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Studies
文章出處
  蔡豐州 醫師
日期&閱覽
 張貼日期:2007/6/28  閱覽次數:3810

                                              Studies

                                                                         作者: 蔡豐州醫師

@ Theorem (M.F. Atiyah and I.M. Singer): Let P(f) = 0 be a system of
differential equations. Then
analytical index(P) = topological index(P)

@ Definition: A meromorphic function f is said to be periodic if and only
if there exists a nonzero w, such that f(z + w) = f(z).
The complex number w is called period.

@ The fact that all Mathematics is Symbolic Logic is one of the
greatest discoveries of our age; and when this fact has been
established, the remainder of the principles of mathematics
consists in the analysis of Symbolic Logic itself.
Bertrand Russell, The Principles of Mathematics (1903)

Le Chatelier's principle
If a chemical system at equilibrium experiences a change in concentration, temperature, volume, or total pressure; the equilibrium will shift in order to partially counter-act the imposed change.

@ Theorem.
If M is the module of periods of a meromorphic function f,
it must have one of the following forms:
 M = {0}
 M = {nw I n belong to Z}, for some nonzero complex value w.
 M = {n1w1 + n2w2 I n1; n2 belong to Z}, for some nonnzero complex values w1; w2 belong to C, whose ratio is not real.

@ If y2 = P(x), where P is any polynomial of degree 3 or 4 in x with no repeated roots, then we obtain a nonsingular plane curve of genus one, which is often also called an elliptic curve. Even more generally, an algebraic curve of genus one, for example from the intersection of two three-dimensional quadric surfaces, is called an elliptic curve.

@ Proposition (Geometric group law). Suppose Pi = (xi, yi), i =
1, 2 are distinct point on an elliptic curve y2 = x3+ax+b, and that x1 is not equal to x2.
Let L be the unique line through P1 and P2. Then L intersects the graph of E at exactly one other point
Q = (J2 − x1 − x2, ¸Jx3 + v)
where J= (y1 − y2)/(x1 − x2) and v = y1 − Jx1.

@
1, Cramer's rule
a11x+a12y+a13z=k1
a21x+a22y+a23z=k2
a31x+a32y+a33z=k3
x=D1/D  y=D2/D   z=D3/D
2, (A-JI)v=0 
A is nxn matrix; v is eigenvector of A (column vector); I is identity matrix
3, A' is transpose of A
A=A' is called "symmetric matrix"----A is square matrix
4, function as a mapping of the domain into the target set.
5, dy=(Jacobian matrix) dx

RAMO-SHOCKLEY THEOREM:
I=1/1volt SUM(Nk Vk Ek) 
Nk: charge carrier number
Vk: drift velocity
Ek: field strength

@ Schrodinger Equation:


@ Symmetry (K. Lother Wolf, Robert Woll):
translation 平行移動
spegelung 鏡照
drehung 迴轉
streckung 擴大或縮小

User posted image


@ Student's t tests:
A test of the null hypothesis that the means of two normally distributed populations are equal. Given two data sets, each characterized by its mean, standard deviation and number of data points, we can use some kind of t test to determine whether the means are distinct, provided that the underlying distributions can be assumed to be normal. All such tests are usually called Student's t tests, though strictly speaking that name should only be used if the variances of the two populations are also assumed to be equal; the form of the test used when this assumption is dropped is sometimes called Welch's t test. There are different versions of the t test depending on whether the two samples are

  • independent of each other (e.g., individuals randomly assigned into two groups), or
  • paired, so that each member of one sample has a unique relationship with a particular member of the other sample (e.g., the same people measured before and after an intervention, or IQ test scores of a husband and wife).

