The enzyme reaction condition, $E + S \leftrightarrow ES \rightarrow E + P$ uses $\kappa_{1}$ (forward reaction), $\kappa_{-1}$ and $\kappa_{2}$ as the rate constants.   E: Enzyme S: Substrate P: Product  What is the relationship between $\kappa_{1}$, $\kappa_{-1}$ and $\kappa_{2}$ to yield Michaelis's constant:    $\kappa_{m}$ = ($\kappa_{-1}$ + $\kappa_{2}$) / $\kappa_{2}$ ?  I understand that $\kappa_{m}$ = $S$ at $1/2 \times V_{max}$, however I do not understand the relationship of the rate constants based on the enzyme reaction condition.   This question has been marked as a repeat of the question "How does pH affect Michaelis' constant". I am not asking for any affect with pH, just a proof for the mathematical relationship of the factors shown above.
122954
I have been trying to confirm the Km of a substrate (which is 34 +/- 4 mM). This value was obtained in 50 mM MOPS, pH 6.3. I conducted my kinetics assay in a buffer of pH 7 and obtained a Km value in the 21.5. According to this , Fig. 2C, the normalized specific activity of the enzyme is about 70% at pH 6.3 and about 47% at pH 7. If I divide 34 mM at pH 6.3 by 0.7 (which should get me the optimum Km at the optimum pH of 5.5) and then multiply by 0.48, then I get 23 mM. However, the paper says Km is between 30 - 38 mM, so if I divide 30 and 38 separately by 0.7 and multiply by 0.47, I get 20 and 25.6 mM respectively. Because my value falls within this range, this then must mean that I have the right enzyme and the same result as the paper.   So my questions are:   When the paper says Km is "34 +/- 4 mM", can I assume that means the Km can be anywhere between 30 - 38 mM? I'm surprised to see how wide the range is. I assumed Km is usually just one value with a deviation of at most 0.1. Do pH change Km values? I understand that pH changes the shape(s) of the enzyme and/or substrate. Therefore that must affect how much it wants to bind to the substrate. If the enzyme's desire to bind to a substrate decreases due to increase in pH, for example, that would mean more substrates are needed to surround the enzyme, thus increasing Km. If pH does change Km, is this how I determine the Km value of a different pH value if the Km value at another pH is already known? I know that specific activity and Michaelis constant are different, but how much product can be converted per minute depends on how much the enzyme likes to bind to a substrate, which is represented by the Michaelis constant. Did my reasoning and calculation arrive at the right conclusion? If not, how is the calculation done?
105988
According to Sedra/Smith Microelectronic Circuits, \$v_{BE}\$ changes by \$-2\text{mV}/\text{°C}\$. I cannot understand how this could possibly be the case given the equations I am familiar with.  With all currents kept constant, we have:  \$\large{i_E = \frac{I_s}{\alpha}e^{v_{BE}/V_T}}\$  To keep \$i_E\$ constant, any change in \$V_T\$ would have to be accompanied by a change of the same factor in \$v_{BE}\$, otherwise \$\alpha\$ or \$I_s\$ would have to change, which as far as I understand is not possible.  So, how can \$v_{BE}\$ be inversely proportional to \$V_T\$?
75561
I read that $U=q+w$ and $H=u+PV$ so aren't $PV$ and $w$ same? If they are, can't we write $H=q+2w$? Also, $dU=$ change in internal energy wrt change in temperature keeping $V$ constant times $dT+$change in internal energy wrt change in volume keeping $T$ constant times $dV$ Here is the second term $=w$? And, $dH=$ change in enthalpy wrt change in temperature keeping $P$ constant times $dT+$ change in enthalpy wrt change in pressure keeping $T$ constant times $dP$. Here is the second term $=PV$?
111067
Just for clarity, this is the specific formulation I am referring to.    $K$: $\Box(p\to q)\to(\Box p\to\Box q)$  Naturally, we might consider the converse of this statement:    $K'$: $(\Box p\to\Box q)\to\Box(p\to q)$  The question is simple. Given $K$, does $K'$ follow? An accompanying proof of the answer would be greatly appreciated.
