaneous differentiation of a column of R8 cells, with repeated column inductions producing approximately 800 units or ommatidia as the furrow moves across the eye.

Development and patterning of the compound eye.

(A) Differentiation is initiated in the developing eye by the MF, which moves across the eye epithelium. On the furrow’s posterior side, G1-arrested progenitor cells undergo differentiation (light blue). On the anterior side, progenitor cells are still proliferating (dark blue). The large grey rectangle outlines the region that was analyzed for Yan levels the small rectangle corresponds to the developmental sequence outlined in panel C (B) A maximal projection of Yan-YFP fluorescence in an eye. Bar = 100 μm. (C) Top, an apical view of the sequential differentiation of eight photoreceptor (R1-R8) and four cone cell types (C1-C4) from multipotent progenitor cells (grey) in an ommatidium. Arrows denote inductive signal transmitted from the R8 to activate EGFR on nearby cells. Bottom, a cross-section view through an eye disc adapted after Wolff and Ready (1993). Note the progenitor cell nuclei are basally positioned, and as they transition into a differentiated state, their nuclei migrate apically. C1/C2 cells are positioned anterior and posterior in the ommatidium while C3/C4 cells are positioned equatorial and polar in the ommatidium. (D) Top, an optical slice of H2Av-mRFP fluorescence in an eye disc at a plane that bisects progenitor cell nuclei. Bottom, the same optical slice imaged for Yan-YFP fluorescence. Bars = 8 μm.

A central tenet of the bistable model of cell differentiation in the eye posits that differentiation is marked by a transition from high Yan protein levels in multipotent progenitor cells to low Yan levels in differentiating cells (Graham et al., 2010). Formulation of this model stemmed from studies of R7 cell differentiation, the final photoreceptor recruited to each ommatidium. Reduced Yan causes inappropriate expression of the R7 determinant pros and ectopic R7 cells in yan hypomorphic mutants (Kauffmann et al., 1996 Lai and Rubin, 1992). Conversely, a Yan isoform that is resistant to MAPK-dependent degradation, blocks R7 differentiation and pros expression (Kauffmann et al., 1996 Rebay and Rubin, 1995).

Quantifying Yan dynamics

To quantitatively test the bistable model, we used BAC recombineering to insert fast-fold yellow fluorescent protein (YFP) in-frame at the carboxy-terminus of the yan coding sequence (Webber et al., 2013). The Yan-YFP transgene fully complemented null mutations in the endogenous yan gene and completely restored normal eye development (Figure 1—figure supplement 1), demonstrating that the YFP tag does not compromise Yan function and that all essential genomic regulatory sequences are included. The pattern of Yan-YFP protein expression was qualitatively similar to that of endogenous Yan (Figure 1—figure supplement 1).

We used histone His2Av-mRFP fluorescence in fixed specimens to mark all eye cell nuclei for automated segmentation (Figure 1D—figure supplement 2). Nuclei were manually classified into the different cell types of the eye, which is possible because every cell undergoing differentiation can be unambiguously identified by its position and nuclear morphology without the need of cell-specific markers (Ready et al., 1976 Tomlinson, 1985 Tomlinson and Ready, 1987 Wolff and Ready, 1993). To validate our identification of all cell types using this method, we cross-checked with specific cell-specific markers, and found that our classification was accurate over 98% of the time (Figure 1—figure supplement 3). Cells were scored for nuclear Yan-YFP concentration and their exact position within 3D coordinate space of each fixed eye sample (Figure 1—figure supplement 2). Yan-YFP protein levels were normalized to His2Av-mRFP, which provided some control over measurement variation (Figure 1—figure supplement 4). We then mapped each cell’s spatial position in the x-y plane of the eye disc onto time. Two reasons made this possible. First, the furrow moves at an approximately constant velocity forming one column of R8 cells every two hours (Basler and Hafen, 1989 Campos-Ortega and Hofbauer, 1977). Second, each column of R8 cells induces the other cell fates at a constant rate (Lebovitz and Ready, 1986). Thus even in a fixed specimen, the temporal dynamics of cell state transitions are visible in the repetitive physical organization of ommatidia relative to the furrow (Figure 1C). Together these features allow the estimation of time based on a cell’s position relative to the furrow (Figure 2—figure supplement 1). This can be applied repeatedly to hundreds of cells in a single sample, creating a composite picture of the dynamics (Figure 2B–F). Although the developmental progression of an individual cell cannot be measured by this approach, it nevertheless provides a dynamic view of hundreds of cells across a developing epithelium as they respond to signaling processes. From this information, individual cell behaviors can be reconstructed and modeled.

