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Mechanisms of Signal Transduction| Volume 284, ISSUE 8, P5277-5288, February 20, 2009

# pH Dependence of the Photoactive Yellow Protein Photocycle Recovery Reaction Reveals a New Late Photocycle Intermediate with a Deprotonated Chromophore*

• Author Footnotes
* The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
The on-line version of this article (available at http://www.jbc.org) contains supplemental Appendices I-IV, Figs. S1-S14, and Tables S1-S5.
Open AccessPublished:December 17, 2008
The recovery reaction of the signaling state of photoactive yellow protein includes the following: (i) deprotonation of the p-coumaryl chromophore, (ii) refolding of the protein, and (iii) chromophore re-isomerization from the cis to the trans configuration. Through analysis of the pH dependence of this recovery reaction, we were able to provide proof for the existence of an additional photocycle intermediate. The spectral similarity between this new intermediate and the dark state indicates that the new intermediate has a deprotonated chromophore, which may facilitate chromophore re-isomerization. This spectral similarity also explains why this new intermediate has not been noticed in earlier studies. For our data analysis we introduce a photocycle model that takes into account the effect of the specific light regime selected, a model that was also used for simulations.
The photoactive yellow protein (PYP)
The abbreviations used are: PYP, photoactive yellow protein; SVD, singular value decomposition.
2The abbreviations used are: PYP, photoactive yellow protein; SVD, singular value decomposition.
from Halorhodospira halophila is a small, water-soluble photoreceptor protein. It makes use of a relatively simple chromophore, p-coumaric acid, which is bound to Cys-69 via a thiol ester bond (
• Baca M.
• Borgstahl G.E.
• Boissinot M.
• Burke P.M.
• Williams D.R.
• Slater K.A.
• Getzoff E.D.
,
• Hoff W.D.
• Dux P.
• Hard K.
• Devreese B.
• Nugteren-Roodzant I.M.
• Crielaard W.
• Boelens R.
• Kaptein R.
• van Beeumen J.
• Hellingwerf K.J.
). Favorable in vitro characteristics have made this protein a popular model system for both experimental and computational studies of enzyme/protein structure and functioning (
• Hellingwerf K.J.
• Hendriks J.
• Gensch T.
). High resolution structures are available, not only of the PYP ground state (
• Borgstahl G.E.
• Williams D.R.
• Getzoff E.D.
,
• Düx P.
• Rubinstenn G.
• Vuister G.W.
• Boelens R.
• Mulder F.A.
• Hard K.
• Hoff W.D.
• Kroon A.R.
• Crielaard W.
• Hellingwerf K.J.
• Kaptein R.
,
• Getzoff E.D.
• Gutwin K.N.
• Genick U.K.
) but also of several of its photocycle intermediates (
• Ihee H.
• Rajagopal S.
• Srajer V.
• Pahl R.
• Anderson S.
• Schmidt M.
• Schotte F.
• Anfinrud P.A.
• Wulff M.
• Moffat K.
,
• Schmidt V.
• Pahl R.
• Srajer V.
• Anderson S.
• Ren Z.
• Ihee H.
• Rajagopal S.
• Moffat K.
) and mutants (
• Anderson S.
• Srajer V.
• Pahl R.
• Rajagopal S.
• Schotte F.
• Anfinrud P.
• Wulff M.
• Moffat K.
,
• Shimizu N.
• Kamikubo H.
• Yamazaki Y.
• Imamoto Y.
• Kataoka M.
). In fact, at least 46 PYP structures have already been deposited in the Protein Data Bank. The PYP photocycle has been intensively investigated in all relevant time domains (
• Genick U.K.
• Devanathan S.
• Meyer T.E.
• Canestrelli I.L.
• Williams E.
• Cusanovich M.A.
• Tollin G.
• Getzoff E.D.
,
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
,
• van Wilderen L.
• Van der Horst M.A.
• van Stokkum I.H.M.
• Hellingwerf K.J.
• van Grondelle R.
• Groot M.L.