116483
I am new to MCMC and reading a  regarding Gibbs sampling. However, there are two parts in the paper I cannot understand and get stuck. The first part is equation 2.3 in page 168. It says that if $X^{'}_{j}$ is drawn from the process of Gibbs sampling, $X^{'}_{j}\sim f(x|Y^{'}_{j}=y^{'}_{j})$. It further claims under &quot;general conditions,&quot; the distribution of $X^{'}_{j}$ converges to $f(x)$ as $n\rightarrow\infty$. Can anyone help explain what these general conditions are and why/how this statement is true. The second part is at the beginning of page 169, expression 2.9 says $\hat{f(x)}=\frac{1}{m}\sum^{m}_{i=1}f(x|y_{i})$, and the theory behind this equation is $E[f(x|Y)]=\int f(x|y)f(y)dy$. This confuses me because my understanding about monte carlo integration is that for example, $E[X]=\frac{1}{m}\sum^{m}_{i=1}X_{i}$ is true due to the law of large number that $\frac{X_{i}}{m}$ converges to $X_{i}\times p(X_{i})$ when $m\rightarrow\infty$. However, I don't see how equation 2.9 can be explained by the concept of monote carlo integration. Can anyone explain the intuition or math behind this equation? Thanks! EDIT 1 (My Answer/Thought for Part 2) I have thought about this question, and since no one responds yet, I will try my answer first and see whether someone has a better explanation. For the second part, instead of using $f(x|y_{i})$, let's focus on the density of a specific value $x^{a}$, which is $f(x^{a}|y_i)$. For simplicity, now assume $y_i$ is randomly drawn from a density $f(y)$ instead of $f(y|x_i)$ but the result could be generalized to $f(y|x_i)$. Both $f(x|y_i)$ and $f(y)$ are know distribution. Now we first randomly draw a $y_i$ from $f(y)$ and once we have $y_i$, we further randomly draw a $x_i$ from $f(x|y_i)$. Suppose we do this process $n$ time, and $n\rightarrow\infty$. Given a specific $y^{a}$ and $x^{a}$, we do know the value of $f(x^{a}|y^{a})$, which is just the density of $x^{a}$ cnditional on $y^{a}$. In other words, by Monte Carlo integration, for a given set of $y_i$'s and $x^{a}$, we can dervive that $\frac{1}{n}\sum^{n}_{i=1}f(x^{a}|y_i)\approx\int f(x^{a}|y_i)f(y_i)dy_i=\int f(x^{a}|y)f(y)dy=\int \dfrac{f(x^{a},y)}{f(y)}f(y)dy=\int f(x^{a},y)dy=f(x^{a})$ We can do the same process to dervie $f(x^{b})$, $f(x^{c})$, and etc. In the end, we will have $f(x^{a})$, $f(x^{b})$, $f(x^{c})$, ..., which all together constrcut $f(x)$. Therefore, $\hat{f(x)}=\frac{1}{m}\sum^{m}_{i=1}f(x|y_{i})$.
33499
We're given a sequence defined by the recursive relation: $$a_n = \sqrt{a_{n-1}a_{n-2}}$$  $a_1$ and $a_2$ are positive constants. We have to show the following:   The sequences $\{ b_n \} = \{ a_{2n-1} \}$ and $\{ c_n \} = \{ a_{2n} \} $ are monotonic, and if one is increasing, the other is decreasing. The limit of the sequence $\{ a_n \} $ is $ \left(a_1a_2^2\right)^{\frac13} $     Now, I have proved the first part. Besides that, I have also proved a few other things:  If $a_1 &gt; a_2$, then:  a. $ \{ b_n \} $ decreases, and $ \{ c_n \} $ increases.  b. $ c_n &lt; b_n $  If $a_1 &lt; a_2$, we just flip $\{ b_n \}$ and $\{c_n\}$  Besides, I have also shown that both the sequences : $\{b_n\}$ and $\{ c_n\}$ have the same limit. What I don't know, is how to evaluate the limit.
107190
I am working on a very large data set, where I am comparing 2 genes. Currently I am trying to test  H0: correlation = 0.64   HA: correlation > 0.64 at the alpha = 0.05 level.   At first I thought I could use cor.test() or cor() with an argument to set the 0.64, but I am not seeing that as an option.  I'm not really sure where to go with this.
108581
I want to know the relationship between the eigensystems of two non-negative-definite (covariance-like) matrices.  Both are derived from X which is a T-by-K real matrix (wlog say K > T).  I avoid subtracting the mean to avoid complicating the main issue which is about the eigensystem structure.  Ct = (1/T) * X' X = Et Lt Et' Ck = (1/K) * X X' = Ek Lk Ek'  .... where Et, Lt are the (column) eigenvectors and diagonal matrix of eigenvalues respectively for Ct, and Ek Lk are the same for Ck.  I remember vaguely from school that there is some kind of duality relationship here (between the two eigensystems) which is particularly interesting because both matrices have rank T &lt; K, and hence only T of the diagonal elements of Lt are nonzero.  Does anyone know this relationship explicitly? Thanks in advance for any light you can shed!
