Why are centrioles aligned at 90 degree with each other?

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The centrioles are aligned at 90 degree with each other. What is the function of this?

As far as I know, the function is not truly known, although there are some seriously interesting guesses. Part of the problem seems, from scanning the literature, to be that it's not easy or obvious to disrupt centriole orientation. This paper in PLoS Biology presents some really interesting results, in particular two piece of evidence:

Centriole orientation, to some degree, is dictated by the "mother" centriole during centriole division and is thus passed on from cell to (daughter) cell.
Defects in centriole orientation can result in organelle localization defects in the cell (e.g. nuclear orientation), and at least some of these defects are genetic.

So it seems that orientation is, in some part, a product simply of centriole division, but that there are some gene products that do help determine orientation and localization. This paper puts forth the hypothesis that centriole orientation determines the axis of cell division which can affect embryo orientation, whereas this suggests that external environmental factors could translate into altered centriole, and therefore cellular, orientation.

Centrioles and Centrosomes

Centrioles are built from a cylindrical array of 9 microtubules, each of which has attached to it 2 partial microtubules.

The photo (courtesy of E. deHarven) is an electron micrograph showing a cross section of a centriole with its array of nine triplets of microtubules. The magnification is approximately 305,000.

When a cell enters the cell cycle and passes through S phase, each centriole is duplicated. A "daughter" centriole grows out of the side of each parent ("mother") centriole. Thus centriole replication &mdash like DNA replication (which is occurring at the same time) &mdash is semiconservative.

Functional microtubules grow out only from the "mother".
When stem cells divide, one daughter cell remains a stem cell the other goes on to differentiate. [Discussion] In two animal systems that have been examined (mouse glial cells and Drosophila male germline cells), the cell that receives the old ("mother") centriole remains a stem cell while the one that receives what had been the original "daughter" centriole goes on to differentiate. (You can read about these findings in Wang, X., et. al., Nature, 15 October 2009.)

most fungi (but not the primitive chytrids)
"higher" plants (but not the more primitive mosses, ferns, and cycads with their motile sperm)
animal eggs lose their centriole during meiosis and must have it restored by the sperm that fertilizes it.

In nondividing cells, the mother centriole can attach to the inner side of the plasma membrane forming a basal body.
In almost all types of cell, the basal body forms a nonmotile primary cilium.
In cells with a flagellum, e.g. sperm, the flagellum develops from a single basal body. (While sperm cells have a basal body, eggs have none. So the sperm's basal body is absolutely essential for forming a centrosome which will form a spindle enabling the first division of the zygote to take place.)

In ciliated cells such as

the columnar epithelial cells of the lungs [View] like the paramecium

The main concept of the universal joint is based on the design of gimbals, which have been in use since antiquity. One anticipation of the universal joint was its use by the ancient Greeks on ballistae. [2] In Europe the universal joint is often called the Cardano joint or Cardan shaft, after the Italian mathematician Gerolamo Cardano however, in his writings, he mentioned only gimbal mountings, not universal joints. [3]

The mechanism was later described in Technica curiosa sive mirabilia artis (1664) by Gaspar Schott, who mistakenly claimed that it was a constant-velocity joint. [4] [5] [6] Shortly afterward, between 1667 and 1675, Robert Hooke analysed the joint and found that its speed of rotation was nonuniform, but that this property could be used to track the motion of the shadow on the face of a sundial. [4] In fact, the component of the equation of time which accounts for the tilt of the equatorial plane relative to the ecliptic is entirely analogous to the mathematical description of the universal joint. The first recorded use of the term universal joint for this device was by Hooke in 1676, in his book Helioscopes. [7] [8] [9] He published a description in 1678, [10] resulting in the use of the term Hooke's joint in the English-speaking world. In 1683, Hooke proposed a solution to the nonuniform rotary speed of the universal joint: a pair of Hooke's joints 90° out of phase at either end of an intermediate shaft, an arrangement that is now known as a type of constant-velocity joint. [4] [11] Christopher Polhem of Sweden later re-invented the universal joint, giving rise to the name Polhemsknut ("Polhem knot") in Swedish.