). These studies have led to many different, and often conflicting, models for this photocycle, with a widely divergent nomenclature for the transient intermediates involved. Nevertheless, a consensus about the main intermediate states is emerging, the essence of which is summarized in Fig. 1. The ground state of PYP, pG (also named P and PYP), in which the chromophore is deprotonated and has a trans configuration, can absorb a blue photon. From the resulting excited state, a very short living ground state intermediate is formed, which relaxes directly back to pG (
• van Wilderen L.
• Van der Horst M.A.
• van Stokkum I.H.M.
• Hellingwerf K.J.
• van Grondelle R.
• Groot M.L.
). The excited state further results in the formation of the ultra-fast intermediate, I0, in which the chromophore has undergone trans to cis isomerization. Next, the pR intermediate (also named I1, and PYPL) is formed on a nanosecond time scale. Transient grating and time-resolved optical rotatory dispersion measurements revealed that further structural relaxation occurs in pR (
• Takeshita K.
• Imamoto Y.
• Kataoka M.
• Mihara K.
• Tokunaga F.
• Terazima M.
,
• Takeshita K.
• Imamoto Y.
• Kataoka M.
• Tokunaga F.
• Terazima M.
,
• Chen E.F.
• Gensch T.
• Gross A.B.
• Hendriks J.
• Hellingwerf K.J.
• Kliger D.S.
), which is accompanied by a subtle change in the absorption spectrum (
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
,
• Chen E.F.
• Gensch T.
• Gross A.B.
• Hendriks J.
• Hellingwerf K.J.
• Kliger D.S.
). Accordingly, pR has been split up into pR1 and pR2 in Fig. 1. Next, the chromophore is protonated, thereby forming the intermediate pB′ (
• Xie A.
• Kelemen L.
• Hendriks J.
• White B.J.
• Hellingwerf K.J.
• Hoff W.D.
) (also named I2), which is in equilibrium with pR (
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
). From pB′, pB (also named I2, I2′, and PYPM) is formed in which the protein structure accommodates the new protonation state of the chromophore. The exact dynamics of the transition between pB′ and pB is still under debate. Both an equilibrium and a unidirectional reaction between pB′ and pB have been proposed. We favor the interpretation of a unidirectional reaction. An equilibrium would result in the presence of pB′ throughout the recovery reaction, which is not supported by this study. Furthermore, the pB intermediate has been shown to have absorption characteristics that depend on conditions such as pH, to the extent that it can be highly similar to pB′. Based on available infrared difference spectra of both pB′ (
• Xie A.
• Kelemen L.
• Hendriks J.
• White B.J.
• Hellingwerf K.J.
• Hoff W.D.
) and a pB intermediate with UV/visible absorption characteristics similar to pB′ (
• Shimizu N.
• Imamoto Y.
• Harigai M.
• Kamikubo H.
• Yamazaki Y.
• Kataoka M.
), we conclude that pB′ and this pB are distinct intermediates. In this study, we will assume a unidirectional reaction between pB′ and pB. In the next step of the photocycle, pB recovers to the ground state pG. In this recovery reaction several alterations in the configuration of PYP and its coumaryl chromophore have to be accomplished as follows: (i) deprotonation of the chromophore; (ii) chromophore re-isomerization from the cis to trans configuration, and (iii) refolding of the polypeptide chain to the pG configuration. Here deprotonation of the chromophore may facilitate chromophore re-isomerization (
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
,
• Sergi A.
• Gruning M.
• Ferrario M.
• Buda F.
).
In this study we focus on the recovery reaction of the PYP ground state (pG) from the putative signaling state pB. Measurement of PYP recovery kinetics has become a standard tool to determine the effect of site-directed mutagenesis (
• Genick U.K.
• Devanathan S.
• Meyer T.E.
• Canestrelli I.L.
• Williams E.
• Cusanovich M.A.
• Tollin G.
• Getzoff E.D.
,
