Conjugate Gradient Method - On My Ph.D.

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How does calculating only a few of eigenvalues out of thousands work with conjugate-gradient method?

I read that if you have a large square matrix, say more than 1000x1000 and you want to get its eigenvalues but all you need is just, say, 10 of them, then there is the so-called conjugate gradient method that can save you a significant amount of time of calculating exactly the number of eigenvalues you want instead of all of them. Can someone point me to existing numerical libraries (does BLAS or LAPACK have it) and references?
EDIT: The matrix can be 10^6 x 10^6.
submitted by whatisa_sky to numerical [link] [comments]

Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, "developed a number of techniques for mathematical calculations ... Lanczos algorithm for finding eigenvalues, Lanczos approximation for the gamma function, conjugate gradient method for solving systems of linear equations"

Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, submitted by spike77wbs to borntoday [link] [comments]

Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, "developed a number of techniques for mathematical calculations ... Lanczos algorithm for finding eigenvalues, Lanczos approximation for the gamma function, conjugate gradient method for solving systems of linear equations"

Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, submitted by spike77wbs to borntoday [link] [comments]

Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, "developed a number of techniques for mathematical calculations ... Lanczos algorithm for finding eigenvalues, Lanczos approximation for the gamma function, conjugate gradient method for solving systems of linear equations"

Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, submitted by spike77wbs to borntoday [link] [comments]

Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, "developed a number of techniques for mathematical calculations ... Lanczos algorithm for finding eigenvalues, Lanczos approximation for the gamma function, conjugate gradient method for solving systems of linear equations"

Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, submitted by spike77wbs to borntoday [link] [comments]

Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, "developed a number of techniques for mathematical calculations ... Lanczos algorithm for finding eigenvalues, Lanczos approximation for the gamma function, conjugate gradient method for solving systems of linear equations"

Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, submitted by spike77wbs to borntoday [link] [comments]

Lupine Publishers | The Creation of C13H20BeLi2SeSi. The Proposal of a Bio- Inorganic Molecule, Using Ab Initio Methods for The Genesis of a Nano Membrane

Lupine Publishers | The Creation of C13H20BeLi2SeSi. The Proposal of a Bio- Inorganic Molecule, Using Ab Initio Methods for The Genesis of a Nano Membrane
Lupine Publishers | An archive of organic and inorganic chemical sciences