In 1841, the English scientist Robert Willis analyzed the motion of the universal joint. [12] By 1845, the French engineer and mathematician Jean-Victor Poncelet had analyzed the movement of the universal joint using spherical trigonometry. [13]

The term universal joint was used in the 18th century [10] and was in common use in the 19th century. Edmund Morewood's 1844 patent for a metal coating machine called for a universal joint, by that name, to accommodate small alignment errors between the engine and rolling mill shafts. [14] Ephriam Shay's locomotive patent of 1881, for example, used double universal joints in the locomotive's drive shaft. [15] Charles Amidon used a much smaller universal joint in his bit-brace patented 1884. [16] Beauchamp Tower's spherical, rotary, high speed steam engine used an adaptation of the universal joint circa 1885. [17]

The term Cardan joint appears to be a latecomer to the English language. Many early uses in the 19th century appear in translations from French or are strongly influenced by French usage. Examples include an 1868 report on the Exposition Universelle of 1867 [18] and an article on the dynamometer translated from French in 1881. [19]

Which description explains why there are two spring tides per month? earth rotates in and out of a tidal bulge created by gravitational forces. earth revolves around the tidal bulges created by gravitational forces. earth moves closer to the sun and moon twice a month. earth, the moon, and the sun are aligned twice a month.

Earth rotates in and out of a tidal bulge twice a day.

High and low tides are caused by the Moon. The Moon's gravitational pull generates something called the tidal force. The tidal force causes Earth—and its water—to bulge out on the side closest to the Moon and the side farthest from the Moon. These bulges of water are high tides.

The first option is correct.

Earth always has two surrounding gravitational forces, one proceeding from the moon and one from the sun.

The sun's pull, despite the fact our sun is further then the moon, is stronger and creates the high tide, and the moon the low tide. But as we know our earth moves, now exposing the other side to the sun which in turn leaves the other side of earth with the high tides and the other with the low tides. Tides never move, what moves is earth.

Happy homework/ study/ exam!

The moon and sun are at 90 degree angles twice a months to earth

Neap tides are also known as moderate tides. The lunar tide is partially cancelled when the moon and the

sun are at a 90-degree angle to Earth, or being at right angles. This event happens twice a month thus creating neap tides twice a month also.

Strike and dip

Strike and dip refer to the orientation or attitude of a geologic feature. The strike line of a bed, fault, or other planar feature, is a line representing the intersection of that feature with a horizontal plane. On a geologic map, this is represented with a short straight line segment oriented parallel to the strike line. Strike (or strike angle) can be given as either a quadrant compass bearing of the strike line (N25°E for example) or in terms of east or west of true north or south, a single three digit number representing the azimuth, where the lower number is usually given (where the example of N25°E would simply be 025), or the azimuth number followed by the degree sign (example of N25°E would be 025°).

The dip gives the steepest angle of descent of a tilted bed or feature relative to a horizontal plane, and is given by the number (0°-90°) as well as a letter (N,S,E,W) with rough direction in which the bed is dipping downwards. One technique is to always take the strike so the dip is 90° to the right of the strike, in which case the redundant letter following the dip angle is omitted (right hand rule, or RHR). The map symbol is a short line attached and at right angles to the strike symbol pointing in the direction which the planar surface is dipping down. The angle of dip is generally included on a geologic map without the degree sign. Beds that are dipping vertically are shown with the dip symbol on both sides of the strike, and beds that are level are shown like the vertical beds, but with a circle around them. Both vertical and level beds do not have a number written with them.

Another way of representing strike and dip is by dip and dip direction. The dip direction is the azimuth of the direction the dip as projected to the horizontal (like the trend of a linear feature in trend and plunge measurements), which is 90° off the strike angle. For example, a bed dipping 30° to the South, would have an East-West strike (and would be written 090°/30° S using strike and dip), but would be written as 30/180 using the dip and dip direction method.

Strike and dip are determined in the field with a compass and clinometer or a combination of the two, such as a Brunton compass named after D.W. Brunton, a Colorado miner. Compass-clinometers which measure dip and dip direction in a single operation (as pictured) are often called "stratum" or "Klar" compasses after a German professor. Smartphone apps are also now available, that make use of the internal accelerometer to provide orientation measurements. Combined with the GPS functionality of such devices, this allows readings to be recorded and later downloaded onto a map. [1]

Any planar feature can be described by strike and dip. This includes sedimentary bedding, faults and fractures, cuestas, igneous dikes and sills, metamorphic foliation and any other planar feature in the Earth. Linear features are measured with very similar methods, where "plunge" is the dip angle and "trend" is analogous to the dip direction value.