• Kumauchi M.
• Sasaki J.
• Tokunaga F.
) and/or of changes in its mesoscopic context (
• Xie A.
• Kelemen L.
• Hendriks J.
• White B.J.
• Hellingwerf K.J.
• Hoff W.D.
,
• Shimizu N.
• Kamikubo H.
• Mihara K.
• Imamoto Y.
• Kataoka M.
) on protein function. However, the way in which this information has been obtained in various studies is far from standardized. Besides the obvious differences in used equipment, also the applied analysis techniques vary significantly, ranging from basic to highly complex analyses.
The fact that several different models have been proposed to describe the PYP photocycle (
• Ihee H.
• Rajagopal S.
• Srajer V.
• Pahl R.
• Anderson S.
• Schmidt M.
• Schotte F.
• Anfinrud P.A.
• Wulff M.
• Moffat K.
,
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
,
• Borucki B.
• Joshi C.P.
• Otto H.
• Cusanovich M.A.
• Heyn M.P.
) indicates that analysis of kinetic data of a photocycle is far from straightforward. The situation is further complicated by the influence of the selected measurement conditions. For instance, ionic strength (
• Hoersch D.
• Otto H.
• Joshi C.P.
• Borucki B.
• Cusanovich M.A.
• Heyn M.P.
), temperature (
• van Brederode M.E.
• Hoff W.D.
• Van Stokkum I.H.
• Groot M.L.
• Hellingwerf K.J.
), pH (
• Genick U.K.
• Devanathan S.
• Meyer T.E.
• Canestrelli I.L.
• Williams E.
• Cusanovich M.A.
• Tollin G.
• Getzoff E.D.
,
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
), and buffer composition (
• Shimizu N.
• Kamikubo H.
• Mihara K.
• Imamoto Y.
• Kataoka M.
) all have an influence on the recovery reaction of PYP. Although this complicates the comparison of data, it also supplies one with a tool to further understand this recovery reaction through systematically changing measurement conditions.
One of the interesting aspects of PYP is its complex pH dependence, which appears in many of its properties. Photocycle kinetics is a clear example of this, where it seems each step in the photocycle has its own unique pH dependence (
• Genick U.K.
• Devanathan S.
• Meyer T.E.
• Canestrelli I.L.
• Williams E.
• Cusanovich M.A.
• Tollin G.
• Getzoff E.D.
,
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
). Significantly, in many cases we are actually dealing with a pOH dependence (
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
), implying a role for the hydroxide ion. Another important characteristic of PYP is that, upon formation of the putative signaling state pB, the protein undergoes a large structural change (
• Xie A.
• Kelemen L.
• Hendriks J.
• White B.J.
• Hellingwerf K.J.
• Hoff W.D.
,
• van Brederode M.E.
• Hoff W.D.
• Van Stokkum I.H.
• Groot M.L.
• Hellingwerf K.J.
). This characteristic, however, is also pH-dependent, such that at lower pH the extent of structural change decreases (
• Shimizu N.
• Imamoto Y.
• Harigai M.
• Kamikubo H.
• Yamazaki Y.
• Kataoka M.
). The two different structural forms of pB have different absorption spectra. The more compactly folded form has an absorption maximum around 370 nm compared with 355 nm for the more unfolded form. At high pH a third form of pB can exist in which the chromophore is deprotonated and which results in a shift of the absorption maximum to ∼430 nm (
• Hendriks J.
• Hoff W.D.
• Crielaard W.
• Hellingwerf K.J.
). From the above it is clear that the pB intermediate can exist in multiple forms.
Here we report on the pH dependence of pG recovery, applying a new approach to analyze this reaction. In doing so we were able to positively identify the actual photocycle intermediate that catalyzes recovery to the ground state. Furthermore, we show the advantages and limitations of the use of photocycle models that incorporate the characteristics of the illumination conditions.