Abstract
The work is an evolution of research already begin and in development. Therefore, we can observe a part that has already been commented that presents the whole development of the research from its beginning. Preliminary bibliographic studies did not reveal any works with characteristics studied here. With this arrangement of atoms and employees with such goals. Going beyond with imagination using quantum chemistry in calculations to obtain probable one new bio-inorganic molecule, to the Genesis of a bioinorganic membrane with a combination of the elements Be, Li, Se, Si, C and H. After calculation a bio-inorganic seed molecule from the previous combination, it led to the search for a molecule that could carry the structure of a membrane. From simple molecular dynamics, through classical calculations, the structure of the molecule was stabilized. An advanced study of quantum chemistry using ab initio, HF (Hartree-Fock) method in various basis is applied and the expectation of the stabilization of the Genesis of this bio-inorganic was promising. The calculations made so far admit a seed molecule at this stage of the quantum calculations of the arrangement of the elements we have chosen, obtaining a highly reactive molecule with the shape polar-apolar-polar. Calculations obtained in the ab initio RHF method, on the set of bases used, indicate that the simulated molecule, C13H20BeLi2SeSi, is acceptable by quantum chemistry. Its structure has polarity at its ends, having the characteristic polar-apolar-polar. Even using a simple base set the polar-apolar-polar characteristic is predominant. The set of bases used that have the best compatible, more precise results are CC-pVTZ and 6-311G (3df, 3pd). In the CC-pVTZ base set, the charge density in relation to 6-311G (3df, 3pd) is 50% lower. The structure of the bio-inorganic seed molecule for a bio-membrane genesis that challenge the current concepts of a protective mantle structure of a cell such as bio-membrane to date is promising, challenging. Leaving to the biochemists their experimental synthesis.
Introduction
The work is an evolution of research already begin and in development. Therefore, we can observe a part that has already been commented that presents the whole development of the research from its beginning. A small review of the main compounds employed some of their known physicochemical and biological properties and the ab initio methods used. Preliminary bibliographic studies did not reveal any works with characteristics studied here. With this arrangement of atoms and employees with such goals. So, the absence of a referential of the theme. The initial idea was to construct a molecule that was stable, using the chemical elements Lithium, Beryllium, alkaline and alkaline earth metals, respectively, as electropositive and electronegative elements - Selenium and Silicon, semimetal and nonmetal, respectively. This molecule would be the basis of the structure of a crystal, whose structure was constructed only with the selected elements. The elements Li, Be, Se and Si were chosen due to their physicochemical properties, and their use in several areas of technology [1-4]. To construct such a molecule, which was called a seed molecule, quantum chemistry was used by ab initio methods [5,6,7]. The equipment used was a cluster of the Biophysics laboratory built specifically for this task. It was simulated computationally via molecular dynamics, initially using Molecular Mechanics [8-24] and ab initio methods [5,6,7]. The results were satisfactory. We found a probable seed molecule of the BeLi2SeSi structure predicted by quantum chemistry [23]. Due to its geometry, it presents a probable formation of a crystal with the tetrahedral and hexahedral crystal structure [23].
The idea of a new molecule for a crystal has been upgraded. Why not build a molecule, in the form of a lyotropic liquid crystal [25] that could be the basis of a new bio-membrane? For this, the molecule should be amphiphilic, with polar head and apolar tail. Are basic requirement of the construction of a bio-membrane [25]. Then it is necessary to add a hydrophobic tail, with atoms of carbon and hydrogen. Therefore, the molecule seed with a polar hydrophilic “head”. So, would a new amphiphilic molecule. Several simulations were performed, always having as initial dynamics the use of Molecular Mechanics [8-24] for the initial molecular structure, moving to ab initio calculations of quantum chemistry. All attempts were thwarted. Quantum calculations of quantum chemistry did not accept the seed molecule as the polar head, even changing its binding structure. The silicon atom binds in double bond with the carbon chain and Selenium. It binds in double with beryllium and is simple with the two lithium atoms, thus making a stable molecular structure for Molecular Mechanics [8-24], Mm+ and Bio+ Charmm [26]. But in quantum calculations the seed molecule changed all its fundamental structure [1]. The linear structure of the tail with the polar head, in the form of a rope climbing hook, collapsed, bending toward a polar tail. In another simulation carried out the Selenium was connected in double bond to two atoms of Carbon added in double bond. As the +6 polarities of the selenium neutralized with the atoms two atoms of lithium, forming a wing. In the double bonded sequence is the Carbon with the Silicon, and this in double bond with the Beryllium. A new structure for a probable lyotropic liquid crystal has now been formed. A polar tail with the seed molecule undone but retaining the five base atoms of its fundamental structure [25]. The structure after Molecular Mechanics, Mm+ and Bio+ Charmm [26], the shape of the molecule obtained had a structure like a boomerang. After calculations ab initio, the polar tail was undone. The Beryllium atom did not remain in the structure of the molecule, releasing itself from it. There is then a new idea. Why not separate the electropositive and electronegative elements in two polar heads? This would completely change the concepts known so far of a biomembrane with a lipid bilayer. The next challenging step of building a bio-membrane that runs away from known concepts, with a single layer, with two polar heads and its non-polar backbone. Would it be a new way to have a bio-membrane? A challenge for quantum chemistry.
Then he concentrated the calculations on the probable structure of the molecule with polar ends. Separately then in pairs the atoms of Selenium with Beryllium and Silicon with the two bonds. Again, the attempt failed, in quantum calculations. Beryllium was disconnected from the basic structure of the new molecule, polarpolar- polar polar structure. They have decided to further innovate the theory and “challenge” quantum chemistry. Add an aromatic ring to the polar head. The polar-polar-polar linear structure was now maintained, with a six-carbon cyclic chain. At a polar end, the Silicon is bonded to three atoms of the Hydrogen and is connected to a Carbon from the central chain. This one connected to the two atoms of the Lithium and a polar central carbon chain. At the other polar end, the six-carbon cyclic chain attached in single bond to the carbonic chain. The cyclic chain with simple bonds, having at its center the Selenium with six bonds to the cyclic chain and a double with the Beryllium, thus forcing two more covalent bonds. Now with a +2 cationic head, the dynamics of the minimization energy with Mm+ and Bio+ Charmm [26] calculations have maintained a stable structure of the molecule. A polar head like a “parabolic antenna”, with folded edges outward with the Hydrogen atoms. The expected, the obvious, Beryllium playing the role of the “LNB (Low Noise Block) receiver”. We then proceeded to the ab initio calculations in several methods and basis, testing various possibilities with ab initio methods. The polar-apolar-polar (parabolic) molecule in ab initio calculation, by RHF [5-6,27-32] in the TZV [33,34] sets basis was shown to be stable by changing its covalent cyclic chain linkages, which was expected, (Figure 2). The set of bases used was that of Ahlrichs and coworker’s main utility are: the SV, SVP, TZV, TZVP keywords refer to the initial formations of the split valence and triple zeta basis sets from this group [33,34]. Calculations continue to challenge concepts, experimenting. Going where imagination can lead us, getting results that challenge concepts.
Selenium
Selenium is found impurely in metal sulfide ores, copper where it partially replaces the sulfur. The chief commercial uses for selenium today are in glassmaking and in pigments. Selenium is a semiconductor and is used in photocells. Uses in electronics, once important, have been mostly supplanted by silicon semiconductor devices. Selenium continues to be used in a few types of DC power surge protectors and one type of fluorescent quantum dot [2]. Although it is toxic in large doses, selenium is an essential micronutrient for animals. In plants, it sometimes occurs in toxic amounts as forage, e.g. locoweed. Selenium is a component of the amino acids selenocys teine and selenomethionine. In humans, selenium is a trace element nutrient that functions as cofactor for glutathione peroxidases and certain forms ofthioredoxin reductase [45]. Selenium-containing proteins are produced from inorganic selenium via the intermediacy of selenophosphate (PSeO3 3−). Selenium is an essential micronutrient in mammals but is also recognized as toxic in excess. Selenium exerts its biological functions through selenoproteins, which contain the amino acid selenocysteine. Twenty-five selenoproteins are encoded in the human genome [46]. Selenium also plays a role in the functioning of the thyroid gland. It participates as a cofactor for the three thyroid hormonedeiodinases. These enzymes activate and then deactivate various thyroid hormones and their metabolites [47]. It may inhibit Hashimotos’s disease, an auto-immune disease in which the body’s own thyroid cells are attacked by the immune system. A reduction of 21% on TPO antibodies was reported with the dietary intake of 0.2 mg of selenium [48]. Selenium deficiency can occur in patients with severely compromised intestinal function, those undergoing total parenteral nutrition, and [49] in those of advanced age (over 90).
Silicon
Silicon is the eighth most common element in the universe by mass, but very rarely occurs as the pure free element in nature. It is most widely distributed in dusts, sands, planetoids, and planets as various forms of silicon dioxide (silica) or silicates. Over 90% of the Earth’s crust is composed of silicate minerals, making silicon the second most abundant element in the Earth’s crust (about 28% by mass) after oxygen [11]. Elemental silicon also has a large impact on the modern world economy. Although most free silicon is used in the steel refining, aluminium-casting, and fine chemical industries (often to make fumed silica), the relatively small portion of very highly purified silicon that is used in semiconductor electronics (<10%) is perhaps even more critical. Because of wide use of silicon in integrated circuits, the basis of most computers, a great deal of modern technology depends on it [2]. Although silicon is readily available in the form of silicates, very few organisms use it directly. Diatoms, radiolaria and siliceous sponges use biogenic silica as a structural material for skeletons. In more advanced plants, the silica phytoliths (opal phytoliths) are rigid microscopic bodies occurring in the cell; some plants, for example rice, need silicon for their growth [50,51,52]. There is some evidence that silicon is important to nail, hair, bone and skin health in humans, [53] for example in studies that show that premenopausal women with higher dietary silicon intake have higher bone density, and that silicon supplementation can increase bone volume and density in patients with osteoporosis [54]. Silicon is needed for synthesis of elastin and collagen, of which the aorta contains the greatest quantity in the human body [55] and has been considered an essential element [56].
Methods
The steric energy, bond stretching, bending, stretch-bend, out of plane, and torsion interactions are called bonded interactions because the atoms involved must be directly bonded or bonded to a common atom. The van der Waals and electrostatic (qq) interactions are between non-bonded atoms [8-24].
Hartree-Fock
The Hartree-Fock self–consistent method [5-6,27- 32] is based on the one-electron approximation in which the motion of each electron in the effective field of all the other electrons is governed by a one-particle Schrodinger¨ equation. The Hartree- Fock approximation considers of the correlation arising due to the electrons of the same spin, however, the motion of the electrons of the opposite spin remains uncorrelated in this approximation. The methods beyond self-consistent field methods, which treat the phenomenon associated with the many-electron system properly, are known as the electron correlation methods. One of the approaches to electron correlation is the Møller-Plesset (MP) [5,6,57,58] perturbation theory in which the Hartree-Fock energy is improved by obtaining a perturbation expansion for the correlation energy [5]. However, MP calculations are not variational and can produce an energy value below the true energy [6]. The exchangecorrelation energy is expressed, at least formally, as a functional of the resulting electron density distribution, and the electronic states are solved for self-consistently as in the Hartree-Fock approximation [27-30]. A hybrid exchange-correlation functional is usually constructed as a linear combination of the Hartree-Fock exact exchange functional,and any number of exchange and correlation explicit density functional. The parameters determining the weight of each individual functional are typically specified by fitting the functional predictions to experimental or accurately calculated thermochemical data, although in the case of the “adiabatic connection functional” the weights can be set a priori [32]. Terms like “Hartree-Fock”, or “correlation energy” have specific meanings and are pervasive in the literature [59]. The vast literature associated with these methods suggests that the following is a plausible hierarchy:
The extremes of ‘best’, FCI, and ‘worst’, HF, are irrefutable, but the intermediate methods are less clear and depend on the type of chemical problem being addressed [4]. The use of HF in the case of FCI was due to the computational cost.
For calculations a cluster of six computer models was used: Prescott-256 Celeron © D processors [2], featuring double the L1 cache (16 KB) and L2 cache (256 KB), Socket 478 clock speeds of 2.13 GHz; Memory DDR2 PC4200 512MB; Hitachi HDS728080PLAT20 80 GB and CD-R. The dynamic was held in Molecular Mechanics Force Field (Mm+), Equation (1), after the quantum computation was optimized via Mm+ and then by RHF [5-6,27-32], in the TZV [33,34] sets basis. The molecular dynamics at algorithm Polak- Ribiere [60], conjugate gradient, at the termination condition: RMS gradient [61] of 0, 1kcal/A. mol or 405 maximum cycles in vacuum [6,41]. The first principles calculations have been performed to study the equilibrium configuration of C13H20BeLi2SeSi molecule using the Hyperchem 7.5 Evaluation [41], Mercury 3.8 a general molecular and electronic structure processing program [18], GaussView 5.0.8 [64] an advanced semantic chemical editor, visualization, and analysis platform and GAMESS is a computational chemistry software program and stands for General Atomic and Molecular Electronic Structure System [7] set of programs. The first principles approaches can be classified in the Restrict Hartree-Fock [5-6,27-32] approach.
Discussions
The Figure 2 shows the final stable structure of the Bioinorganic molecule obtained by an ab initio calculation with the method RHF [5-6,27-32], in several sets of basis such as: STO-3G [7,30,60,71,83,84, 85,86]; 3-21G [7,30,60,71,83,84,85,86]; 6-31G [7,30,60,71,83,84,85,86]; 6-31(d’) [7,30,60,71,83,84,85,86]; 6-31(d’,p’) [7,30,60,71,83,84,85,86]; 6-311G [7,30,60,71,83,84,85,86]; 6-311G(3df,3pd) [7,30,60,71,83, 84,85,86]; SV [81,82]; SDF [71,72]; SDD [71,72]; SDDAll [71,72]; TZV [81,82]; CC-pVDZ [66,67,68,69,70]; CC-pVTZ [66-70]; CEP- 31G [66-70]; CEP-121G [66-70]; LanL2DZ [71,78,79,80]; LanL2MB [71,78,79,80], starting from the molecular structure of (Figure 1) obtained through a molecular mechanical calculation, method Mm+ and Bio+ Charmm [8-24,26,65].
The molecular structure shown in Figure 2 of the bio-inorganic molecule C13H20BeLi2SeSi, is represented in structure in the form of the van der Walls radius [4,5,6]. As an example of analysis, the set of bases TZV [81,82]. with the charge distribution (Δδ) through it, whose charge variation is Δδ = 4.686 au of elemental charge. In green color the intensity of positive charge displacement. In red color the negative charge displacement intensity. Variable, therefore, of δ- = 2,343 a.u. negative charge, passing through the absence of charge displacement, represented in the absence of black - for the green color of δ+ = 2.343 a.u. positive charge. The electric dipole moment () total obtained was p = 5.5839 Debye, perpendicular to the main axis of the molecule, for sets basis TZV [81,82]. By the distribution of charge through the bio-inorganic molecule it is clear that the molecule has a polar-apolar-polar structure, with neutral charge distributed on its main axis, the carbonic chain. A strong positive charge displacement (cation) at the polar ends of the molecule, in the two lithium and silicon atoms, bound to the carbon atom with strong negative (anion). Therefore, there is a displacement of electrons from the two lithium and silicon atoms towards the carbon attached to them. At the other end of the cyclic chain, attached to it is the totally neutral Selenium atom, while the beryllium is extremely charged with positive charge (cationic), represented in green color. While the two carbon atoms of the cyclic chain connected to Beryllium, with negatively charged (anionic), represented in red color. It happened, therefore, a displacement of electrons of the Beryllium atom towards the Carbons connected to it. An analysis of the individual charge value of each atom of the molecule could be made, but here it was presented only according to (Figure 2), due to the objective being to determine the polarpolar- polar, the polar characteristic of the molecule, whose moment of dipole is practically perpendicular to the central axis of the molecule. In Figure 2 the dipole moment is visualized in all the base sets, being represented by an arrow in dark blue color, with their respective values in Debye. This also presents the orientation axes x, y and z and the distribution of electric charges through the molecule. Analyzing the charge distribution through the molecule.
In all the sets of bases used, the Silicon atom presents a strong positive charge, that is, cationic form, represented in green color, except for the LanL2MB base, which presents a strong negative charge displacement, represented in red color. The two Lithium atoms accompany the cationic tendency of Silicon, but with less intensity. The Carbon atom connected to the central chain, and to Silicon and the two Lithiums, presents a strong negative charge, that is, anionic form, represented in red color. There is, therefore, a shift of the electric charges of the silicon atom and of the two Lithiums towards the Carbon. This charge displacement is evident in all the base sets studied, except for the base STO-3G and LanL2MB, which present almost neutral charge for the said Carbon atom.
The backbone of the molecule, that is, its central axis which has a chain of seven aligned Carbon atoms, has a homogeneous charge distribution, with approximately neutral polarity, represented by the absence of color (black). This charge neutrality is observed in the set of bases: STO-3G; 6-31 (d ‘, p’); TZV; SDD; CEP-31G; CCcVDZ; SV and CEP-121G. In the set of bases: 3-21G; 6-31G; 6-31 (d ‘); 6-311G; SDF; LanL2DZ and LanL2MB, the central axis of the molecule has a small distribution of negative charge throughout its length, due to the negative charge displacement of Hydrogen atoms (seen slightly in blackish green, tending to black) connected to each of their respective Carbon atoms, whose charge is slightly negative (visualized in blackish red color, tending to black). At the other end of the molecule is the cyclic chain of six Carbon atoms. Which has only one double connection. The cyclic chain is attached to the Beryllium atom and to two Carbon atoms, symmetrical and central to the cyclic chain. The Selenium atom is connected to two carbon atoms of the cyclic chain, the first Carbon atom being connected to the central axis of the molecule and the second atoms in sequence, being opposed to the double bonded cyclic chain atoms. The Beryllium atom presents a strong positive charge, cationic character, visualized in green color, in the set of bases: 3-21G; 6-31G; 6-311G; 6-311G (3df, 3pd); SV and TZV. Beryllium presents almost totally neutral charge in the set of bases: 6-31 (d ‘); 6-31 (d, p ‘); CC-pVDZ; cc-pVTZ; CEP-31G and CEP-121G. And charge, slightly positive in another basis studied. The Selenium atom is visualized in Figure 2, as seen always behind the cyclic chain. This presents a neutral charge distribution in all basis studied, with the exception of CCpVTZ and LanL2MB. The Table 1 presents the Molecular parameters of the atoms of the molecule C13H20BeLi2SeSi seed, obtained through computer via ab initio calculation method RHF [5-6,27-32] in base 6-311G**(3df,3pd) [7,30,60,71,83,84,85], obtained using computer programs GAMESS [7]. end software [64], (Figure 1) the right. The distance between the atoms is measured in Ångstron, as well as the position of the atoms in the coordinate axes x, y and z. The angles formed, and the angles formed in the dihedral are given in degrees. In the Table 2 containing the electric dipole moments, in the directions of the coordinate axes axes x, y and z, given in Debye, are presented in all the sets of bases studied. The minimum and maximum charge distributed through the molecule and the variation of the charge (in a.u.) by the extension of the molecule (C13H20BeLi2SeSi). They are represented by the variation of the intensities of the green color (positive charge), through black (zero charge) and red (negative charge), evenly distributed according to the basic functions used in quantum calculations allowed by quantum chemistry. The largest distributed charge variation (Δδ) per molecule was calculated on the base set TZV, with Δδ = 4.686 a.u., and the lowest in the CC-pVTZ set, with Δδ = 0.680 a.u., (Table 2). The highest total electric dipole moment () was obtained using the CEP-31G method, with p = 6.0436 Debye, with Δδ = 1.860 a.u., and the lowest electric dipole moment in the STO-3G method, with p = 4.2492 Debye, with Δδ = 1.510 a.u.
Conclusion
Calculations obtained in the ab initio RHF method, on the set of bases used, indicate that the simulated molecule, C13H20BeLi2SeSi, is acceptable by quantum chemistry. Its structure has polarity at its ends, having the characteristic polar-apolar-polar. Even using a simple base set the polar-apolar-polar characteristic is predominant. From the set of bases used in the RHF, based on 6-311G (3df, 3pd), the Silicon atoms, the two Lithium, have a strong density of positive charge, cationic, from the displacement of charges of these atoms towards the atom which Carbon are connected, which consequently exhibits strong negative charge density, anionic. It is observed a cyclic displacement and constant electric charges originating from the sp orbitals of the Carbon atom, (Figure 2). At the other end of the molecule, a similar situation occurs. The Beryllium atom presents a high density of positive charge, cationic character, due to the displacement of the electronic cloud of that one towards the Carbon atoms that is connected. These Carbon atoms also receive a displacement of negative charges, originating from the two Carbon atoms that are linked in the cyclic chain, in covalent double bonds. Now presenting these latter a strong density of positive, cationic charges, such as Beryllium, leaving the anionic Beryllium bound Carbon. The Selenium atom has a small anionic character. Among all simulated base assemblies, 6-311G (3df, 3pd), is unique that exhibits the characteristic of the central chain, with a small density of negative charges, near the ends of the Carbons of this.
In the CC-pVTZ base set, the charge density in relation to 6-311G (3df, 3pd) is 50% lower, with characteristics like those shown in the Silicon and the two Lithium atoms. However, the central chain presents an anionic feature, for all its extension, originating from the displacement of charges of the Hydrogen atoms connected to them. At the other end of the cyclic chain, the Selenium atom presents high density of negative charges, anionic, as well as in the cyclic chain the Carbon atoms present anionic characteristics, with little intensity, distributed proportionally by these atoms, originating from the displacement of charges of the Hydrogens linked to these. Except for the Carbon atom, connected to the central axis of the molecule that is not bound to Hydrogens atoms. The structure of the Bio-inorganic seed molecule for a bio-membrane genesis that defies the current concepts of a protective mantle structure of a cell such as bio-membrane to date is promising, challenging. Leaving to the Biochemists their experimental synthesis. The quantum calculations must continue to obtain the structure of the bioinorganic bio-membrane. The following calculations, which are the computational simulation via Mm+, QM/MM, should indicate what type of structure should form. Structures of a liquid crystal such as a new membrane may occur, micelles.
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Why do we have so many Matrix Invertibility formulas?