Apparent dip is the name of any dip measured in a vertical plane that is not perpendicular to the strike line. True dip can be calculated from apparent dip using trigonometry if the strike is known. Geologic cross sections use apparent dip when they are drawn at some angle not perpendicular to strike.

The Resultant of Two Forces

Videos, worksheets, games and activities to help PreCalculus students learn to obtain the resultant of two forces using vectors.

The Resultant of Two Forces
When vectors represent forces, their sum is called the resultant. The resultant of two forces can be found using the methods for adding vectors when the vectors are a geometric representation. When using methods for the algebraic representation to find the resultant of two forces, it can be helpful to understand the components of a force.

The following diagram shows the resultant of two forces. Scroll down the page for more examples and solutions on how to obtain the resultant of two forces using vectors.

Resultant of two forces at right angles
How to calculate the resultant force from two forces acting at right angles to each other?

Example:
Two forces 300 N at 0 degrees and 400 N at 90 degrees pull on an object.
Answer the following (Use the tail-tip method):

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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The Best Resistance Band Exercises for People Over 60

Loop the resistance band around your wrists. Position your arms at a 90-degree angle with your palms facing each other and fingers pointing to the ceiling.
Brace your core and slowly rotate your elbows outward, squeezing your shoulder blades together. Be careful not to arch your back and keep your tummy tucked tight.
Squeeze your shoulder blades for two seconds before releasing and returning your arms to the starting position. Do 10 to 15 reps and rest for 30 to 60 seconds. Complete 2 to 3 sets.

Why Scapular Retractions Work

Your scapula is the bone that connects your humerus (upper arm bone) to your clavicle (collarbone), also known as your shoulder blade, according to the American Council on Exercise. The scapulae and arms are connected by various muscles, ligaments and tendons, and when those tissues get weak, it can lead to poor posture, Scanzillo says.

A common sign of weak scapular muscles is rounded shoulders and a protruding neck (also known as forward head posture), according to a June 2018 study in the Journal of Physical Therapy Science.

Poor posture can also lead to pain in the upper back, neck and shoulders, per the Cleveland Clinic.

Aligning circles to non-90-degree corners

I am trying to achieve perfect circular corners for some shapes. This seems really easy for 90° angles, but for more complicated corners, I can't seem to figure out how to get the circle to align perfectly with the lines.

I have found solutions by using Astute Graphics SubScribe, but I cannot afford that at the moment, and would assume that this would be possible in Vanilla AI.

4 Answers 4

Consider this diagram where $|R| = |X_L|$ : -

The following has to be true: -

The voltage across the resistor (red arrow) and the voltage across the inductor (blue arrow) have to spatially add-up to equal the supply voltage (black arrow). That addition is shown by the dotted lines.

The red and blue voltage arrows have to be 90 degrees apart because both inductor (blue) and resistor (red) share the same current.

The current through both components (orange arrow) has to be 90 degrees lagging the inductor voltage (blue) AND simultaneously in phase with resistor voltage (red).

When |R| = | $X_L$ |, there can be no other diagram that complies with the above requirements.

If L dominates or R dominates you get these scenarios: -

If you take these two scenarios to the extremes, you should see that: -

Current lags supply voltage by 90 degrees (no resistance)
Current becomes in phase with supply voltage (no inductance)

If we set the combined series impedance magnitude of R and L, $sqrt$ to be a constant, we will see that the trajectories for $V_R$ and $V_L$ follow circular paths: -

For the scenarios above, the current will have a constant amplitude and be in phase with $V_R$ .

In case you didn't realize, these are called phasor diagrams and can be re-arranged like so (for example): -

This should allow you to see why the impedance magnitude of a series inductor and resistor (same current) is $sqrt$ . This is because the angle inside a semi-circle is always 90°.

There isn't a great answer to this question. But I will point out that the voltage across an inductor and the current through the inductor are ALWAYS 90 degrees out of phase. That is the rule that the inductor enforces.

Likewise, the voltage across a resistor and the current through the resistor are always in phase. The resistor enforces this condition by its nature, just like the inductor.

If you lump them together in series this is still true. What happens is, the supply ends up supplying voltage and current which are not in phase when measured at the supply. But the angle between voltage and current at the resistor will still be 0 degrees, and 90 degrees measured at the inductor.