@ cadherin:
1, superfamily of calcium-dependent cell adhesion proteins
2, multiple functions which include adhesion, selectivity, clustering, signaling
--- development, tissue morphogenesis, apoptosis, cell differentiation, tissue polarity
3, classical cadherins consist of an ectodomain, a single-pass transmembrane domain, a cytoplasmic domain
4, ectodomain
--- the bidirectional force transducer at these junctions---transmitting mechanical stimuli to cytoplasm
--- determine cadherin specificity
--- 4 N-terminal EC domain junctions binds three calcium ions(位於間隔)
    at low calcium, the ectodomain is flexible, but it becomes a semiflexible, elongated rod at Ca>40uM. Only Ca>0.5mM activates local, adhesive function
5, two models:
(1) on the same cell surface to form lateral or cis dimers
(2) in turn adhere to dimers on the adjacent cell to form trans adhesive bonds
@ Integrins
(1) are the major transmembrane component of both costameres and FAs where they serve as a link between cytoplasmic
proteins and proteins in the extracellular matrix (ECM) (Spence
et al., 2002; Carragher and Frame, 2004). Proteins in the ECM
that are known to bind to specific integrins and that align with
costameres include laminin-2 (Imanaka-Yoshida et al., 1999;
Bezakova and Lomo, 2001), collagens (Borg et al., 1983) and
fibronectin (Kami et al., 1993). Based on their location and
associations, it has been proposed that the functions of costameres are to transmit contractile force laterally from the myofibrils across the sarcolemma to the ECM, to maintain the spatial organization of the myofibrils, and to insure the integrity of the muscle fibers during cycles of contraction and relaxation (Pardo et al., 1983; Shear and Bloch, 1985; Danowski et al., 1992).
the normally transverse banded pattern of several costameric proteins is disrupted in skeletal muscle after 3 days of denervation, but is restored when the muscles are electrically stimulated (Bezakova and Lomo, 2001).
<<Developmental Biology 293 (2006) 38–52>>
(2) The extracellular binding activity of integrins is regulated from the inside of the cell (inside-out signaling), while the binding of the ECM elicits signals that are transmitted into the cell (outside-in signaling).
(3) integrin clustering: The cytoplasmic tails of integrins are generally short and always devoid of enzymatic features. Hence, integrins transduce signals by associating with adapter proteins that connect the integrin to the cytoskeleton, kinases, and transmembrane growth factor receptors.
As integrins bind to ECM, they become clustered in the plane of the cell membrane and associate with a cytoskeletal and signaling complex that promotes the assembly of actin filaments. In this manner, integrins serve as integrators of the ECM and cytoskeleton.
(4) FAK pathways: integrins activate various protein tyrosine kinases, including focal adhesion kinases(FAK), Src-family kinases, and Abl, seronine-threonine kinase, integrin-linked kinase(ILK). These interactions link FAK to signaling pathways that modify the cytoskeleton and activate mitogen-activated protein kinase(MAPK) cascades.
(5) Control of cell cycle: Cells require anchorage to ECM to proliferate. In one pathway, integrins activate the MAPK c-Jun NH2-terminal kinase(JNK) which regulates progression through the G1 phase of the cell cycle. Activated JNK enters the nucleus and phosphorylates the transcription factor c-Jun, which combines with c-Fos to form the AP-1 transcription factor complex. Progression through G1 phase of the cell cycle requires the sequential activation of the cyclin-dependent kinases(Cdk): Cdk4/6 and Cdk2.  Because most growth factors are poor activators of JNK, the ability of integrins to activate kinase may explain why cell proliferation requires integrin-mediated adhesion.
(6) protecive effect against Anoikis (a Greek word meaning "homelessness"): Loss of attachment to the matrix causes apoptosis in many cell types. This phenomenon, refered to as "anoikis". It would prevent cells that have lost contact with their surroundings from establishing themselves at inappropriate locations. FAK plays a major role in conveying survival signals from the ECM. Because FAK binds PI 3-kinase, the protective effect against anoikis may be the result of PI 3-kinase-mediated activation of protein kinase B/Akt. Akt promotes survival and inactivates two proapoptotic proteins, Bad and caspase-9.  Like cell growth, anoikis can be controlled by the ECM in an integrin-specific manner. The alpha5beta1 integrin, which binds to fibronectin, induces expression of the antiapoptotic protein Bcl-2.
<Science 285 (1999) 1028>>


@ Binding of calcium to troponin C exposes a hydrophobic pocket that binds to the switch segment of troponin I . This interaction moves the inhibitory loop of troponin I away from the actin filament; as a result, there is a cooperative transition of tropomyosin along the actin filament permitting muscle contraction

@ Chemodenervation: The light chain BoNT-A cleaves the synaptosomal-associated protein of 25-kDa (SNAP-25), one of the 3 components of the synaptic fusion complex, a set of SNARE proteins(Soluble NSF-attachment Protein Receptor)---- blocks the membrane of the synptic vesicle containing acetylcholine from fusing with the neural cell membrane---- acetylcholine cannot be released into synaptic cleft to bind to the nicotinic acetylcholine receptors(nAChRs) on the muscle following an action potential.
nAChR: hetero-oligomeric membrane protein composed of alpha, beta, delta, and either gamma or epsilon subunits--- pentamers activated by the agonists acetylcholine and nicotine
Lyophyllized Botox was reconstituted with the instillation of 2ml of saline in a 100-unit vial and was injected into the muscle at a dosage of 6 units/kg body weight. An  injection volume of 10 -20ul was injected into the gastrocnemius m. with a Hamilton syringe.
Two stages: aneural stage (2 to 3 weeks) and neural stage (begin 4 to 6 weeks after injection)---
IGF-1 ---- GAP-43
p21 (satellite cell survival)
MuSK (NMJ stabilization)
Muscle nAchR inhibitor= choline
Marcaine

@ mRNA expression of muscle creatine kinase (MCK) does not change after denervation.

@ Type I, type IIA, type IIB fibers were differentiated on the basis of myofibrillar ATPase activity at PH4.5 as paper ( Brooke MH and Kaiser KK Muscle fiber types: how many and what kind? Arch Neurol 23:369-379, 1970 )

@ interstitial connective tissue was identified in muscle sections by Trichrome Masson staining

@ GAPDH, one of the glycolytic pathway enzymes often used as a marker of the glycolytic capacity, provided additional evidence of the decrease in fast glycolytic (fast IIb) fibers.