We have:
Newton's method
Cayley–Hamilton method
Eigendecomposition
Cholesky decomposition
Analytic solution
Inversion of 2 × 2 matrices
Inversion of 3 × 3 matrices
Inversion of 4 × 4 matrices
Blockwise inversion
By Neumann series
P-adic approximation,
formulas/methods to invert a matrix; why do we have so many? How did we get to so many?
submitted by sumanth_lazarus to math [link] [comments]

AP Bio Guide (Units 8 in comments)

AP Bio Guide (Units 8 in comments)

1) Chemistry of Life

Content

  • Transpiration
    • Hydrogen bonds pull water up like string and leave through stoma
    • Stomata: leaf pores that allow gas exchange, most are on bottom side of leaf
    • Xylem: tube-shaped, nonlining, vascular system, carries water from roots to rest of plant
    • Epidermis: outer layer, protects plant
    • Phloem: transports food
    • Parenchyma: stores food
    • Transpiration: evaporation of water from leaves
    • Adhesion: polar water molecules adhere to polar surfaces (sides of xylem)
    • Cohesion: polar water molecules adhere to each other
    • Guard cells: cells surrounding stoma, regulate transpiration through opening and closing stoma
    • Turgid vs flaccid guard cells
      • Turgid swell caused by potassium ions, water potential decreases, water enters vacuoles of guard cells
      • Swelling of guard cells open stomata
    • High light levels, high levels of water, low temperature, low CO2 causes opening of stomata
    • Water potential: transport of water in plant governed by differences in water potential
      • Affected by solute concentration and environmental conditions
    • High water potential (high free energy and more water) travels to low water potential
    • Hydrophilic = attracts water, hydrophobic = repels water
  • Water and its Properties
    • Polar molecule due to positive hydrogen and negative oxygen regions
    • Negative oxygen of one molecule to positive hydrogen of another water molecule forms a hydrogen bond, which are weak individually but strong together
    • Important physical properties of water:
      • Cohesion and adhesion: cohesion creates surface tension and they both allow for transpiration
      • High specific heat: enables water to absorb and lose heat slowly
      • High heat of vaporization: allows much of it to remain liquid
      • Nearly universal polar solvent: dissolves a lot of stuff
      • Flotation of ice: insulates, transportation
  • Biological Macromolecules
    • Polymer: long molecule consisting of many similar building blocks linked by covalent bonds
    • Monomer: building block of a polymer
    • ATP - adenosine triphosphate, energy carrier that uses bonds between phosphates to store energy
      • Similar in structure to a ribonucleotide
    • Four Types
      • Carbohydrates
      • Lipids
      • Proteins
      • Nucleic Acids
https://preview.redd.it/xp12oli61w451.png?width=1098&format=png&auto=webp&s=cc897738989258c67bcc760ba040e2cee8f7875c
  • Functional groups
    • Hydroxyl - carbs, alcohols - OH-, O-
    • Amino - proteins - NH2, NH3+
    • Carboxyl - weak acids - COOH, COO-
    • Sulfhydryl - proteins - SH
    • Phosphatic - salts, strong acids - PO
  • Directionality:
    • ex: glucose alpha and beta
    • ex: DNA and RNA 5’ and 3’ ends
  • Identification of Macromolecules
https://preview.redd.it/cb3oau2j1w451.png?width=1089&format=png&auto=webp&s=409e26f32c9996a3649bad81d17ed72769955ce9