Each element in the circuit has its rule that it tries to enforce. Fortunately, the supply is flexible about phase angle. It only enforces voltage, and the phase angle of the current at the supply is free to be whatever it needs to be to make the rest of the circuit happy.

Why are centrioles aligned at 90 degree with each other? - Biology

Actin polymerized from actin protein (binds ATP)
(first found in muscle cells, but most kinds of cells have it)

Microtubules polymerized from tubulin (binds GTP)
(hollow, rather stiff)
(found inside cilia & flagella but are in cytoplasm too)

Intermediate filaments polymerized from keratin, etc. etc.
(really include several different kinds of proteins,
that don't actually have that much to do with each other)

II) actin fibers and microtubules exert forces and create patterns
in two different kinds of ways:
A) active sliding B) polymerization (assembling into fibers)

A) because certain other proteins ("motor proteins")
slide along them:
(myosin slides actively along actin fibers)
(dynein slides actively along microtubules)
(kinesin (also) slides actively along microtubules)

This active sliding uses the energy from ATP
For example, in muscle, myosin slides along actin
and in cilia & flagella, dynein slides along microtubules
And there are many other examples of each:
for example, amoeboid locomotion mostly due to actomyosin
another example, mitotic spindles are made of microtubules
and dividing cells pinch in two using belts of actomyosin

B) By assembling monomers onto the ends of actin fibers and microtubules. NOTE how important it is that they add and lose monomers only at the ends of fibers

III) Procaryotes & arhcaea don't have actin or tubulin
but they do have a tubulin homolog called FtsZ
(pronounced Futz-Zee) filamentous mutant # Z

A thought question if bacteria can't divide, they form filaments
but why should mutation of this gene cause filament formation?

IV) Actin protein binds ATP (or ADP) tubulin binds GTP (or GDP)

Being bound to the tri-phosphate promotes assembly into fibers

Being bound to the di-phosphate promotes DISassembly,
from fibers back into monomers

Imagine one of these proteins diffusing around the cytoplasm, binding to an ATP, then assembling onto the end of a fiber
(but while in the fiber, cleaving the ATP to ADP thus tending
to disassemble back out of the fiber again

V) Microtubules and actin fibers ends differ
Each microtubule has a plus end and a minus end
Each actin fiber has a plus end and a minus end
THIS HAS NOTHING TO DO WITH + OR - CHARGES

Plus ends where addition (and also removal) of subunits favored
Minus ends are where assembly & disassembly are slower.

VI) Two related kinds of dynamic behavior result from this:

A) Treadmilling (on one end (+), off other end (-))
B) Dynamic instability (irregularly on & off same end, mostly+)

microtubules frequently have dynamic instability
actin fibers frequently treadmill (in cell locomotion, for example)

(in principle, however, either could do either) (I think!?)

In the time lapse video that you saw showing microtubules
they were elongating and shortening by dynamic instability
(at their + ends)
And they were being pushed rearward by a constant wind of treadmilling actin fibers (+ end at cell edge, - end in center)

VII) There are lots of special proteins whose function is to bundle actin fibers together other special proteins to cut them
other special proteins to nucleate their formation,
or to stabilize their ends, in the sense of blocking disassembly
and likewise for microtubules
(& that is how cells control their shapes & movements, etc.)

VIII) There are special poisons that bind tubulin & actin
These are useful for experimenters, & also used to treat cancer.

vinblastine, taxol, nocodazole & colchicine
are some of the main tubulin poisons

phalloidin and cytochalasin are actin poisons,
(phalloidin is from deadly nightshade mushrooms, & they also have RNA polymerase poisons)

IX) Myosin is a + end directed motor protein (along actin)
(Besides the 2-headed kind of myosin found in muscles
are other myosins that transport stuff around the cytoplasm)
UNC Prof. Dick Cheney

Kinesin is a + end directed motor protein along microtubules

Dynein is a - end directed motor protein along microtubules

. all 3 of these proteins are ATPases

Dynein drives bending of cilia and flagella:
& also drives transport of membranes, organelles etc.
toward the middle of cells (- ends of microtubules)

Kinesins drive transport of organelles toward cell edges
(where the + ends of the microtubules are.
especially important in carrying stuff to ends of nerve fibers.