105804
In a question I am doing it says:     Show explicitly that the function $$y(t)=\frac{-gt^2}{2}+\epsilon t(t-1)$$ yields an action that has no first order dependency on $\epsilon$.   Also my textbook says that     [...] if a certain function $x_0(t)$ yields a stationary value of $S$ then any other function very close to $x_0(t)$ (with the same endpoint values) yields essentially the same $S$, up to first order of any deviations.   I am confused about the first order bit? In the first case does it mean that $\frac{\partial S}{\partial \epsilon}=0$ or that it does not depend of $\epsilon$ but may take some other constant value. In the second case does it mean likewise or something different, please explain?
114912
Supose Caspase-1 is allosterically inhibited. Since the inhibitor is not binding in the active site but instead in the allosteric binding site, can I conclude it is a non-competitive inhibitor?
99768
I would like some help with this exercise.  Suppose that $f_1,\ f_2 \in V^*$ and that $\text{Ker} f_1 = \text{Ker} f_2$. Show that $f_1 = k f_2$ for some scalar $k$.  I expect your suggestions.  Thanks!
90672
I am trying to understand the . It considers the Lagrangian   $$L = \frac{1}{2} \left(\dot{Q}^2 - \left(\Omega^2-\Delta \Omega^2\right)Q^2\right) - Q \sum_{i} f_iq_i + \sum_{i}\frac{1}{2} \left(\dot{q}^2 - \omega_i^2q^2\right)$$   where $Q$ is the generalised coordinate of the macro variable (an oscillator with natural frequency $\Omega$), $q_i$ are variables related to an array of harmonic oscillators each with natural frequency $\omega_i$.The first term describes the potentail and kinetic energies related to the macro degree of freedom, the second term describes the coupling using constants $f_i$, the third again describes potential and kinetic energies of the array of oscillators,   $$\Delta \Omega^2 = -\sum_i \left( \frac{f_i}{\omega_i} \right)^2$$   is the ad hoc term my first question relates to. The explanation I found goes      the quantity is inserted to cancel the frequency shift    $$\Omega^2 \to \Omega^2 - \sum_{i} \left(\frac{f_i}{\omega_i}\right)^2$$    [...] the shift arises because a static Q displaces the bath oscillators so that $$f_i q_i = - \left(f_i^2 / \omega_i^2\right)Q$$    Substituing these values for the $f_i q_i$ into the potential terms shows that the effective potentail seen by $Q$ would have a "shifted" frequency.   I regretfully do not get it. I tried to get to the equation   $$f_i q_i = - \left(f_i^2 / \omega_i^2\right)Q$$   by considering the equation of motion in equilibrium, without success. Why would $\Omega$ be affected by the coupling? Also from the intuitive point of view, I fail to see how a "pre-load" would affect the natural frequency of a harmonic oscillator. Any hint would be so appreciated.
38934
Suppose $k$ is a field, and $k(t)$ is the rational function field. If $f(t)=P(t)/Q(t)$ for some polynomials $P(t)$ and $Q(t)\neq 0$, then the map $t\mapsto t+1$ sends $f(t)$ to $f(t+1)$.   So the fixed field for this map the set of rational functions $P(t)/Q(t)$ such that $$ \frac{P(t)}{Q(t)}=\frac{P(t+1)}{Q(t+1)}\text{ or }P(t)Q(t+1)=P(t+1)Q(t). $$ This does not seem awfully descriptive. Is there a more explicit description of what the fixed field looks like, (perhaps in terms of the form of $P$ and $Q$?), in order to get a better handle on it?   Later: With the wonderful help of Professor Suárez-Alvarez, I understand that the $\sigma$-invariant polynomials in $k[t]$ have all factors of form $\phi_\alpha$ where $\phi_\alpha=\prod_{i=1}^{p-1} (t-\alpha-i)$ for $\alpha$ a root of $f$ in $k$ when $k$ is algebraically closed.  What happens when $k$ is not algebraically closed? I think you can still factor out polynomials of form $\phi_\alpha$ for $\alpha$ roots of $f$ in $k$. But if $f$ does not split completely then there will be some $\sigma$-invariant factor with not roots in $k$ in the factorization of $f$. Is there anything more one can say to describe this remaining part? Or is that about the best we can do? Thank you for your time.
