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it is now thought that perhaps a quarter of all eukaryotic proteins can adopt structures that are mostly disordered, fluctuating rapidly between many different conforma- tions.

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it is now thought that perhaps a quarter of all eukaryotic proteins can adopt structures that are mostly disordered, fluctuating rapidly between many different conforma- tions.

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lysozyme—an enzyme in tears that dissolves bacterial cell walls—retains its antibacterial activity for a long time because it is stabilized by such cross-linkages.

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lysozyme—an enzyme in tears that dissolves bacterial cell walls—retains its antibacterial activity for a long time because it is stabilized by such cross-linkages.

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lysozyme—an enzyme in tears that dissolves bacterial cell walls—retains its antibacterial activity for a long time because it is stabilized by such cross-linkages.

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The use of smaller subunits to build larger structures has several advantages: 1. A large structure built from one or a few repeating smaller subunits requires only a small amount of genetic information. 2. Both assembly and disassembly can be readily controlled reversible pro- cesses, because the subunits associate through multiple bonds of relatively low energy. 3. Errors in the sy

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The use of smaller subunits to build larger structures has several advantages: 1. A large structure built from one or a few repeating smaller subunits requires only a small amount of genetic information. 2. Both assembly and disassembly can be readily controlled reversible pro- cesses, because the subunits associate through multiple bonds of relatively low energy. 3. Errors in the synthesis of the structure can be more easily avoided, since correction mechanisms can operate during the course of assembly to exclude malformed subunits.</s

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eating smaller subunits requires only a small amount of genetic information. 2. Both assembly and disassembly can be readily controlled reversible pro- cesses, because the subunits associate through multiple bonds of relatively low energy. <span>3. Errors in the synthesis of the structure can be more easily avoided, since correction mechanisms can operate during the course of assembly to exclude malformed subunits.<span><body><html>

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These principles are dramatically illustrated in the protein coat or capsid of many simple viruses, which takes the form of a hollow sphere based on an icosahedron

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The first large macromolecular aggregate shown to be capable of self-as- sembly from its component parts was tobacco mosaic virus (TMV ).

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the simplest case, a long core protein or other macromolecule provides a scaffold that determines the extent of the final assembly. This is the mechanism that deter- mines the length of the TMV particle, where the RNA chain provides the core. Similarly, a core protein interacting with actin is thought to determine the length of the thin filaments in muscle.</bo

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er macromolecule provides a scaffold that determines the extent of the final assembly. This is the mechanism that deter- mines the length of the TMV particle, where the RNA chain provides the core. Similarly, a core protein interacting with <span>actin is thought to determine the length of the thin filaments in muscle.<span><body><html>

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Typically, hundreds of monomers will aggregate to form an unbranched fibrous structure that is several micrometers long and 5 to 15 nm in width

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Typically, hundreds of monomers will aggregate to form an unbranched fibrous structure that is several micrometers long and 5 to 15 nm in width

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A surprisingly large fraction of pro- teins have the potential to form such structures, because the short segment of the polypeptide chain that forms the spine of the fibril can have a variety of different sequences and follow one of several different paths (

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A surprisingly large fraction of pro- teins have the potential to form such structures, because the short segment of the polypeptide chain that forms the spine of the fibril can have a variety of different sequences and follow one of several different paths (Figure 3–32). However, very few proteins will actually form this structure inside cells

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A set of closely related diseases—scra- pie in sheep, Creutzfeldt–Jakob disease (CJD) in humans, Kuru in humans, and bovine spongiform encephalopathy (BSE) in cattle—are caused by a misfolded, aggregated form of a particular pr

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A set of closely related diseases—scra- pie in sheep, Creutzfeldt–Jakob disease (CJD) in humans, Kuru in humans, and bovine spongiform encephalopathy (BSE) in cattle—are caused by a misfolded, aggregated form of a particular protein called PrP</spa

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A set of closely related diseases—scra- pie in sheep, Creutzfeldt–Jakob disease (CJD) in humans, Kuru in humans, and bovine spongiform encephalopathy (BSE) in cattle—are caused by a misfolded, aggregated form of a particular protein called PrP

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A set of closely related diseases—scra- pie in sheep, Creutzfeldt–Jakob disease (CJD) in humans, Kuru in humans, and bovine spongiform encephalopathy (BSE) in cattle—are caused by a misfolded, aggregated form of a particular protein called PrP