X) Most animal cells can crawl: especially in tissue culture,
during embryonic development, in cancer invasion

The reaching, sticking, contracting sequence (fig 16-85) is wrong
really, they use actin treadmilling to pull material rearward
past their upper & lower plasma membranes
Traction forces: which can distort rubber & carry particles
fig 16-94 & 16-91
& can also reorganize extracellular collagen to make tendons
fig 16-95 (which by some accident is the same as fig 19-50! rotated 90 degrees!)

XI) Certain GTP-binding proteins: Rho, Rac & Cdc42
serve to control actin fiber organization
and are very important in cancer research. ("Rac and Rho are here to stay")

Chapter 17: programmed cell death and the cell cycle

Programmed cell death = "apoptosis"
#1) Is NOT the same thing as ordinary cell death ="necrosis"
instead, apoptosis is a deliberate self-destruction, in which
special enzymes digest the cell from the inside out.

The classic example of programmed cell death is the destruction of the tail of tadpoles, as they change themselves into frogs

#2) The main mechanism is activation of special proto-enzymes
in the cytoplasm: called caspases
(because they have aspartic acid and cysteine at their active sites)
The active site is blocked by part of the protein chain
but once this part is digested away, then caspases become active
activated caspases can unblock other caspases: a domino effect

all our cells (including all cancer cells!) already have this self-destruct system ready and waiting inside them, if you can set it off!>

#3) Other examples of apoptosis include
A* elimination of

1/2 of embryonic nerve cells.
B* elimination of webbing between vertebrate toes.
C* getting rid of many specific cells in nematode embryos
D* weeding out lymphocytes that make anti-self antibodies
E* killing virus-infected cells (even in colds, etc.)
F* much of the cell death after strokes & heart attacks.
(& also nerve cell death in severed spinal cords!)
Therefore, it might be a big medical advance if somebody were
to discover a pill etc. to prevent all apoptosis for a few days
that might prevent up to 90% of the long-term damage.

The same basic mechanism of apoptosis is used in all these cases
(but plants have a different system to protect from virus infection)

#4) There are some special cell proteins whose function is to initiate apoptosis: Fas and Fas ligand (produced by killer lymphocytes)

#5) A special set of (mitochondrial !) proteins serve to inhibit apoptosis Bcl-2 and the Bcl-2 family (Bad, Bax etc.)
These were discovered in human B-cell lymphoma cancer patients,
because chromosome breaks that put the Bcl-2 gene next to active promoters cause too much of this protein to be made,
& such lymphocytes accumulate & can't die causing lymphoma
(but such cases of lymphoma are still treated with anti-DNA growth poisons which logically should be no help! But do cure some!?

Several viruses make their own Bcl-2 like proteins, to help protect themselves from self destruction of infected host cells.

A certain gene discovered in nematode worms resembles bcl2
so much that substitution of the human gene into worms results in a normal phenotype. Nobody tried the reverse experiment!!

The cell cycle: the Nobel Prize for last fall was for cyclins, etc.

#6) DNA synthesis is done during a specific period between the times of actual cell division (mitosis). (in eucaryotes!)
This was an early discovery of radioactive precursors

M period
G1 period
S period this is called "the cell cycle"
G2 period
M period

#7) Progress from one phase to the next is controlled by several kinds of "checkpoint mechanisms" (often important to cancer etc.)
There are mitotic checkpoints - that halt mitosis if chromosomes aren't lined up
There are DNA damage checkpoints that halt DNA synthesis until damage is fixed
If damage can't be fixed, checkpoint mechanisms set off apoptosis

#8) Timing of each phases of the cycle turns out to be
controlled by synthesis & accumulation of specific proteins.
these proteins are named cyclins
(there are different cyclins for the G1 period, etc.)
this is sort of analogous to an hour-glass, except the cyclins are made & destroyed

When the concentration of a cyclin becomes high enough, this sets off two events:
one-> going on to the next stage
two-> digestion of all of that accumulated cyclin protein

These events are set off by cyclin-dependent kinases (enzymes)
(tell the story about how cyclins were accidentally discovered by a lab project in the Woods Hole physiology course: short term labeling)

#9) When cells halt growth, it is (almost?) always at the boundaries
between phases of the cell cycle usually at the G1-S boundary
G-0 pronounced Gee-Zero

Certain proteins called growth factors, and other "mitogens"
can push cells through this boundary.
including "platelet derived growth factor" (derived from blood platelets)

The internal mechanisms of this G1-S checkpoint
include RB protein, and others that check for DNA damage

#10) frogs & flies (but NOT humans or other mammals)
turn off their checkpoint controls during early cell cycles
(so their embryos have no G1 or G2
they just alternate between M and S, until

Some unsolved problems: related to sizes of individual cells

Tetraploid cells become exactly twice the volume of diploid cells
and haploid cells become exactly half the size of diploids.