Calculations

  • Number of bonds
    • # of molecules - 1
    • i.e. 20 glucose molecules linked together would have 19 bonds
  • Molecular formula
    • # of molecules * molecular formula - number of bonds * H20 (from hydrolysis)
    • i.e. when you bond 5 glucose molecules together you have to subtract 4H2O
  • pH/pOH
    • -log[H+] = pH
    • -log[OH-] = pOH
    • pH + pOH = 14
  • Leaf surface area
    • i.e. using graph paper to find surface area
  • Transpiration rate
    • Amount of water used / surface area / time

Labs

  • Transpiration Lab
    • Basically you take this potometer which measures the amount of water that gets sucked up by a plant that you have and you expose the plant to different environmental conditions (light, humidity, temperature) and see how fast the water gets transpired
    • Random stuff to know:
      • It’s hard to get it to work properly
      • A tight seal of vaseline keeps everything tidy and prevents water from evaporating straight from the tube, also allows for plant to suck properly
      • Water travels from high water potential to low water potential

2) Cell Structure & Function

Content

  • Cellular Components
    • Many membrane-bound organelles evolved from once free prokaryotes via endosymbiosis, such as mitochondria (individual DNA)
    • Compartmentalization allows for better SA:V ratio and helps regulate cellular processes
    • Cytoplasm: thick solution in each cell containing water, salts, proteins, etc; everything - nucleus
      • Cytoplasmic streaming: moving all the organelles around to give them nutrients, speeds up reactions
    • Cytosol: liquid of the cytoplasm (mostly water)
    • Plasma Membrane: separates inside of cell from extracellular space, controls what passes through amphipathic area (selectively permeable)
      • Fluid-Mosaic model: phospholipid bilayer + embedded proteins
      • Aquaporin: hole in membrane that allows water through
    • Cell Wall: rigid polysaccharide layer outside of plasma membrane in plants/fungi/bacteria
      • Bacteria have peptidoglycan, fungi have chitin, and plants have cellulose and lignin
      • Turgor pressure pushes the membrane against the wall
    • Nucleus: contains genetic information
      • Has a double membrane called the nuclear envelope with pores
    • Nucleolus: in nucleus, produces ribosomes
    • Chromosomes: contain DNA
    • Centrioles: tubulin thing that makes up centrosome in the middle of a chromosome
    • Smooth Endoplasmic Reticulum: storage of proteins and lipids
    • Rough Endoplasmic Reticulum: synthesizes and packages proteins
    • Chloroplasts: photosynthetic, sunlight transferred into chemical energy and sugars
      • More on this in photosynthesis
    • Vacuoles: storage, waste breakdown, hydrolysis of macromolecules, plant growth
    • Plasmodesmata: channels through cell walls that connect adjacent cells
    • Golgi Apparatus: extracellular transport
    • Lysosome: degradation and waste management
      • Mutations in the lysosome cause the cell to swell with unwanted molecules and the cell will slow down or kill itself
    • Mitochondria: powerhouse of the cell
      • Mutations in the mitochondria cause a lack of deficiency of energy in the cell leading to an inhibition of cell growth
    • Vesicles: transport of intracellular materials
    • Microtubules: tubulin, stiff, mitosis, cell transport, motor proteins
    • Microfilaments: actin, flexible, cell movement
    • Flagella: one big swim time
    • Cilia: many small swim time
    • Peroxisomes: bunch of enzymes in a package that degrade H202 with catalase
    • Ribosomes: protein synthesis
    • Microvilli: projections that increase cell surface area like tiny feetsies
      • In the intestine, for example, microvilli allow more SA to absorb nutrients
    • Cytoskeleton: hold cell shape
  • Cellular Transport
    • Passive transport: diffusion
      • Cell membranes selectively permeable (large and charged repelled)
      • Tonicity: osmotic (water) pressure gradient
    • Cells are small to optimize surface area to volume ratio, improving diffusion
    • Primary active transport: ATP directly utilized to transport
    • Secondary active transport: something is transported using energy captured from movement of other substance flowing down the concentration gradient
    • Endocytosis: large particles enter a cell by membrane engulfment
      • Phagocytosis: “cell eating”, uses pseudopodia around solids and packages it within a membrane
      • Pinocytosis: “cell drinking”, consumes droplets of extracellular fluid
      • Receptor-mediated endocytosis: type of pinocytosis for bulk quantities of specific substances
    • Exocytosis: internal vesicles fuse with the plasma membrane and secrete large molecules out of the cell
    • Ion channels and the sodium potassium pump
      • Ion channel: facilitated diffusion channel that allows specific molecules through
      • Sodium potassium pump: uses charged ions (sodium and potassium)
    • Membrane potential: voltage across a membrane
    • Electrogenic pump: transport protein that generates voltage across a membrane
    • Proton pump: transports protons out of the cell (plants/fungi/bacteria)
    • Cotransport: single ATP-powered pump transports a specific solute that can drive the active transport of several other solutes
    • Bulk flow: one-way movement of fluids brought about by pressure
    • Dialysis: diffusion of solutes across a selective membrane
  • Cellular Components Expanded: The Endomembrane System
    • Nucleus + Rough ER + Golgi Bodies
      • Membrane and secretory proteins are synthesized in the rough endoplasmic reticulum, vesicles with the integral protein fuse with the cis face of the Golgi apparatus, modified in Golgi, exits as an integral membrane protein of the vesicles that bud from the Golgi’s trans face, protein becomes an integral portion of that cell membrane

Calculations

  • Surface area to volume ratio of a shape (usually a cube)
  • U-Shaped Tube (where is the water traveling)
    • Solution in u-shaped tube separated by semi-permeable membrane
    • find average of solute (that is able to move across semi permeable membrane)
    • add up total molar concentration on both sides
    • water travels where concentration is higher
  • Water Potential = Pressure Potential + Solute Potential
    • Solute Potential = -iCRT
      • i = # of particles the molecule will make in water
      • C = molar concentration
      • R = pressure constant (0.0831)
      • T = temperature in kelvin

Labs

  • Diffusion and Osmosis
    • Testing the concentration of a solution with known solutions
    • Dialysis bag
      • Semipermeable bag that allows the water to pass through but not the solute
    • Potato core
      • Has a bunch of solutes inside

Relevant Experiments

  • Lynne Margolis: endosymbiotic theory (mitochondria lady)
  • Chargaff: measured A/G/T/C in everything (used UV chromatography)
  • Franklin + Watson and Crick: discovered structure of DNA; Franklin helped with x ray chromatography