## EXPERIMENTAL PROCEDURES

Materials—Recombinant wild type and H108F apoPYP were produced heterologously in Escherichia coli, as described previously (
• Kort R.
• Hoff W.D.
• Van West M.
• Kroon A.R.
• Hoffer S.M.
• Vlieg K.H.
• Crielaand W.
• Van Beeumen J.J.
• Hellingwerf K.J.
). The apoPYP was reconstituted with a derivative of p-coumaric acid as described previously (
• Hendriks J.
• Gensch T.
• Hviid L.
• van Der Horst M.A.
• Hellingwerf K.J.
• van Thor J.J.
). The genetically introduced hexa-histidine containing the N-terminal tag was not removed for these experiments. The purity index (A278/A446) of the samples was <0.5.
Sample/Buffer Preparation—Buffer solutions with defined ionic strength were prepared by mixing 1 m acetic acid, 0.2 m Na2HPO4, 1 m Trizma® base, 0.5 m boric acid, 4 m NaCl, 1 m HCl, and 1 m NaOH. The required amount of these stock solutions was calculated (see supplemental material for details). Minor pH adjustments were made with either 1 m HCl or NaOH as necessary.
Transient ms/s UV/Visible Spectroscopy—UV/visible spectra were recorded using an HP8453 UV/visible spectrophotometer. Time-resolved spectra were recorded from 250 to 600 nm, using an integration time of 0.1 s. A “Kraayenhof” cuvette (
• Kraayenhof R.
• Schuurmans J.J.
• Valkier L.J.
• Veen J.P.C.
• Van Marum D.
• Jasper C.G.G.
) was used to thermostat the sample at 25 °C and allowed measurement of the sample pH in the measurement setup. A Schott KL1500 LCD lamp was used to illuminate the sample. Actual sample illumination was computer-controlled via a shutter system, allowing synchronization of sample illumination with the spectroscopic measurement. A 2-s delay was used between the start of the measurement and the 5-s illumination period of the sample.
Light Titration of H108F PYP—Transient ms/s UV/visible spectroscopy was performed on a sample containing H108F PYP in 20 mm Tris/HCl with NaCl of pH 8 and an ionic strength of 0.25 m. The sample was illuminated with varying light intensities by changing settings on the Schott KL1500 LCD lamp. To minimize the effect of probe light, a 17.3% glass gray filter was placed between the probe lamp and the sample. The sample was illuminated until a light-induced steady state was reached. For the majority of the tested light intensities, the computer-controlled shutter system was used for sample illumination. However, to reach the higher light intensities, the shutter system was removed, and illumination was regulated by manually inserting/removing the light guide in/from the light source. See under “Lamp Characterization” for a description of how light intensities were determined.
pH Titration of PYP Ground State—The same spectrophotometer setup was used as described above. A buffer containing 5 mm each of acetic acid, phosphate, Trizma® base, and boric acid with an ionic strength of ∼0.25 m was used. To a PYP solution of pH ∼ 7.3 small aliquots of either NaOH or HCl was added repeatedly. After each addition, the pH and spectrum of the sample were recorded. Especially at extreme pH values, measurements were performed as quickly as possible to minimize the effect of chromophore dissociation. Also, sample illumination was kept at a minimum to minimize light-induced effects at the pH extremes.
Lamp Characterization—Light intensities were measured with a LI-COR LI-250 light meter in combination with a LI-COR Quantum sensor, which measures light intensity from 400 to 700 nm. The lamp spectrum of the HP8453 UV/visible spectrophotometer was measured using the diagnostics mode of the spectrometer. The lamp spectrum of the Schott KL1500 LCD lamp was measured with an Ocean Optics USB4000 miniature fiber optic spectrometer. Light intensities for wavelength ranges other than 400-700 nm were extrapolated based on the measured lamp spectra.
Data Analysis—Analysis of two-dimensional datasets, such as the pH dependence of rate constants, was performed in OriginPro 8 (OriginLab Corp.). Three-dimensional datasets, such as time-resolved spectra and pH-dependent spectra, were analyzed in MatLab (The MathWorks™), using techniques described elsewhere (
• Hendler R.W.
• Shrager R.I.
).
Prior to analysis of time-resolved spectra, each spectrum in the series was corrected by subtracting the average absorbance in the wavelength range from 550 to 600 nm from the entire spectrum. The wavelength range from 250 to 550 nm was subsequently used in the analysis. In the time domain the data were generally analyzed from 6.5 s upward. This allowed for inclusion of a small part of the light-induced steady state in the fit. Equation 1 describes the model used to analyze the data.
Here fpB and fpG are the calculated fractions of pB and pG, respectively; fpB,a is the assumed steady state fraction of pB; k1 and k2 are rate constants; fk1 equals the fraction that the exponent with rate constant k1 contributes; t0 is the time at which the excitation light is turned off. Essentially, the model describes a bi-exponential recovery from pB to pG. As the value for fpB,a has no influence on the quality of the fit nor on the obtained rate constants (see “Results”), this parameter was set to 0.5 for all analyses.
From the above analysis a pG spectrum and a pG/pB mixed spectrum was obtained. The mixed spectrum (Smix) is related to the spectra for pG and pB (SpG and SpB) according to Equation 2.
$Smix = (fpB,a - fpB,ss)fpB,a·SpG + fpB,ssfpB,a·SpB$
(Eq. 1)