@ Three major families of neurotrophic factors:
they all function through RTKs(receptor tyrosine kinase)---they are activated when two receptor monomers are brought together by dimeric excellular ligands, bringing intracellualr kinase domains in close proximity for effective activation by transphosphorylation---its excellular domain is cysteine-rich
(1) NTs (nerve growth factor, brain-derived neurotrophic factor, NT-3, NT-4)
(2) Neurokines (ciliary neurotrophic factors, interleukin-6, cardiotrophin-1, etc)
(3) GDNF family  ligands (GFL) (GDNF, NRTN, TRTN, PSPN)

@ 5 Ca compartments:
1, mitochondrial matrix;
2, dyadic subspace;
3, junctional SR;
4, network SR;
5, myoplasmic compartments
@ Excitation Contraction Coupling (EC Coupling):
The train of physiological events that run from the moment the brain
decides to initiate a muscle contraction to the actual contraction are termed Excitation Contraction Coupling (EC Coupling). The term "excitation" refers to the transmission of a neural impulse (an Action Potential, AP) through the peripheral nerves and eventually along the sarcolemma and t-tubules. "Contraction" obviously refers to the muscle contraction and the biochemical events responsible for the contraction. Coupling refers to the linking of these two
processes.

When acetylcholine attaches to these sites, it causes a shape change in the receptor which leads to a voltage change on the muscle fiber's sarcolemma. This voltage change is called an excitatory post-synaptic potential (or EPSP) for short.
@
Stein & Padykula (1962) determined the distribution of fibres with
different histochemical properties in rat fast and slow muscles and suggested that type A fibres are the classical 'white' muscle fibres and that B and C fibres are two kinds of 'red' muscle fibre. This view is supported by the work of Henneman & Olson (1965) which showed that A fibres predominate in cat gastrocnemius muscle and most of the motor units are fast (Wuerker, McPhedran & Henneman, 1965) whereas cat soleus muscle is composed entirely of B fibres and all the motor units are slow (McPhedran, Wuerker & Henneman, 1965). The existence of three kinds of motor unit and three kinds of muscle fibre in rat limb muscles points to the possibility that each motor unit is made up of one kind of muscle fibre. Moreover, the estimated sizes of the intermediate and slow motor unit components given above correspond approximately with the sizes of C and B fibre components
shown in Fig. 13 of Stein & Padykula (1962). Their photomicrograph
shows that 30 % of the muscle fibres present in the section were C
fibres and that these constituted 20 % of the total cellular cross-sectional area. Furthermore, Stein & Padykula (1962) found higher sarcoplasm/fibril ratios for C fibres than for B fibres and this probably means that the tetanic tension developed per unit cross-sectional area is less for C fibres than for B fibres. In other words the C-fibre component probably contributes less than 20 % and the B-fibre component more than 80 % of the maximum isometric tetanic tension of the soleus muscle. These estimates of the tension developed by the C and B fibre components are approximately
the same as those given above for the tension developed by the
intermediate and slow motor unit components respectively.
@ According to the major MHC isoforms found in adult
mammalian skeletal muscles, the following pure fiber types exist: slow type I with MHCIb, and three fast types, namely type IIA with MHCIIa, type IID with MHCIId (MHCIId and fiber type IID are considered to be equivalent to MHCIIx and fiber type IIX, respectively; Pette andStaron, 1990; Schiaffino and Reggiani, 1994, 1996), and
type IIB with MHCIIb. The coexpression of specific pairs of these major MHC isoforms results in the formation of hybrid fibers, which can be subdivided based on the predominant MHC isoform. Accordingly, the following hybrid fiber types can be distinguished: type I/IIA, also
termed IC (MHCIb.MHCIIa); type IIA/I, also termed IIC (MHCIIa.MHCIb); type IIAD (MHCIIa.MHCIId); type IIDA (MHCIId.MHCIIa); type IIDB (MHCIId .MHCIIb), and type IIBD (MHCIIb.MHCIId).
@
Fast-twitch MHCIIb Fiber types IIB, IIBD, IIAB
Fast-twitch MHCIId Fiber types IID, IIBD, IIDA
Fast-twitch MHCIIa Fiber types IIA, IIAB, IIDA, IIC, IC
Fast-twitch MHCeom Extraocular and laryngeal muscles
Fast-twitch MHCIIm Masticatory muscles
Slow-twitch MHCIb Fiber types I, IC, IIC
Slow-twitch MHCIa Extraocular, diaphragm, masseter muscles, fast-to-slow transforming fibers
Slow-twitch MHCIa Plantaris, soleus, slow-to-fast transforming fibers
Slow-tonic MHCIton Extraocular, laryngeal, and tensor tympani muscles
Embryonic MHCemb Extraocular muscles
Neonatal MHCneo Extraocular, masseter muscles


@



@ cycloid:

A cycloid generated by a rolling circle.

@ Logarithmic spiral




@ Static problems: for which time is not a variable
    Dynamic problems: for which time is a variable


 


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