3) Cellular Energetics

Content

  • Reactions and Thermodynamics
    • Baseline: used to establish standard for chemical reaction
    • Catalyst: speeds up a reaction (enzymes are biological catalysts)
    • Exergonic: energy is released
    • Endergonic: energy is consumed
    • Coupled reactions: energy lost/released from exergonic reaction is used in endergonic one
    • Laws of Thermodynamics:
      • First Law: energy cannot be created nor destroyed, and the sum of energy in the universe is constant
      • Second Law: energy transfer leads to less organization (greater entropy)
      • Third Law: the disorder (entropy) approaches a constant value as the temperature approaches 0
    • Cellular processes that release energy may be coupled with other cellular processes
    • Loss of energy flow means death
    • Energy related pathways in biological systems are sequential to allow for a more controlled/efficient transfer of energy (product of one metabolic pathway is reactant for another)
    • Bioenergetics: study of how energy is transferred between living things
    • Fuel + 02 = CO2 + H20
      • Combustion, Photosynthesis, Cellular Respiration (with slight differences in energy)
  • Enzymes
    • Speed up chemical processes by lowering activation energy
    • Structure determines function
    • Active sites are selective
    • Enzymes are typically tertiary- or quaternary-level proteins
    • Catabolic: break down / proteases and are exergonic
    • Anabolic: build up and are endergonic
    • Enzymes do not change energy levels
    • Substrate: targeted molecules in enzymatic
    • Many enzymes named by ending substrate in “-ase”
    • Enzymes form temporary substrate-enzyme complexes
    • Enzymes remain unaffected by the reaction they catalyze
    • Enzymes can’t change a reaction or make other reactions occur
    • Induced fit: enzyme has to change its shape slightly to accommodate the substrate
    • Cofactor: factor that help enzymes catalyze reactions (org or inorg)
      • Examples: temp, pH, relative ratio of enzyme and substrate
      • Organic cofactors are called coenzymes
    • Denaturation: enzymes damaged by heat or pH
    • Regulation: protein’s function at one site is affected by the binding of regulatory molecule to a separate site
    • Enzymes enable cells to achieve dynamic metabolism - undergo multiple metabolic processes at once
    • Cannot make an endergonic reaction exergonic
    • Steps to substrates becoming products
      • Substrates enters active site, enzyme changes shape
      • Substrates held in active site by weak interactions (i.e. hydrogen bonds)
      • Substrates converted to product
      • Product released
      • Active site available for more substrate
    • Rate of enzymatic reaction increases with temperature but too hot means denaturation
    • Inhibitors fill the active site of enzymes
      • Some are permanent, some are temporary
      • Competitive: block substrates from their active sites
      • Non competitive (allosteric): bind to different part of enzyme, changing the shape of the active site
    • Allosteric regulation: regulatory molecules interact with enzymes to stimulate or inhibit activity
    • Enzyme denaturation can be reversible
  • Cellular Respiration
    • Steps
      • Glycolysis
      • Acetyl co-A reactions
      • Krebs / citric acid cycle
      • Oxidative phosphorylation
    • Brown fat: cells use less efficient energy production method to make heat
    • Hemoglobin (transport, fetal oxygen affinity > maternal) and myoglobin (stores oxygen)
  • Photosynthesis
    • 6CO2 + 6H20 + Light = C6H12O6 + 6O2
    • Absorption vs action spectrum (broader, cumulative, overall rate of photosynthesis)
    • Components
      • Chloroplast
      • Mesophyll: interior leaf tissue that contains chloroplasts
      • Pigment: substance that absorbs light
    • Steps
      • Light-Dependent Reaction
      • Light-Independent (Dark) Reaction (Calvin Cycle)
  • Anaerobic Respiration (Fermentation)
    • Glycolysis yields 2ATP + 2NADH + 2 Pyruvate
    • 2NADH + 2 Pyruvate yields ethanol and lactate
    • Regenerates NAD+

Calculations

  • Calculate products of photosynthesis & cellular respiration

Labs

  • Enzyme Lab
    • Peroxidase breaks down peroxides which yields oxygen gas, quantity measured with a dye
    • Changing variables (i.e. temperature) yields different amounts of oxygen
  • Photosynthesis Lab
    • Vacuum in a syringe pulls the oxygen out of leaf disks, no oxygen causes them to sink in bicarbonate solution, bicarbonate is added to give the disks a carbon source for photosynthesis which occurs at different rates under different conditions, making the disks buoyant
  • Cellular Respiration Lab
    • Use a respirometer to measure the consumption of oxygen (submerge it in water)
    • You put cricket/animal in the box that will perform cellular respiration
    • You put KOH in the box with cricket to absorb the carbon dioxide (product of cellular respiration)-- it will form a solid and not impact your results

Relevant Experiments

  • Engelmann
    • Absorption spectra dude with aerobic bacteria

4) Cell Communication & Cell Cycle

Content

  • Cell Signalling
    • Quorum sensing: chemical signaling between bacteria
      • See Bonnie Bassler video
    • Taxis/Kinesis: movement of an organism in response to a stimulus (chemotaxis is response to chemical)
    • Ligand: signalling molecule
    • Receptor: ligands bind to elicit a response
    • Hydrophobic: cholesterol and other such molecules can diffuse across the plasma membrane
    • Hydrophilic: ligand-gated ion channels, catalytic receptors, G-protein receptor
  • Signal Transduction
    • Process by which an extracellular signal is transmitted to inside of cell
    • Pathway components
      • Signal/Ligand
      • Receptor protein
      • Relay molecules: second messengers and the phosphorylation cascade
      • DNA response
    • Proteins in signal transduction can cause cancer if activated too much (tumor)
      • RAS: second messenger for growth factor-- suppressed by p53 gene (p53 is protein made by gene) if it gets too much
    • Response types
      • Gene expression changes
      • Cell function
      • Alter phenotype
      • Apoptosis- programmed cell death
      • Cell growth
      • Secretion of various molecules
    • Mutations in proteins can cause effects downstream
    • Pathways are similar and many bacteria emit the same chemical within pathways, evolution!
  • Feedback
    • Positive feedback amplifies responses
      • Onset of childbirth, lactation, fruit ripening
    • Negative feedback regulates response
      • Blood sugar (insulin goes down when glucagon goes up), body temperature
  • Cell cycle
    • Caused by reproduction, growth, and tissue renewal
    • Checkpoint: control point that triggers/coordinates events in cell cycle
    • Mitotic spindle: microtubules and associated proteins
      • Cytoskeleton partially disassembles to provide the material to make the spindle
      • Elongates with tubulin
      • Shortens by dropping subunits
      • Aster: radial array of short microtubules
      • Kinetochores on centrosome help microtubules to attach to chromosomes
    • IPMAT: interphase, prophase, metaphase, anaphase, telophase
      • PMAT is mitotic cycle
    • Steps
      • Interphase
      • Mitosis
      • Cytokinesis
    • Checkpoints
      • 3 major ones during cell cycle:
      • cyclin-cdk-mpf: cyclin dependent kinase mitosis promoting factor
      • Anchorage dependence: attached, very important aspect to cancer
      • Density dependence: grow to a certain size, can’t hurt organs
      • Genes can suppress tumors
    • G0 phase is when cells don’t grow at all (nerve, muscle, and liver cells)

Calculations

Relevant Experiments

  • Sutherland
    • Broke apart liver cells and realized the significance of the signal transduction pathway, as the membrane and the cytoplasm can’t activate glycogen phosphorylase by themselves

5) Heredity

Content

  • Types of reproduction
    • Sexual: two parents, mitosis/meiosis, genetic variation/diversity (and thus higher likelihood of survival in a changing environment)
    • Asexual: doesn’t require mate, rapid, almost genetically identitical (mutations)
      • Binary fission (bacteria)
      • Budding (yeast cells)
      • Fragmentation (plants and sponges)
      • Regeneration (starfish, newts, etc.)
  • Meiosis
    • One diploid parent cell undergoes two rounds of cell division to produce up to four haploid genetically varied cells
    • n = 23 in humans, where n is the number of unique chromosomes
    • Meiosis I
      • Prophase: synapsis (two chromosome sets come together to form tetrad), chromosomes line up with homologs, crossing over
      • Metaphase: tetrads line up at metaphase plate, random alignment
      • Anaphase: tetrad separation, formation at opposite poles, homologs separate with their centromeres intact
      • Telophase: nuclear membrane forms, two haploid daughter cells form
    • Meiosis II
      • Prophase: chromosomes condense
      • Metaphase: chromosomes line up single file, not pairs, on the metaphase plate
      • Anaphase: chromosomes split at centromere
      • Telophase: nuclear membrane forms and 4 total haploid cells are produced
    • Genetic variation
      • Crossing over: homologous chromosomes swap genetic material
      • Independent assortment: homologous chromosomes line up randomly
      • Random fertilization: random sperm and random egg interact
    • Gametogenesis
      • Spermatogenesis: sperm production
      • Oogenesis: egg cells production (¼ of them degenerate)
  • Fundamentals of Heredity
    • Traits: expressed characteristics
    • Gene: “chunk” of DNA that codes for a specific trait
    • Homologous chromosomes: two copies of a gene
    • Alleles: copies of chromosome may differ bc of crossing over
    • Homozygous/Heterozygous: identical/different
    • Phenotype: physical representation of genotype
    • Generations
      • Parent or P1
      • Filial or F1
      • F2
    • Law of dominance: one trait masks the other one
      • Complete: one trait completely covers the other one
      • Incomplete: traits are both expressed
      • Codominance: traits combine
    • Law of segregation (Mendel): each gamete gets one copy of a gene
    • Law of independent assortment (Mendel): traits segregate independently from one another
    • Locus: location of gene on chromosome
    • Linked genes: located on the same chromosome, loci less than 50 cM apart
    • Gene maps and linkage maps
    • Nondisjunction: inability of chromosomes to separate (ex down syndrome)
    • Polygenic: many genes influence one phenotype
    • Pleiotropic: one gene influences many phenotypes
    • Epistasis: one gene affects another gene
    • Mitochondrial and chloroplast DNA is inherited maternally
  • Diseases/Disorders
    • Genetic:
      • Tay-Sachs: can’t break down specific lipid in brain
      • Sickle cell anemia: misshapen RBCs
      • Color blindness
      • Hemophilia: lack of clotting factors
    • Chromosomal:
      • Turner: only one X chromosome
      • Klinefelter: XXY chromosomes
      • Down syndrome (trisomy 21): nondisjunction
  • Crosses
    • Sex-linked stuff
    • Blood type
    • Barr bodies: in women, two X chromosomes; different chromosomes expressed in different parts of the body, thus creating two different phenotype expressions in different places