Dynamics of Yan-YFP in eye cells.

(A) Average time at which initiation of differentiation is first detected by apical migration of committing cell nuclei. Time zero is set to when R8 differentiation initiates. Differentiation proceeds over a time course after commitment is initiated (horizontal arrows) (B) Yan-YFP fluorescence in multipotent progenitor cells. We fit a Hill function (blue curve) to the inductive phase and an exponential decay (black curve) to the decay phase. (C-F) Scatter plots of Yan-YFP levels in individual cells for R2/R5 (C), R3/R4 (D), R7 (E), and C3/C4 (F) cells. These are overlaid with scatter plots of Yan-YFP in multipotent cells at times preceding and coincident with the appearance of the relevant differentiated cells. Note the similar Yan-YFP levels between multipotent cells and differentiating cells at their first appearance. (G) Moving averages of Yan-YFP levels for multipotent and photoreceptor cells. Gaps between the multipotent and photoreceptor curves are due to the window size for line averaging. (H) Moving averages of Yan-YFP in multipotent and cone cells.

Yan-YFP expression in multipotent progentior cells was biphasic (Figure 2B). Cells anterior to the furrow expressed a basal level of Yan-YFP, but this level dramatically increased in cells immediately anterior to the furrow, starting eight hours before the first R8 cells were identifiable. Approximately eight hours after R8 definition, Yan-YFP levels peaked, and thereafter gradually decayed until Yan-YFP approached its basal level again. The results are surprising in two ways. First, Yan-YFP is not maintained at a stable steady state within progenitor cells, which would have been predicted by the bistable model. Rather, its dynamics are reminiscent of monostable responses to transient stimuli, with a single basal steady state. Second, at the peak of Yan-YFP expression, there is remarkably large heterogeneity in Yan-YFP levels across cells.

We also followed Yan-YFP dynamics in cells as they transited into a differentiated state and thereafter. Again, the results did not follow the expectations predicted by the bistable model. First, progenitors transited to a differentiated state at levels of Yan-YFP that varied, depending upon the type of differentiated state being adopted (Figure 2C–H). Cells entering the R3/R4 and R1/R6 states began with Yan-YFP levels that were approximately two-fold greater than cells entering the R2/R5 states. Cells entering the R7 state were intermediate between these two extremes. Despite these differences at the transition point, Yan-YFP levels decayed to a similar basal steady state irrespective of the photoreceptor type, and this basal level was at least three-fold lower than that which the progenitor cells achieved (Figure 2G). Thus, rather than the single high Yan progenitor state previously modeled, our results suggest a dynamic range of high Yan states from which different cell fates are specified according to the spatio-temporal sequence of differentiation.

We noted that for most photoreceptors, it took approximately 20 hr for Yan-YFP to decay to the basal steady state (Figure 2G), significantly longer than had been previously modeled (Graham et al., 2010). Since expression of several neural-specific genes is detected 2–8 hr after the transition (Tomlinson and Ready, 1987 Van Vactor et al., 1988), the slower than anticipated Yan decay indicates that early differentiation does not require cells to have assumed a basal steady-state level of Yan-YFP.