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A set of closely related diseases—scra- pie in sheep, Creutzfeldt–Jakob disease (CJD) in humans, Kuru in humans, and bovine spongiform encephalopathy (BSE) in cattle—are caused by a misfolded, aggregated form of a particular protein called PrP

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protein hormones that they will secrete in specialized “secretory granules,” which package a high concentra- tion of their cargo in dense cores with a regular structure (see Figure 13–65). We now know that these structured cores consist of <span>amyloid fibrils, which in this case have a structure that causes them to dissolve to release soluble cargo after being secreted by exocytosis to the cell exterior<span><body><html>

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nules,” which package a high concentra- tion of their cargo in dense cores with a regular structure (see Figure 13–65). We now know that these structured cores consist of amyloid fibrils, which in this case have a structure that causes them <span>to dissolve to release soluble cargo after being secreted by exocytosis to the cell exterior<span><body><html>

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When A has more columns than rows, then solving a linear equation using the pseudoin v erse provides one of the man y p ossible solutions. Speciﬁcally , it pro vides the solution x = A + y with minimal Euclidean norm ||

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When A has more columns than rows, then solving a linear equation using the pseudoin v erse provides one of the man y p ossible solutions. Speciﬁcally , it pro vides the solution x = A + y with minimal Euclidean norm || || x 2 among all p ossible solutions

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When A has more columns than rows, then solving a linear equation using the pseudoin v erse provides one of the man y p ossible solutions. Speciﬁcally , it pro vides the solution x = A + y with minimal Euclidean norm ||x|| 2 among all p ossible solutions

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the trace op erator provides an alternativ e w a y of writing the F rob enius norm of a matrix: || || A F = T r( AA )

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the trace op erator provides an alternativ e w a y of writing the F rob enius norm of a matrix: || || A F = T r( AA )

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the trace op erator provides an alternativ e w a y of writing the F rob enius norm of a matrix: ||A|| F = Tr( AA T )

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the trace op erator is in v arian t to the transp ose op erator: T r( ) = T r( A A )

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the trace op erator is in v arian t to the transp ose op erator: T r( ) = T r( A A )

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the trace op erator is in v arian t to the transp ose op erator: T r(A) = T r(A T )

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This inv ariance to cyclic p erm utation holds even if the resulting pro duct has a diﬀeren t shap e. F or example, for A ∈ R m n × and B ∈ R n m × , w e ha v e T r(AB ) = T r( BA)

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One simple mac hine learning algorithm, principal components analysis or PCA can b e deriv ed using only knowledge of basic linear algebra

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Lossy compression means storing the p oints in a wa y that requires less memory but ma y lose some precision

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Lossy compression means storing the p oints in a wa y that requires less memory but ma y lose some precision

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T o k eep the enco ding problem easy , PCA constrains the colum ns of D to b e orthogonal to eac h other.

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There are three p ossible sources of uncertain t y: 1. Inheren t stochasticit y in the system b eing mo deled. F or example, most in terpretations of quantum mechanics describ e the dynamics of subatomic particles as b eing probabilistic. W e can also create theoretical scenarios that w e p ostulate to ha v e random dynamics, such as a hypothetical card game where w e assume that the cards are truly sh uﬄed in to a random order. 2. Incomplete observ ability . Ev en deterministic systems can app ear sto chastic when w e cannot observ e all of the v ariables that drive the b ehavior of the system. F or example, i

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of subatomic particles as b eing probabilistic. W e can also create theoretical scenarios that w e p ostulate to ha v e random dynamics, such as a hypothetical card game where w e assume that the cards are truly sh uﬄed in to a random order. <span>2. Incomplete observ ability . Ev en deterministic systems can app ear sto chastic when w e cannot observ e all of the v ariables that drive the b ehavior of the system. F or example, in the Mont y Hall problem, a game sho w con testan t is ask ed to choose b etw een three do ors and wins a prize held b ehind the c hosen do or. T w o do ors lead to a goat while a third leads to a car. The outcome giv en the contestan t’s c hoice is deterministic, but from the con testan t’s p oin t of view, the outcome is uncertain. 3. Incomplete mo deling. When we use a mo del that must discard some of the information we hav e observ ed, the discarded information results in uncertain t y in the mo del’s prediction

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s and wins a prize held b ehind the c hosen do or. T w o do ors lead to a goat while a third leads to a car. The outcome giv en the contestan t’s c hoice is deterministic, but from the con testan t’s p oin t of view, the outcome is uncertain. <span>3. Incomplete mo deling. When we use a mo del that must discard some of the information we hav e observ ed, the discarded information results in uncertain t y in the mo del’s predictions.<span><body><html>

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A random v ariable is a v ariable that can take on diﬀerent v alues randomly

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On its o wn, a random v ariable is just a description of the states that are p ossible; it m ust b e coupled with a probability distribution that sp eciﬁes how likely each of these states are.