Calculations

  • Pedigree/Punnett Square
  • Recombination stuff
    • Recombination rate = # of recombinable offspring/ total offspring (times 100) units: map units

Relevant Experiments

  • Mendel

6) Gene Expression and Regulation

Content

  • DNA and RNA Structure
    • Prokaryotic organisms typically have circular chromosomes
    • Plasmids = extrachromosomal circular DNA molecules
    • Purines (G, A) are double-ringed while pyrimidines (C, T, U) have single ring
    • Types of RNA:
      • mRNA - (mature) messenger RNA (polypeptide production)
      • tRNA - transfer RNA (polypeptide production)
      • rRNA - ribosomal RNA (polypeptide production)
      • snRNA - small nuclear RNA (bound to snRNPs - small nuclear ribonucleoproteins)
      • miRNA - microRNA (regulatory)
  • DNA Replication
    • Steps:
      • Helicase opens up the DNA at the replication fork.
      • Single-strand binding proteins coat the DNA around the replication fork to prevent rewinding of the DNA.
      • Topoisomerase works at the region ahead of the replication fork to prevent supercoiling.
      • Primase synthesizes RNA primers complementary to the DNA strand.
      • DNA polymerase III extends the primers, adding on to the 3' end, to make the bulk of the new DNA.
      • RNA primers are removed and replaced with DNA by DNA polymerase I.
      • The gaps between DNA fragments are sealed by DNA ligase.
  • Protein Synthesis
    • 61 codons code for amino acids, 3 code as STOP - UAA, UAG, UGA - 64 total
    • Transcription Steps:
      • RNA polymerase binds to promoter (before gene) and separate the DNA strands
      • RNA polymerase fashions a complementary RNA strand from a DNA strand
      • Coding strand is same as RNA being made, template strand is complementary
      • Terminator on gene releases the RNA polymerase
    • RNA Processing Steps (Eukaryotes):
      • 5’ cap and 3’ (poly-A tail, poly A polymerase) tail is added to strand (guanyl transferase)
      • Splicing of the RNA occurs in which introns are removed and exons are added by spliceosome
      • Cap/tail adds stability, splicing makes the correct sequence (“gibberish”)
    • Translation Steps:
      • Initiation complex is the set up of a ribosome around the beginning of an mRNA fragment
      • tRNA binds to codon, amino acid is linked to other amino acid
      • mRNA is shifted over one codon (5’ to 3’)
      • Stop codon releases mRNA
  • Gene Expression
    • Translation of mRNA to a polypeptide occurs on ribosomes in the cytoplasm as well as rough ER
    • Translation of the mRNA occurs during transcription in prokaryotes
    • Genetic info in retroviruses is an exception to normal laws: RNA to DNA is possible with reverse transcriptase, which allows the virus to integrate into the host’s DNA
    • Regulatory sequences = stretches of DNA that interact with regulatory proteins to control transcription
    • Epigenetic changes can affect expression via mods of DNA or histones
    • Observable cell differentiation results from the expression of genes for tissue-specific proteins
    • Induction of transcription factors during dev results in gene expression
    • Prokaryotes: operons transcribed in a single mRNA molecule, inducible system
    • Eukaryotes: groups of genes may be influenced by the same transcription factors to coordinate expression
    • Promoters = DNA sequences that RNA polymerase can latch onto to initiate
    • Negative regulators inhibit gene expression by binding to DNA and blocking transcription
    • Acetylation (add acetyl groups)- more loosely wound/ less tightly coiled/compressed
    • Methylation of DNA (add methyl groups) - less transcription- more tightly wound
  • Mutation and Genetic Variation
    • Disruptions in genes (mutations) change phenotypes
    • Mutations can be +/-/neutral based on their effects that are conferred by the protein formed - environmental context
    • Errors in DNA replication or repair as well as external factors such as radiation or chemical exposure cause them
    • Mutations are the primary source of genetic variation
    • Horizontal acquisition in prokaryotes - transformation (uptake of naked DNA), transduction (viral DNA transmission), conjugation (cell-cell DNA transfer), and transposition (DNA moved within/between molecules) - increase variation
    • Related viruses can (re)combine genetic material in the same host cell
    • Types of mutations: frameshift, deletion, insertion
  • Genetic Engineering
    • Electrophoresis separates molecules by size and charge
    • PCR magnifies DNA fragments
    • Bacterial transformation introduces DNA into bacterial cells
  • Operons
    • Almost always prokaryotic
    • Promoter region has operator in it
    • Structural genes follow promoter
    • Terminator ends operon
    • Regulatory protein is active repressor
    • Active repressor can be inactivated
    • Enhancer: remote gene that require activators
    • RNAi: interference with miRNA
    • Anabolic pathways are normally on and catabolic pathways are normally off

Calculations

  • Transformation efficiency (colonies/DNA)
  • Numbers of base pairs (fragment lengths)
  • Cutting enzymes in a plasmid or something (finding the lengths of each section)

Labs

  • Gel Electrophoresis Lab
    • Phosphates in DNA make it negative (even though it’s an acid!), so it moves to positive terminal on the board
    • Smaller DNA is quicc, compare it to a standard to calculate approx. lengths
  • Bacterial Transformation Lab
    • Purpose of sugar: arabinose is a promoter which controls the GFP in transformed cells, turns it on, also green under UV
    • Purpose of flipping upside down: condensation forms but doesn’t drip down
    • Purpose of heat shock: increases bacterial uptake of foreign DNA
    • Plasmids have GFP (green fluorescent protein) and ampicillin resistance genes
    • Calcium solution puts holes in bacteria to allow for uptake of plasmids
  • PCR Lab
    • DNA + primers + nucleotides + DNA polymerase in a specialized PCR tube in a thermal cycler
    • Primers bind to DNA before it can repair itself, DNA polymerase binds to the primers and begins replication
    • After 30 cycles, there are billions of target sequences

Relevant Experiments

  • Avery: harmful + harmless bacteria in mice, experimented with proteins vs DNA of bacteria
  • Griffith: Avery’s w/o DNA vs protein
  • Hershey and Chase: radioactively labeled DNA and protein
  • Melson and Stahl: isotopic nitrogen in bacteria, looked for cons/semi/dispersive DNA
  • Beadle and Tatum: changed medium’s amino acid components to find that a metabolic pathway was responsible for turning specific proteins into other proteins, “one gene one enzyme”
  • Nirenberg: discovered codon table