The last group of progenitors to differentiate form the non-neuronal cone cells. We also measured Yan-YFP in those cells. Yan-YFP dynamics in cone cells were more similar to progenitor cells over the same time period (Figure 2F,H). This behavior was in contrast to photoreceptors, which exhibited different decay dynamics from progenitor cells. Thus, accelerated degradation of Yan-YFP is not essential for cells to transition to all retinal cell states.

EGFR-ras signaling regulates Yan-YFP dynamics

The bistable model posits that the switch from one state to another is triggered by a signal that progenitor cells receive though the EGFR protein. Given the unanticipated Yan-YFP dynamics, we wanted to ask whether and how they were influenced by EGFR signaling. EGFR null mutants are inviable however, a temperature sensitive (ts) allele of EGFR enables controlled inactivation of the RTK’s activity (Kumar et al., 1998). We grew EGFR ts mutant larvae at a restrictive temperature for 18 hr before analyzing effects on Yan-YFP. Surprisingly, progenitor cells exhibited biphasic expression of Yan-YFP over time, but the amplitude of the pulse in expression was significantly reduced (Figure 3A). This suggests that EGFR signaling contributes to the stimuli that induce the Yan-YFP peak. To further test this hypothesis, we misexpressed a constitutively active form of Ras protein in eye cells. The peak of Yan-YFP in progenitors was now prolonged (Figure 3B). Together, these results suggest that EGFR-Ras signaling stimulates the transient appearance of Yan-YFP in progenitor cells, and that the decline in Yan-YFP within older progenitor cells is linked to a loss of signal reception by these cells over time.

EGFR/Ras and Pnt regulate Yan-YFP levels.

(A–D) Moving averages of Yan-YFP in different cell types. Shown with shading is the standard error of the mean for each moving average. (A,C) Wildtype and EGFR ts mutants incubated at the non-permissive temperature and analyzed for progenitors (A) and R2/R5 cells (C). (B,D) Wildtype and sev>Ras v12 mutants were analyzed for progenitors (B) and R2/R5 cells (D). (E) Optical slice through progenitor cell nuclei in a disc that contains clones of pnt 2 mutant cells. Left, fluorescence of RFP, which positively marks wildtype cells and not pnt 2 mutant cells. Middle, Yan-YFP fluorescence, showing reduced levels in pnt 2 mutant clones. Arrows highlight apoptotic nuclei. Right, merged image with Yan-YFP in green and RFP in purple. Clone borders are outlined. (F,G) Moving averages of Yan-YFP in R3/R4 cells that ectopically express PntP1 (F) or PntP2-VP16 (G) due to LongGMR-Gal4 driving the UAS transgenes. Since PntP2 requires MAPK phosphorylation to become transcriptionally active, we misexpressed a VP16 fusion of PntP2. PntP1 is constitutively active (Brunner et al., 1994 Gabay et al., 1996).

We next examined the effects of EGFR and Ras on Yan-YFP dynamics in cells as they differentiate. The bistable model predicts that EGFR is required for the loss of Yan-YFP in photoreceptors. Indeed, EGFR ts mutant R2/R5 cells delayed their initial decline in Yan-YFP levels (Figure 3C). Conversely, misexpression of constitutively active Ras caused a premature decline in Yan-YFP (Figure 3D). These results are consistent with EGFR-Ras stimulating the degradation of Yan-YFP as cells switch their states. However, Yan-YFP dropped to below-normal levels in EGFR ts mutant R2/R5 cells (Figure 3C). These complex effects suggest a dual role for EGFR signaling in photoreceptors. In the first few hours as cells transit to a photoreceptor state, EGFR stimulates the accelerated decay of Yan-YFP. Thereafter, EGFR inhibits the decay of Yan-YFP in a manner that might be related to that role that EGFR plays in boosting Yan-YFP levels in progenitor cells.