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On its o wn, a random v ariable is just a description of the states that are p ossible; it m ust b e coupled with a probability distribution that sp eciﬁes how likely each of these states are.

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T o b e a probability mass function on a random v ariable x , a function P m ust satisfy the follo wing prop erties: • The domain of P must b e the set of all p ossible states of x. • ∀x ∈ x , 0 ≤ P ( x ) ≤ 1 . An imp ossible ev en t has probabilit y and no state can 0 b e less probable than that. Likewise, an ev en t that is guaran teed to happ en has proba

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T o b e a probability mass function on a random v ariable x , a function P m ust satisfy the follo wing prop erties: • The domain of P must b e the set of all p ossible states of x. • ∀x ∈ x , 0 ≤ P ( x ) ≤ 1 . An imp ossible ev en t has probabilit y and no state can 0 b e less probable than that. Likewise, an ev en t that is guaran teed to happ en has probabilit y , and no state can ha v e a greater c hance of o ccurring. 1 • ∑x∈xP(x)=1∑x∈xP(x)=1 . W e refer to this prop erty as b eing normalized . Without this prop ert y , we could obtain probabilities greater than one by computing the probabilit y o

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no state can 0 b e less probable than that. Likewise, an ev en t that is guaran teed to happ en has probabilit y , and no state can ha v e a greater c hance of o ccurring. 1 • ∑x∈xP(x)=1∑x∈xP(x)=1 . W e refer to this prop erty as b eing <span>normalized . Without this prop ert y , we could obtain probabilities greater than one by computing the probabilit y of one of man y ev en ts o ccurring.<span><body><html>

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ewise, an ev en t that is guaran teed to happ en has probabilit y , and no state can ha v e a greater c hance of o ccurring. 1 • ∑x∈xP(x)=1∑x∈xP(x)=1 . W e refer to this prop erty as b eing normalized . Without this prop ert y , we could <span>obtain probabilities greater than one by computing the probabilit y of one of man y ev en ts o ccurring.<span><body><html>

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W e can place a uniform distribution on x —that is, make each of its states equally lik ely—b y setting its probabilit y mass function to P (x = x i ) = 1/k

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The probability distribution o v er the subset is kno wn as the marginal probability distribution.

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The name “marginal probabilit y” comes from the pro cess of computing marginal probabilities on pap er. When the v alues of P ( x y , ) are written in a grid with diﬀeren t v alues of x in rows and diﬀerent v alues of

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The name “marginal probabilit y” comes from the pro cess of computing marginal probabilities on pap er. When the v alues of P ( x y , ) are written in a grid with diﬀeren t v alues of x in rows and diﬀerent v alues of y in columns, it is natural to sum across a row of the grid, then writ

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me “marginal probabilit y” comes from the pro cess of computing marginal probabilities on pap er. When the v alues of P ( x y , ) are written in a grid with diﬀeren t v alues of x in rows and diﬀerent v alues of y in columns, it is natural to <span>sum across a row of the grid, then write P ( x ) in the margin of the pap er just to the righ t of the ro w<span><body><html>

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The exp ectation or exp ected v alue of some function f ( x ) with resp ect to a probabilit y distribution P ( x ) is the a v erage or mean v alue that f tak es on when x is dra wn from . F or discrete v ariables this can be computed with a summation: P E x ∼ P [ ( )] = f x x P x f x , ( ) ( )

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pan>The exp ectation or exp ected v alue of some function f ( x ) with resp ect to a probabilit y distribution P ( x ) is the a v erage or mean v alue that f tak es on when x is dra wn from . F or discrete v ariables this can be computed with <span>a summation: E x~P[f(x)] =∑xP(x)f(x)∑xP(x)f(x)<span><body><html>