Species with lots of junk DNA have proportionately bigger cells!

Organ sizes and shapes seem to be independent of cell size.
fig 17-52

Chapter 18: Cell Division - today's amino acid: threonine (almost like serine, but with a methyl)

#II) Animals, plants & all other eucaryotes form a special structure
The mitotic spindle (made mostly out of microtubules)
microtubules radiate out from 2 asters (MTOCs) centrioles

and their chromosomes condense
(DNA strands coil up and become visible
(and DNA becomes inactive, transcriptionally)

Mitosis is divided into the following stages, that you should learn..

prophase (chromosomes condense, spindle forms)
metaphase (chromosome pairs, kinetochores line up along midline
(where they line up is called "metaphase plate")
anaphase (chromosome pairs pulled apart, toward opposite poles)

telophase (chromosomes "decondense", new nuclei form, etc.)

There are some eucaryotes with interesting variations on this, such as that the nuclear membranes sometimes remain intact, etc.
Dinoflagellates, diatoms, yeasts and other interesting variations. 18-41

#III) Animal cells split in two by contraction of a contractile ring of actin and myosin that forms just under the plasma membrane, around the "equator" usually around where the metaphase plate was.

Plant cells separate by building a new cell wall (exocytosis) by a
"phragmoplast" that forms (usually) where the metaphase plate was
the cell plate, etc. but the future location of the phragmoplast
is where the preprophase band of microtubules had formed in prophase?

So: no contractile ring in plants no phragmoplasts in animal cells
& neither one in bacteria and no mitosis or spindle in bacteria.

#IV) There is a lot of research about what signals control where the contractile ring forms. "cleavage furrow"
A) Signals come from the poles (not the equator)
Ray Rappaport's classic "doughnut" experiment fig 18-31
B) The contractile ring is a concentration of actin and myosin
C) 100s of papers have been published debating subjects as whether
the signals from the poles cause stronger or weaker contraction.

Julie Canman in this dept, THIS WEEK, may have discovered the answer to several of these questions, about how asters control the location where the contractile ring!! (in PTK1 cell line:rat kangaroo)
She used a certain kinase inhibitor to cause cells to form
monopolar spindles (only one aster, instead of 2)
and she injected fluorescent tubulin, and made video time lapse
to record the cells' behavior: The new discovery is that cleavage furrows formed and pinched off the ends of cells, further away than the lengths of aster microtubules.
This implies that something released from or at the ends of aster microtubules is what causes acto-myosin to aggregate into the ring.

#V) Anaphase A is pulling of chromosomes toward the poles

Anaphase B is elongation of the poles = spindle elongation

the relative degree of A vs. B varies widely between species

#VI) Poleward flux of microtubules, from kinetochore toward pole
during metaphase (look at fig # 18-21)

#VII) balance of forces mechanism (Ostergren, Hays, Skibbens)
causes chromosomes to gravitate to metaphase plate.

#VIII) Spindle attachment checkpoint delays anaphase until all kinetochore pairs are stretched at metaphase plate.

#IX) Two main hypotheses about the mechanical force that pulls the kinetochores toward the poles in anaphase:
* some kinesin-like motor, using ATP, in the kinetochore
* simply the depolymerization of the microtubules

#X) in early embryos of some species (flies, etc.)
mitoses occur without cytokinesis fig 18-36
13 rounds of cell cycles: mitoses produce

Chapter 19: Extracellular Matrix and Adhesions Proline

1) Cell walls are made of secreted polysaccharides (chains of sugars)
(& are outside the plasma membrane)

2) Turgor pressure: osmotic pressure, of water "trying" to diffuse into cells, pushes plasma membranes outward against the cell walls.
This force is often very large: 22 atmospheres (700 foot depth in water)
burst bacteria, penicillin blocks enzymes that synthesize cell walls