The source of EGFR ligand originates from the R8 cell (Freeman, 1996). If this diffusive ligand is responsible for controlling Yan-YFP levels in other photoreceptors as they are recruited to an ommatidium, we would predict a correlation between Yan-YFP levels in differentiating cells and their distances from adjacent R8 cells. Indeed, at the times when R2/R5 cells differentiate (

0–15 hr), their Yan-YFP levels are positively correlated (p<0.01) with their physical distance to the nearest R8 cells (Figure 3—figure supplement 1). These correlations are absent in EGFR ts mutants, providing evidence that R8 cells act through EGFR to control Yan-YFP dynamics in differentiating cells.

Pnt regulates Yan dynamics

Pnt proteins have been hypothesized to cross-repress Yan expression, and this double negative feedback loop is thought to be necessary for bistability (Graham et al., 2010 Shilo, 2014). At odds with this view, Pnt and Yan proteins are co-expressed in progenitor and differentiating cells of the eye (Boisclair Lachance et al., 2014). Since Pnt proteins act downstream of many RTKs including EGFR, we wondered if Pnt mediated the positive effects of EGFR-Ras signaling on Yan-YFP in progenitor cells. Null mutants of the pnt gene are embryonic inviable therefore we generated clones of eye cells that were null mutant for pnt in an otherwise wildtype animal. Mutant progenitor cells showed a reduction in Yan-YFP levels as they progressed through their biphasic expression pattern (Figure 3E). Thus, Pnt possibly mediates the positive effect of EGFR signaling on Yan-YFP expression in progenitors. We also wished to know if Pnt mediates the complex effects of EGFR in differentiating photoreceptors. Pnt proteins are rapidly cleared in photoreceptors (Boisclair Lachance et al., 2014) and so loss-of-function mutant analysis would be uninformative. Therefore, we overexpressed PntP1 or constitutively-active PntP2 in cells as they transited into a photoreceptor state and beyond. The early phase of Yan-YFP decay was accelerated while the later phase of Yan-YFP decay was inhibited (Figure 3F,G). These complex effects are precisely the opposite to those caused by loss of EGFR signaling, as would be expected if Pnt mediated EGFR’s complex effects on Yan-YFP dynamics in photoreceptor cells.

Cells are indifferent to absolute levels of Yan

The bistable model predicts that different cell states depend upon discrete absolute concentration of Yan present inside cells. To test this idea, we varied the number of Yan-YFP gene copies. In general, protein output is proportional to gene copy number in Drosophila (Lucchesi and Rawls, 1973). We increased Yan-YFP copy number from its normal diploid number to tetraploid, and monitored Yan-YFP in progenitors and differentiating cells. As expected, four copies caused a higher steady-state level of Yan-YFP in progenitor cells, though this increase was less than two-fold (Figure 4A). The amplitude of the Yan-YFP pulse was also increased as progenitor cells aged. Strikingly, as four-copy progenitor cells transited to a differentiated state, the onset of Yan-YFP decay occurred at the same time as it occurred for two-copy cells (Figure 4A–C). Yan-YFP levels were much greater in four-copy cells compared to two-copy cells making their transit into the identical cell states. To confirm that absolute Yan-YFP concentration had little effect on cell state transitions, we examined expression of a direct target of Yan in R7 cells: the pros gene (Kauffmann et al., 1996 Xu et al., 2000). Expression was monitored with an antibody specific for Pros protein. We observed at most a one hour delay in the onset of Pros expression in R7 cells containing four copies of Yan-YFP (Figure 4D), far less than the ten-hour delay predicted if absolute concentration of Yan-YFP dictated when Pros expression begins (Figure 4D).

Cell state transitions are unaffected by Yan-YFP gene copy number.

(A–C) Moving averages of Yan-YFP in eye discs containing two versus four copies of the Yan-YFP transgene. (A) R2/R5 and progenitor cells. (B) R3/R4 and progenitor cells. (C) R7 and progenitor cells. (D) Moving averages of Yan-YFP and Pros proteins in R7 cells containing either two vs. four copies of the Yan-YFP transgene.