7) Natural Selection

  • Scientific Theory: no refuting evidence (observation + experimentation), time, explain a brand/extensive range of phenomena
  • Theory of Natural Selection
    • Definition
      • Not all offspring (in a population) will survive
      • Variation among individuals in a population
      • Some variations were more favourable than others in a particular environment
      • Those with more favourable variations were more likely to survive and reproduce.
      • These favourable variations were passed on and increased in frequency over time.
  • Types of Selection:
    • Directional selection: one phenotype favored at one of the extremes of the normal distribution
      • ”Weeds out” one phenotype
      • Ony can happen if a favored allele is already present
    • Stabilizing Selection: Organisms within a population are eliminated with extreme traits
      • Favors “average” or medium traits
      • Ex. big head causes a difficult delivery; small had causes health deficits
    • Disruptive Selection: favors both extremes and selects against common traits
      • Ex. sexual selection (seems like directional but it’s not because it only affects one sex, if graph is only males then directional)
  • Competition for limited resources results in differential survival, favourable phenotypes are more likely to survive and produce more offspring, thus passing traits to subsequent generations.
    • Biotic and abiotic environments can be more or less stable/fluctuating, and this affects the rate and direction of evolution
      • Convergent evolution occurs when similar selective pressures result in similar phenotypic adaptations in different populations or species.
      • Divergent evolution: groups from common ancestor evolve, homology
      • Different genetic variations can be selected in each generation.
      • Environments change and apply selective pressures to populations.
    • Evolutionary fitness is measured by reproductive success.
    • Natural selection acts on phenotypic variations in populations.
      • Some phenotypic variations significantly increase or decrease the fitness of the organism in particular environments.
    • Through artificial selection, humans affect variation in other species.
      • Humans choose to cause artificial selection with specific traits, accidental selection caused by humans is not artificial
    • Random occurrences
      • Mutation
      • Genetic drift - change in existing allele frequency
      • Migration
    • Reduction of genetic variation within a given population can increase the differences between populations of the same species.
    • Conditions for a population or an allele to be in Hardy-Weinberg equilibrium are
      • Large population size
      • Absence of migration
      • No net mutations
      • Random mating
      • Absence of selection
    • Changes in allele frequencies provide evidence for the occurrence of evolution in a population.
    • Small populations are more susceptible to random environmental impact than large populations.
    • Gene flow: transference of genes/alleles between populations
  • Speciation: one species splits off into multiple species
    • Sympatric (living together i.e. disruption) Allopatric (physically separate, i.e. founder effect) Parapatric (habitats overlapping)
      • Polyploidy (autopolyploidy), sexual selection
    • Species: group of populations whose members can interbreed and produce healthy, fertile offspring but can’t breed with other species (ex. a horse and donkey can produce a mule but a mule is nonviable, so it doesn’t qualify)
      • Morphological definition: body shape and structural characteristics define a species
      • Ecological species definition: way populations interact with their environments define a species
      • Phylogenetic species definition: smallest group that shares a common ancestor is a species
    • Prezygotic barriers: barriers to reproduction before zygote is formed
      • Geographical error: two organisms are in different areas
      • Behavioural error (i.e. mating rituals aren’t the same)
      • Mechanical error: “the pieces don’t fit together”
      • Temporal error (i.e. one organism comes out at night while the other comes out in the day)
      • Zygotic/Gametic isolation: sperm and egg don’t physically meet
    • Postzygotic barriers: barriers to reproduction after zygote is formed
      • Hybrid viability: developmental errors of offspring
      • Hybrid fertility: organism is sterilized
      • Hybrid breakdown: offspring over generations aren’t healthy
    • Hybrid zone: region in which members of different species meet and mate
      • Reinforcement: hybrids less fit than parents, die off, strength prezygotic barriers
      • Fusion: two species may merge into one population
      • Stability: stable hybrid zones mean hybrids are more fit than parents, thus creating a stable population, but can be selected against in hybrid zones as well
    • Punctuated equilibria: long periods of no or little change evolutionarily punctuated by short periods of large change, gradualism is just slow evolution
    • Evidence of evolution
      • Paleontology (Fossils)
      • Comparative Anatomy
      • Embryology: embryos look the same as they grow
      • Biogeography: distribution of flora and fauna in the environment (pangea!)
      • Biochemical: DNA and proteins and stuff, also glycolysis
    • Phylogenetic trees
      • Monophyletic: common ancestor and all descendants
      • Polyphyletic: descendants with different ancestors
      • Paraphyletic: leaving specifies out of group
    • Out group: basal taxon, doesn’t have traits others do
    • Cline: graded variation within species (i.e. different stem heights based on altitude)
    • Anagenesis: one species turning into another species
    • Cladogenesis: one species turning into multiple species
    • Taxon: classification/grouping
    • Clade: group of species with common ancestor
    • Horizontal gene transfer: genes thrown between bacteria
    • Shared derived characters: unique to specific group
    • Shared primitive/ancestral characters: not unique to a specific group but is shared within group
  • Origins of life
    • Stages
      • Inorganic formation of organic monomers (miller-urey experiment)
      • Inorganic formation of organic polymers (catalytic surfaces like hot rock or sand)
      • Protobionts and compartmentalization (liposomes, micelles)
      • DNA evolution (RNA functions as enzyme)
    • Shared evolutionary characteristics across all domains
      • Membranes
      • Cell comm.
      • Gene to protein
      • DNA
      • Proteins
    • Extant = not extinct
    • Highly conserved genes = low rates of mutation in history due to criticalness (like electron transport chain)
    • Molecular clock: dating evolution using DNA evidence
    • Extinction causes niches for species to fill
    • Eukaryotes all have common ancestor (shown by membrane-bound organelles, linear chromosomes, and introns)

Calculations

  • Hardy-Weinberg
    • p + q = 1
    • p^2 + 2pq +q^2 = 1
  • Chi Squared

Labs

  • Artificial Selection Lab
    • Trichrome trait hairs
    • Anthocyanin for second trait (purple stems)
    • Function of the purple pigment?
    • Function of trichome hairs?
  • BLAST Lab
    • Putting nucleotides into a database outputs similar genes

Relevant Experiments

  • Darwin
  • Lamarck
  • Miller-Urey
    • Slapped some water, methane, ammonia, and hydrogen is some flasks and simulated early earth with heat and stuff and it made some amino acids.
submitted by valiantseal to u/valiantseal [link] [comments]

Can we try to reduce overfitting by analyzing for "outliers" in gradients?

This is just a hypothetical thought that came across my mind and I was wondering if it has any potential.
TLDR: Skip to the Solution section
TLDR 2: What if instead of performing gradient descent the traditional way, we calculate two gradients g_A and g_B from two separate partitions of the training data and backpropagate g_A⋅g_B / (norm(g_A) norm(g_B)) * g_A instead of using just the gradients from one batch.

Background

Typically, in most deep learning methods, we compute the gradient of some cost function J over all samples in a batch, average those gradients, and update our parameters θ by subtracting these gradients (scaled by a learning rate). Obviously, there are many different optimizers that do this update step differently, but irregardless, they're all some form of gradient descent.

Motivation

When we perform gradient descent, if I understand correctly, we're trying to maximize some form of the log likelihood of the training data (i.e. maximize how well our model parameters explain the dataset). However, we can sometimes overfit the training data if our gradient descent brings us to a solution that's too well fit for the training data, and thus is no longer the optimal solution for "real world" data (like the test set).

Solution...?

What if during our training process, we account for the fact that our training set doesn't completely represent the "real world" (test) data; that although we can have very similar distributions between the training data and test data, they're not identical.
More specifically, what I'm proposing is that we could split our training dataset into two partitions. Let's call it training set A and training set B. At each step, we compute the gradients of a batch from A (let's call this g_A) and gradients of a batch from B (g_B). Next, instead of just backpropagating these gradients, we analyze the gradients between g_A and g_B and discard any gradients that seem "out of place". One proposal that I was thinking of is what if we backpropagate g_A⋅g_B / (norm(g_A) norm(g_B)) * g_A, where g_A⋅g_B is the inner product of g_A and g_B. Therefore, this is kind of like backpropagating g_A (like normal gradient descent), but scaled by how much this "conjugate gradient" g_B agrees with g_A (which is the g_A⋅g_B / (norm(g_A) norm(g_B)) term).
Using this method, let's say we're near the end of training. If we only had one training set, we would just keep getting our model closer and closer to the optimal set of parameters for that training set. With this new method, I believe that the g_A⋅g_B / (norm(g_A) norm(g_B)) term would prevent the gradients g_A from overfitting training set A, and maybe it will help with overfitting?
Edit: One other thing that I realized is that if there's the issue of computation cost when calculating g_A⋅g_B, we could instead maybe perform this type of update on only a random subset of gradients, while keeping the other gradients updating the same as standard backpropagation.
Edit 2: I was able to code up a short snippet using the MNIST database. Ironically, this method overfits a LOT more than standard SGD. See results here. Basically, I took the first 50000 samples in the MNIST dataset, and trained it with batch size 500 using SGD. Next, I took the same 50000 samples (as set A) and the remaining datapoint (as set B) and retrained using this proposed method. Both methods had the same batch size of 500 and learning rate of 0.5. Both were trained for 100 epochs, and clearly there's overfitting in the proposed method. I guess my new question now is why does this happen and why this drastically?
submitted by yliu1021 to MLQuestions [link] [comments]

Looking for iterative search algorithm for local minimum, given a differentiable function and sets of measured values

This is for a research project.
My differentiable function is unfortunately not solvable for the variable I am interested in, and I do not know it's true value, either. However, I do know my physical system and its interactions well.
I want to be able to (iteratively) calculate my target value, given a continuous stream of measured values, which will be tainted by noise.
I was thinking of the gradient method, or the conjugated gradient method - how do i use them in this setting? where can i find tutorials for this?
submitted by bremen15 to ECE [link] [comments]

Looking for iterative search algorithm for local minimum, given a differentiable function and sets of measured values

My differentiable function is unfortunately not solvable for the variable I am interested in, and I do not know it's true value, either. However, I do know my physical system and its interactions well.
I want to be able to (iteratively) calculate my target value, given a continuous stream of measured values, which will be tainted by noise.
I was thinking of the gradient method, or the conjugated gradient method - how do i use them in this setting? where can i find tutorials for this?
submitted by bremen15 to askmath [link] [comments]

Derivative free optimization. Is there any methods that use the extra points to estimate the derivative?

I'm doing the optimization of a function that takes some time to evaluate (around 40 seconds). Analytical derivatives are not available (Dynamical parameter estimation). I'm currently using Nielder-Meads. When the function is close to the optimum the space between the vertices is quite small, but not "alligned" to the coordinates. Is there anyway to use these calculated vertices to estimate the jacobian and improve convergence?
My colleagues suggest Nielder-Meads until close to the optimum and then use some derivative based method (like conjugate gradient descent) to "fine tune". I'm imagine somebody invented a method that somehow use boths.
EDIT: I'm thinking about doing a linear regression and take the jacobian of the linear function, Does it makes sense? I'm not really formally trained in math.
submitted by FellowOfHorses to math [link] [comments]