3) In higher (multicellular) plants, the mitotic divisions are concentrated (only) in certain small zones = meristems
shoot meristems, root meristems, cambium (sideways meristem)
but divisions exert negligible (if any) pushing forces.
Plant shape is caused by controlled osmotic swelling
by controlled weakening of cells walls: certain places & directions

4) Cellulose: polymer of glucose:
enzyme that synthesizes it is actually in the plasma membrane
and (apparently is guided by a layer of cortical microtubules
Cellulose fibers are often lined up all in one direction
microtubules just under plasma membrane aligned in same direction
Later MTs (somehow) become reoriented perpendicular to this
and then the cellulose fibers are laid down parallel to this new direction
?What would you expect to happen in plant cells treated with colchicine?

Plants also secrete many other structural polysaccharides:
Lignin, pectins swelling: jelly

Cells in higher plants are connected by narrow cytoplasmic connections
called "plasmadesmata"

A thought question: what do you suppose was the evidence behind our textbook's statement that at least 700 of Arabidopsis' genes serve to control some aspect of cell wall morphogenesis?

5) Animal cells (no cell walls. )
extracellular matrix collagen (jello glue Elmer's cement
Glycine every 3rd amino acid almost 1/3 proline
triple helix (NOT alpha helix pattern) long, stiff, side links
Type I collagen, type II collagen, type IV collagen, into the 20s
67 nm repeat striations
tendons, dermis, organ capsules, walls of arteries, tooth-jaw connections
often form alternating perpendicular layers: cornea, intervertebral discs

vitamin C needed for secretion of collagen scurvy
collagenous structures therefore can't be renewed
the ones with the fastest turnover degenerate first.

6) Integrin (transmembrane connection) from actin to fibronectin
integrin a dimer of alpha & beta components, many interesting facts
fibronectin is an extracellular protein: connects collagen to integrin
RGD peptide snake venom "disintegrin"
(there are lots of other interesting proteins in snake venoms)

Focal adhesions: focal adhesion kinases, etc.

7) mechanical interactions between fibroblast cells and collagen
note figure # 19-50 which is also fig # 16-95 (rotated 90 degrees)
The theory that fibroblast traction is the mechanism that causes
formation of tendons, capsules, wraps blood vessels, etc.
But what could be the mechanism that forms the 90 degree/layers?

A brief explanation of the physics (thermodynamics) of rubber
medical-biomechanical questions about embolisms

9) Epithelia basement lamella laminin type IV collagen
differences between apical versus baso-lateral plasma membranes

10) Different kinds of cell-cell adhesions (especially in epithelia)
tight junctions (occluding junctions: also control membrane proteins)

Gap junctions connexin proteins
(mention electrotonic coupling of heart muscle cells to each other)

Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. [9] They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method. [10]

In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC, Hipparchus (from Nicaea, Asia Minor) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry. [11] In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest. [12] Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. [13] (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) in Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds.

The modern sine convention is first attested in the Surya Siddhanta, and its properties were further documented by the 5th century (AD) Indian mathematician and astronomer Aryabhata. [14] These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry. [15] [16] The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right. [17] [18] [19] Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form. [20] He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws. [21] Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi. [22] One of the earliest works on trigonometry by a northern European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus, who was encouraged to write, and provided with a copy of the Almagest, by the Byzantine Greek scholar cardinal Basilios Bessarion with whom he lived for several years. [23] At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond. [24] Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts.

Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics. [25] Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. [26] Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. [27] Also in the 18th century, Brook Taylor defined the general Taylor series. [28]

Trigonometric ratios are the ratios between edges of a right triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:

Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

Cosine function (cos), defined as the ratio of the adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.

Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

The hypotenuse is the side opposite to the 90 degree angle in a right triangle it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. See below under Mnemonics.

Since any two right triangles with the same acute angle A are similar, [29] the value of a trigonometric ratio depends only on the angle A.

The reciprocals of these functions are named the cosecant (csc), secant (sec), and cotangent (cot), respectively:

The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-". [30]

With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. [31] These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known.