68567
\begin{table}     \begin{tabular}{|c|c|c|c|c|}     \hline     \multicolumn{5}{|c|}{\textbf{Table 1:}Unfiltered Water}\\     Concentration of analyte in sample   &amp; Avg &amp; Std Dev &amp;  RSD &amp; 90\% CI \\      \hline     Total hardness (ppm $CaCo_3$ equiv)  &amp; 97  &amp;    2   &amp;  2\%  &amp; \pm 5   \\     Calcium hardness (ppm $Ca^{2+}$)     &amp; 57  &amp;    3   &amp;  5\%  &amp;  \pm 5   \\     Magnesium hardness( ppm $Mg^{2+}$)   &amp; 40  &amp;    3   &amp; 7.5\% &amp;  \pm 5 \\     \hline     \end{tabular}   This table is giving me an error. That says   ! Missing $ inserted. &lt;inserted text&gt; $ l.41 ...Co_3 equiv\) &amp; 97 &amp; 2 &amp; 2\% &amp; \pm 5 \I've inserted a begin-math/end-math symbol since I think   you left one out. Proceed, with fingers crossed.  What is the cause of this error?  I can't seem to find any thing that is mismatched.
61903
Combining the first and second law of thermodynamics we can get the following equation  $$TdS=dU-P_{ext}dV$$  First question: Is this equation applicable for irreversible processes such that that $dS≠\dfrac{dQ}{T}$?  Second question:If the system temperature $T_{sys}$ is smaller than the surrounding temperature $T_{sur}$, which temperature should we put in the equation?   I have this question because sometimes people use $T_{sur}$ instead of $T_{sys}$ (e.g. , ) but the equation is supposed to describe changes in the system.
27333
In a scientific paper, I've seen the following $$\frac{\delta K^{-1}}{\delta p} = -K^{-1}\frac{\delta K}{\delta p}K^{-1}$$ where $K$ is a $n \times n$ matrix that depends on $p$. In my calculations I would have done the following $$\frac{\delta K^{-1}}{\delta p} = -K^{-2}\frac{\delta K}{\delta p}=-K^{-T}K^{-1}\frac{\delta K}{\delta p}$$ Is my calculation wrong? Note: I think $K$ is symmetric.
88883
Consider a  $\phi(x)$, and let its  be $$ \frac{1}{p^2-m^2-\Pi(p^2)}=\int_0^\infty\mathrm d\mu^2\ \rho(\mu^2)\ \frac{1}{p^2-\mu^2} $$  We usually have $\Pi(m^2)=0$, which implies that there is a one-particle state with mass $m^2$. In terms of the , this reads $$ \rho(\mu^2)=\delta(\mu^2-m^2)+\sigma(\mu^2) $$ where the support of $\sigma(\mu^2)$ is disconnected from $m^2$.  On the other hand, the continuum contribution usually begins at $(2m)^2$, meaning that $\Pi(k^2)$ has a branch cut from $(2m)^2$ to $\infty$. Again, in terms of the spectral function, this is written as $\text{supp}(\sigma)=[(2m)^2,\infty)$.  My question  I would expect that the first bound state to lie close to $(2m)^2$, but slightly below, something around $(2m)^2-\mathcal O(\alpha^n)$, where $\alpha$ is the coupling constant and $n\in\mathbb N$. For example, in a simplified QED model, the bound state of an  has binding energy $\sim 7\mathrm{eV}\sim m\alpha^2$, and so I would expect the branch point to be somewhere around $(2m)^2-m^2\alpha^4$.  I know the value of the self-energy to one loop, and the branch point is indeed exactly at $(2m)^2$. I don't know where to find higher loop corrections, so I cannot conclude whether   1) at higher loops, the branch point moves a bit closer to $m^2$, or 2) it stays at $(2m)^2$.   I would expect that the correct behaviour is 1), but this contradicts the fact that all books I've read always depict the threshold of pair production as an hyperboloid with its bottom sitting exactly at $(2m)^2$. On the other hand, if it turns out that the correct option is 2), I am lead to ask why is perturbation theory unable to correctly predict the position of the branch point (I know this is a subtle issue because bound states are, to some extent, non-perturbative entities; but AFAIK, perturbative calculations do contain a lot of information about bound states).