Possibly, absolute concentration of Yan is unimportant when a cell transits to a different state, but the switch to a constant basal Yan level is robustly maintained regardless of starting concentration. An examination of Yan-YFP decay in photoreceptor cells makes that possibility less likely Yan-YFP decays to different basal levels in two- versus four-copy differentiated cells (Figure 4A–C). To further test this notion, we fit the data to several plausible functional forms. We found that exponentially decaying functions systematically best-fit to the data (Figure 4—figure supplement 1). Thus, for each cell state we fit an exponential decay function to its Yan-YFP temporal profile (Figure 4—figure supplement 2). From these fits, we derived the average decay rate constants and half-lives of Yan-YFP for cells carrying two, four, or six copies of yan. As expected, we found that Yan-YFP half-life was different between progenitors and differentiating photoreceptors (Figure 5—figure supplement 1). The half-life in photoreceptors was two-fold lower than in progenitors, accounting for the more rapid loss of Yan-YFP in the former cells. Strikingly, Yan-YFP half-life was not significantly affected by yan copy number within either progenitor or photoreceptor cells (Figure 5). Thus, Yan-YFP concentration only affected its decay rate as a first-order function, implying that there is no higher order mechanism to accelerate Yan-YFP decay when cells contain greater concentrations of Yan-YFP.

Yan protein half-life is largely unaffected by yan gene copy number.

Exponential decay of Yan-YFP levels. Note the robustness of Yan-YFP half-lives across replicates and yan gene copy number. Note also how half-lives are nearly twice as long for progenitor cells versus photoreceptor neurons.

As a final test of the effects of Yan-YFP levels on cell states, we allowed 4X and 6X yan animals to complete eye development and then examined the external patterning of the fully differentiated compound eye. The compound eyes were completely normal in appearance (Figure 5—figure supplement 2), implying that differentiation was unaffected by the absolute concentrations of Yan inside eye cells.

Yan expression noise spikes during cell state transitions

Our results indicate that Yan’s effects on retinal cell states are not dictated by uniform and stable concentrations of Yan protein. One potential explanation is that Yan’s effect on cell states actually depends on variability in Yan protein levels. A growing body of evidence is pointing to the importance of transient fluctuations in expression of factors to control cell states (Cahan and Daley, 2013 Torres-Padilla and Chambers, 2014). Key regulators of stem cells fluctuate in abundance over time, and this is correlated with stem cells existing in multiple connected microstates, with the overall structure of the population remaining in a stable pluripotent macrostate (MacArthur and Lemischka, 2013). Heterogeneity among cells is not simply due to independent noise in expression of individual genes but is due to fluctuating networks of regulatory genes (Chang et al., 2008 Kumar et al., 2014). Such fluctuations appear to be stochastic in nature, priming cells to transit into differentiated states with a certain probability that is guided by extrinsic signals.

Our data revealed considerable heterogeneity in the level of Yan-YFP among cells at similar developmental times (Figure 2B). To quantify the noise, we used developmental time to bin cells of similar age, and measured detrended fluctuations of Yan-YFP by calculating residuals to the fitted function and normalizing binned residuals to the function (Goldberger et al., 2002). Progenitor cells showed a spike in Yan-YFP noise as they began to induce Yan-YFP expression (Figure 6A). The noise spike was short-lived (approximately 17 hr), and noise thereafter returned to a basal level with secondary minor spikes. The major peak in noise magnitude coincided with the time at which R8 cells are formed.

Noise in Yan-YFP expression is highly dynamic.