Mnemonics

A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them and their corresponding sides as strings of letters. For instance, a mnemonic is SOH-CAH-TOA: [32]

Sine = Opposite ÷ Hypotenuse Cosine = Adjacent ÷ Hypotenuse Tangent = Opposite ÷ Adjacent

One way to remember the letters is to sound them out phonetically (i.e., SOH-CAH-TOA, which is pronounced 'so-ka-toe-uh' / s oʊ k æ ˈ t oʊ ə / ). Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid". [33]

The unit circle and common trigonometric values

Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane. [34] In this setting, the terminal side of an angle A placed in standard position will intersect the unit circle in a point (x,y), where x = cos ⁡ A and y = sin ⁡ A . [34] This representation allows for the calculation of commonly found trigonometric values, such as those in the following table: [35]

Using the unit circle, one can extend the definitions of trigonometric ratios to all positive and negative arguments [36] (see trigonometric function).

Graphs of trigonometric functions

The following table summarizes the properties of the graphs of the six main trigonometric functions: [37] [38]

Function	Period	Domain	Range
sine	2 π	( − ∞ , ∞ )	[ − 1 , 1 ]
cosine	2 π	( − ∞ , ∞ )	[ − 1 , 1 ]
tangent	π	x ≠ π / 2 + n π	( − ∞ , ∞ )
secant	2 π	x ≠ π / 2 + n π	( − ∞ , − 1 ] ∪ [ 1 , ∞ )
cosecant	2 π	x ≠ n π	( − ∞ , − 1 ] ∪ [ 1 , ∞ )
cotangent	π	x ≠ n π	( − ∞ , ∞ )

Inverse trigonometric functions

Because the six main trigonometric functions are periodic, they are not injective (or, 1 to 1), and thus are not invertible. By restricting the domain of a trigonometric function, however, they can be made invertible. [39] : 48ff

The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table: [39] : 48ff [40] : 521ff

Power series representations

When considered as functions of a real variable, the trigonometric ratios can be represented by an infinite series. For instance, sine and cosine have the following representations: [41]

With these definitions the trigonometric functions can be defined for complex numbers. [42] When extended as functions of real or complex variables, the following formula holds for the complex exponential:

This complex exponential function, written in terms of trigonometric functions, is particularly useful. [43] [44]

Calculating trigonometric functions

Trigonometric functions were among the earliest uses for mathematical tables. [45] Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. [46] Slide rules had special scales for trigonometric functions. [47]

Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses). [48] Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions. [49] The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions. [50]

Other trigonometric functions

Astronomy

For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions, [53] predicting eclipses, and describing the orbits of the planets. [54]

In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars, [55] as well as in satellite navigation systems. [16]

Navigation

Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation. [56]

Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles. [57]

Surveying

In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects. [58]

On a larger scale, trigonometry is used in geography to measure distances between landmarks. [59]

Periodic functions

The sine and cosine functions are fundamental to the theory of periodic functions, [60] such as those that describe sound and light waves. Fourier discovered that every continuous, periodic function could be described as an infinite sum of trigonometric functions.

Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics [61] and communications, [62] among other fields.

Optics and acoustics

Trigonometry is useful in many physical sciences, [63] including acoustics, [64] and optics. [64] In these areas, they are used to describe sound and light waves, and to solve boundary- and transmission-related problems. [65]

Other applications

Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs. [80]

Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities, [81] relate both the sides and angles of a given triangle.

Triangle identities

In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram). [82]

Law of sines

The law of sines (also known as the "sine rule") for an arbitrary triangle states: [83]

Law of cosines

The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles: [83]

Law of tangents

The law of tangents, developed by François Viète, is an alternative to the Law of Cosines when solving for the unknown edges of a triangle, providing simpler computations when using trigonometric tables. [84] It is given by:

Given two sides a and b and the angle between the sides C, the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides: [83]

Heron's formula is another method that may be used to calculate the area of a triangle. This formula states that if a triangle has sides of lengths a, b, and c, and if the semiperimeter is

then the area of the triangle is: [85]

where R is the radius of the circumcircle of the triangle.

Trigonometric identities

Pythagorean identities

The following trigonometric identities are related to the Pythagorean theorem and hold for any value: [86]

The second and third equations are derived from dividing the first equation by cos 2 ⁡ A > and sin 2 ⁡ A > , respectively.

Euler's formula

Euler's formula, which states that e i x = cos ⁡ x + i sin ⁡ x =cos x+isin x> , produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:

Other trigonometric identities

Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities. [29]