Stochastic gradient descent outperforming L-BFGS

I've been hitting my head against this problem for a while now, and I'm about to give up and just use the method which I have that performs. However, I think I also have evidence that something about my implementation is broken. I'm asking for help here because my options for soliciting feedback/advice are pretty limited, so apologies for the multiple posts on the same subject matter.
Here's an album of some simple experimental results based on building a 25 hidden layer unit autoencoder, and training it with 8x8 grayscale images from Bruno Olshausen's whitened natural images dataset:
http://imgur.com/a/zuzJO
Ideally, such an autoencoder should resolve 25 edge detectors in this configuration. The first image shows this, and it's the result of training the network with "stochastic gradient descent", i.e. simple fixed-step gradient descent wherein the batch size is low (100 training examples), and only one step is taken per batch. The second figure shows the objective function versus the training iteration, and you can see the random walk downwards over 24,000 batch iterations. This took a little over 2 minutes to run.
The last picture is a typical example of the results I get from running any of three algorithms in a more typical fashion (i.e. with a batch size equal to the training set size, with multiple steps taken on the batch). Both L-BFGS and Conjugate Gradient Descent manage to quickly (within 50 iterations) find a minima on the order of 0.5 (equivalent to the finishing value of stochastic gradient descent), but the result looks like the third figure. Standard gradient descent with a large batch also does this. L-BFGS in particular (I'm using the implementation from the RISO project) will iterate a few times and then fail when it has a nonzero gradient but ends up taking a step of length 0.
My gradient calculation has been tested and I have high confidence that it is working properly. My objective function calculation seems to be the only thing separating CGD and L-BFGS from fixed-step gradient descent, but I've been staring at it for many hours now and it just isn't complex enough to convince me that there's a bug hidden in there. I would blame the data, but this exact experiment is solved using L-BFGS in Andrew Ng's tutorial here.
I'm about to use this code on some much larger experiments and I don't want to start off with a buggy implementation, but I can't nail down where my method might be diverging from Ng's example. Any thoughts or suggestions would be appreciated.
submitted by eubarch to MachineLearning [link] [comments]

Question about the behavior of conjugate gradient descent optimization

So I'm playing around with sparse autoencoders, and I'm trying to train a simple example with conjugate gradient descent. I just witnessed some behavior I can't explain and I'm hoping someone here can help me understand what's going on.
The neural network I'm training is small, and meant to solve the XOR problem. It has two inputs plus a bias on the input layer, two hidden units (plus a bias), and a single output. This creates 3*2 + 3 = 9 total weights to be trained. I have confidence that my gradient calculations are correct, because they pass the gradient estimation check described here, and are used to generate edge detectors for natural images with the backpropagation algorithm as described here. It should be a short couple of steps to train this network to solve XOR with conjugate gradient descent using my already-coded gradient calculation plus an erf() function that calculates overall network error. I'm using the Polak-Ribiere method to generate the Beta coefficient. My erf() function is more of less exactly as described at the UFLDL site.
Finally, the problem: My CGD algorithm seems to be sensitive to the magnitude of the weights that I initialize the network with. When I initialize the weights with uniform random numbers in the range of [-0.1 0.1], the algorithm reliably converges on a bad local minima (all inputs result in an output of 0.5). If I hange the weight initialization to uniform random numbers of the range [-0.3 0.3], then the network converges to a state that solves XOR.
What's the principle at work here? Is this kind of weight sensitivity something specific to CGD?
Thanks!
submitted by eubarch to MachineLearning [link] [comments]

An efficient choice (for stokes flow, artificial compressibility)?

Hello numerical!
I work in the realm of geodynamics and so I have a smattering of training in applied math but mostly geology, so I apologize in advance for oversimplistic/incorrect terminology.
Quick version of my question: Choice 1: solve Ax=b, where A is of size 2N by 2N. Choice 2: solve A_1 x_1 = b_1 and solve A_2 x_2 = b_2, where A_1 and A_2 are both of size N by N. In all cases I would use an indirect solver to get x (Conjugate Gradient, GMRES, or something else included in the PETSc library) and A is a banded, tridiagonal, sparse matrix. Is there any way to tell which choice is faster other than trying both?
Background: I'm solving the 2D stokes flow system via an artificial compressibility method. There are 3 variables, pressure P, horizontal velocity U and vertical velocity W. In other iterative approaches to solving the stokes system, the 3 variables are solved for simultaneously so that if there are N grid points, the operator A will be 3N by 3N in size. In the artificial compressibility method, P is held fixed while U and W are calculated then P is updtaded and fed back into a new calculation of U and W. Because P is fixed when solving for U and W, I can envision two ways to solve for U and W at each iteration. I could solve for U and W simultaneously, so that there would be two degrees of freedom at every gridpoint and A would be 2N by 2N. Or I could solve for U then W and each solution would involve an operator N by N in size. A typical size for N in my application is ~500. Any advise on which might be a better choice?
I have a vague recollection from a class I took years ago that in the worst case, solving Ax=b will take N2 operations. If that's true then choice 1 would take (2N)2 while choice 2 would take 2 * N2 so choice 2 would take less operations than choice 1 (for the typical values of N that I use). So I'm leaning towards choice 2, but I'm not confident in this.
Thanks for reading! And let me know if you need clarification or more information!
submitted by wrinkledknows to numerical [link] [comments]

conjugate gradient method calculator video

Numerical Methods for Linear Systems - SOR - YouTube The Organic Chemistry Tutor - YouTube Stochastic Gradient Descent - YouTube Mod-01 Lec-05 Linear Systems - III Calculating a Double Integral - YouTube Maths Genie - YouTube

The conjugate gradient method can be applied on the normal equations. The CGNE and CGNR methods are variants of this approach that are the simplest methods for nonsymmetric or indefinite systems. Since other methods for such systems are in general rather more complicated than the conjugate gradient method, transforming the system to a symmetric definite one and then applying the conjugate gradient method is attractive for its coding simplicity. CGNE solves the system (AA^(T))y=b ... The conjugate gradient method is an algorithm for finding the nearest local minimum of a function of n variables which presupposes that the gradient of the function can be computed. It uses conjugate directions instead of the local gradient for going downhill. If the vicinity of the minimum has the shape of a long, narrow valley, the minimum is reached in far fewer steps than would be the case using the method of steepest descent. For a discussion of the conjugate gradient method on vector... 3. Method of Conjugate Gradients (cg-Method) The present section will be devoted to a description of a method of solving a system of linear equations Ax=k. This method will be called the conjugate gradient method or, more briefly, the cg-method, for reasons which will unfold from the theory developed in later sections. For the moment, we shall ... Conjugate gradient method. The steepest descent method is great that we minimize the function in the direction of each step. But it doesn’t guarantee that the direction we are going to minimize the function from all the previous directions. Here we introduce a very important term A conjugate directions. Directions p are A conjugate directions if they have the following property (note A is ... Conjugate Gradient Method consists in building a vectorial sequence $(p_k)_{k\in\mathbf{N}^*}$ of $n$ $A$-conjugate vectors . Consequently, the sequence $p_1,p_2,\ldots,p_n$ form a basis of $\mathbf{R}^n$. The exact solution $x_\star$ can be expanded like follows: Abstract The Conjugate Gradient Method is an iterative technique for solving large sparse systems of linear equations. As a linear algebra and matrix manipulation technique, it is a useful tool in approximating solutions to linearized partial dierential equations. conjugate gradient method implemented with python Raw. cg.py # -*- coding: utf-8 -*-import numpy as np: from scipy. sparse. linalg import cg: import tensorflow as tf: import time: def conjugate_grad (A, b, x = None): """ Description-----Solve a linear equation Ax = b with conjugate gradient method. Parameters -----A: 2d numpy.array of positive semi-definite (symmetric) matrix: b: 1d numpy ... However, finding a conjugate basis is not a trivial thing. Conjugate gradient by iteration. In practice, the conjugate gradient is computed as an iterative method. Starting with an initial guess of the solution, we calculate the gradient to find a first direction to move. During the next iterations, the directions have to be conjugate to the previous ones, thereby assuring the conditions of the previous section. 6.2 In this example, the conjugate gradient method also converges in four total steps, with much less zig-zagging than the gradient descent method or even Newton’s method.77 7.1 The steps of the DFP algorithm applied to F(x;y).84 7.2 The steps of the DFP algorithm applied to F(x;y).91 8.1 A comparison of the BFGS method using numerical gradients vs. exact gradients.97 8.2 Powell’s ... Conjugate gradient method in Python With the conjugate_gradient function, we got the same value (-4, 5) and wall time 281 μs, which is a lot faster than the steepest descent. Visualizing steepest ...

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Numerical Methods for Linear Systems - SOR - YouTube

Share your videos with friends, family, and the world Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! Calculating a Double Integ... This channel focuses on providing tutorial videos on organic chemistry, general chemistry, physics, algebra, trigonometry, precalculus, and calculus. TI Calculator Tutorial: ... Gauss-Seidel Method of Solving Simultaneous Linear Equations: ... Mod-01 Lec-18 Conjugate Gradient Method - Duration: 55:24. nptelhrd 2,655 views. In this video we are going to look at the SOR (Successive Over-Relaxation) improvement over the Gauss-Seidel. For early access to new videos and other perks: https://www.patreon.com/welchlabsWant to learn more or teach this series? Check out the Imaginary Numbers are... This video is part of the Udacity course "Deep Learning". Watch the full course at https://www.udacity.com/course/ud730 Gradient Descent is the workhorse behind most of Machine Learning. When you fit a machine learning method to a training dataset, you're probably using Gradie... Free Math Video with Practice Test Problems. Register for FREE Online Classes: https://forms.gle/P5zGpffSnQgE1mRf6

conjugate gradient method calculator

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