Moving averages of Yan-YFP levels and noise (detrended fluctuations) for (A) progenitors, (B) R2/R5, (C) R3/R4, (D) R1/R6, and (E) R7 cells. (F) Comparative noise dynamics for all cells analyzed in (AE). (G) Moving averages of Yan-YFP noise (coefficient of variation) in R2/R5 cells sampled from wildtype and EGFR ts mutant eyes at the non-permissive temperature. Shown with shading is the standard error of the mean for each moving average. (H) Moving averages of Yan-YFP noise (coefficient of variation) in R2/R5 cells sampled from wildtype and sev>Ras v12 mutant eyes. Shown with shading is the standard error of the mean for each moving average.

Photoreceptor cells showed a large spike in Yan-YFP noise as they began to differentiate (Figure 6B–E). The magnitude of each noise peak varied with the photoreceptor cell state R3 and R4 cells exhibited the greatest amplitude in noise (Figure 6F). These noise spikes showed a distinct temporal relationship, with spikes coinciding with the times at which individual cell states were switched (Figure 6F). Thus, the noise spikes are not a simple consequence of a global stimulus synchronously affecting noise in all cells. Thereafter, all cells reduced Yan-YFP noise to a basal level that was comparable to basal noise in the progenitor cells. However, each cell type exhibited a secondary minor spike at 30–35 hr, which might reflect a synchronous stimulus.

Detrended fluctuation is one method to measure expression variability, but it can suffer from distortion if the model fitting is not adequately weighted. Therefore, we also calculated the coefficient of variation, that is, the standard deviation of Yan-YFP intensity within a sliding window divided by its mean. This method yielded noise profiles with transient spikes for each cell type that was consistent with calculations using detrended fluctuation (Figure 6—figure supplements 1 and 2). However, while the coefficient of variation yielded results that varied strongly with bin width, the detrended fluctuations yielded profiles that were generally robust against changes in bin width (Figure 6—figure supplement 1).

To rule out the possibility that these unexpectedly dynamic features of Yan-YFP were caused by its transgenic origins or fusion with YFP, we compared Yan-YFP dynamics to those of endogenous Yan protein that was bound with an anti-Yan antibody. The profiles of Yan-YFP protein levels and noise were highly similar to endogenous Yan protein levels and noise, in both multipotent and differentiating cells (Figure 6—figure supplement 3). Thus, transient spikes of expression heterogeneity are a fundamental feature of Yan protein.

Since Yan regulates Pros expression in R7 cells, it was possible that Pros showed a transient noise spike as a consequence. Therefore, we measured Pros protein heterogeneity and found that its dynamics did not resemble that of Yan (Figure 6—figure supplement 4). Instead, Pros noise was high starting at the onset of expression, and thereafter gradually declined as Pros protein levels increased in R7 cells. We conclude that noise spikes are not a general feature of gene expression in the developing eye but might reflect unique roles of Yan in mediating cell state transitions.

We wondered what might cause these spikes in Yan-YFP noise during cell state transitions. Because EGFR signaling is important for regulating Yan-YFP concentration during these transitions (Figure 3), we analyzed Yan-YFP noise when EGFR signaling was inhibited in EGFR ts mutant animals raised at a non-permissive temperature. The noise spike in progenitor cells was not significantly affected by loss of EGFR signaling (data not shown). We also examined the effects of EGFR ts on noise in differentiating photoreceptor cells. Interestingly, noise increased at the normal time of transition but the elevated noise did not quickly drop to basal levels (Figure 6G). Rather, high noise was extended for an additional 10 to 15 hr. Conversely, misexpressing constitutively active Ras within differentiating cells caused a premature dropdown in Yan-YFP noise (Figure 6H). These results indicate that EGFR/Ras signaling is required for the rapid drop in Yan-YFP noise after it has peaked, creating a transient spike.


Which is the bigger number 10 or 10+i?

The word "bigger" is an english word with no mathematical meaning. A common substitute is a norm, which does have a precise definition. The most common norm on the complex numbers is the euclidean norm or magnitude, in which case 10 + i has norm sqrt(101) and 10 has norm 10. So 10 + i wins.

For complex numbers, there is no meaningful notion of greater than or less than.

We could define an order (meaning a notion of "greater than") on the set of complex numbers. For example, we could take the lexicographic order on the pair (Re(z) , Im(z) ), where you first compare the real parts and then the imaginary part. In the setting 10 is smaller than 10+i and will be bigger than 9+3i

You can imagine many different way to compare two complex numbers.

The problem is that none of these orders will be compatible with the operations of addition and multiplication. That's just because we can write -1 as the square of a complex number, and squares have to be positive . So it is mostly useless to define such a notion.

Which is the bigger direction up or right?

The choice of i and -i is largely a matter of convention. If you had some way of ordering 10, 10+i and 10-i it should be able to be mirrored via the real axis and still remain the same if the ordering should be independent of the choice of i and -i. Thus i can't be atleast bigger or smaller than -i.

Unlike the reals, the complex numbers can not be ordered.

For any complex numbers z and w, we can never say that z<w or w>z.

We can say that |z| < |w| or z=w, but that is about it.

I recall someone saying you can order the complex numbers if you introduce some sort of ordering property. For instance, w<z if and only if Re(w) < Re(z) or Im(w) < Im(z). That's about it though.

You can absolutely order the complex numbers! Lexicographically for example is what you are taking about. Or you can use modulus and if two numbers have the same modulus say the one with the larger real (or imaginary) part is larger. There's a lot of ways to do it.

There isn't really any good way to say that one of those is larger than the other, but you can compare a kind of 'magnitude'. 10+i has a real part of 10 and an imaginary part of 1, while 10 has a real part of 10 and an imaginary part of 0. So you can concretely say that the imaginary part of 10+i is larger than the imaginary part of 10, and that the real parts of both are equal, however since there isn't a proper ordering in the complex field you can't accurately say that one is larger than the other.

The complex numbers are not what is called totally ordered. The real numbers are totally ordered, it is one of the defining features of the reals (among other things, plenty of sets that are not the real numbers are totally ordered).

Total order means that if I pick two random elements of the set, in this case two random complex numbers, then it makes sense to say one is bigger than or less than the other, or they are the same element. Clearly, with the real numbers this is the case, but it isn't so with complex numbers. What is bigger, 1 or i? It's fuzzy. Instead we use the modulus of a complex number, which is a real number, so the moduli of complex numbers are totally ordered, and it makes sense to compare them like that.

The modulus of a complex number x + iy is sqrt(x 2 + y 2). Geometrically this is the distance from the origin in the complex plane (easily verifiable with the Pythagorean theorem).

The notion of magnitude of complex numbers is extremely important in complex analysis. If you want to study the complex numbers this geometric interpretation of "greater than or less than" is essential.


The masses and spins of neutron stars and stellar-mass black holes

Stellar-mass black holes and neutron stars represent extremes in gravity, density, and magnetic fields. They therefore serve as key objects in the study of multiple frontiers of physics. In addition, their origin (mainly in core-collapse supernovae) and evolution (via accretion or, for neutron stars, magnetic spindown and reconfiguration) touch upon multiple open issues in astrophysics.

In this review, we discuss current mass and spin measurements and their reliability for neutron stars and stellar-mass black holes, as well as the overall importance of spins and masses for compact object astrophysics. Current masses are obtained primarily through electromagnetic observations of binaries, although future microlensing observations promise to enhance our understanding substantially. The spins of neutron stars are straightforward to measure for pulsars, but the birth spins of neutron stars are more difficult to determine. In contrast, even the current spins of stellar-mass black holes are challenging to measure. As we discuss, major inroads have been made in black hole spin estimates via analysis of iron lines and continuum emission, with reasonable agreement when both types of estimate are possible for individual objects, and future X-ray polarization measurements may provide additional independent information. We conclude by exploring the exciting prospects for mass and spin measurements from future gravitational wave detections, which are expected to revolutionize our understanding of strong gravity and compact objects.


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