Here, fpB,ss is the actual steady state fraction of pB, and fpB,a is the assumed steady state fraction of pB used to obtain Smix (see Equation 1).
An initial estimation for fpB,ss was made, based on a simple photocycle model with a single-step recovery from pB to pG, using k1 as recovery rate. Although this is not the optimal model to describe the data, it is able to provide a reasonable estimation of the steady state fractions of pB and pG (see under “Results”). In this way, the recovery rate is linked to the steady state fractions as described by Equation 3,
$fpB,ss = Qy·I0·∊·In(10)Qy·I0·∊·In(10) + kre$
(Eq. 2)

Here fpB,ss is the steady state fraction of pB, and kre is the photocycle recovery rate. Qy, I0, and ∊ are constants describing the photocycle quantum yield (0.35) (
• van Brederode M.E.
• Gensch T.
• Hoff W.D.
• Hellingwerf K.J.
• Braslavsky S.E.
), light intensity for the wavelength range of 300 to 550 nm (2.154·10-4 mmol·cm-2·s-1 = 2154 μmol·m-2·s-1), and extinction coefficient for the wavelength range 300-550 nm (5179 m-1·cm-1), respectively, as also described elsewhere (
• Hendriks J.
• Avila-Perez M.
• Bruggeman F.
• Hellingwerf K.J.
). For a detailed derivation, see supplemental material. Note that the extinction coefficient is specific for the lamp and wavelength range used, as it was calculated from the lamp spectrum in combination with the absorption spectrum of the PYP ground state at pH 8, using an ∊446 of 45,500 m-1·cm-1.
For the analysis of the pH dependence of a rate constant, with a single pH transition, use was made of the Henderson-Hasselbalch equation in the form of Equation 4. Here klow pH and khigh pH are the values of k at low and high pH, respectively.
$k = khigh pH - klow pH1 + 10n(pk - pH) + klow pH$
(Eq. 3)

For the analysis of multiple pH transitions within a single molecule Equation 5 was used,
Here fi is the fraction of species i, where the sum of all fractions is 1; nt is the number of pH transitions; i, j are species number indicators; k, l are pH transition number indicators; pK is the pK of a pH transition, where transition 1 has the lowest pK, and the pK increases with increasing transition number; n is a cooperativity constant for the pH transition. See supplemental material for buffer calculations for an example on how to use Equation 5.

## RESULTS

In this study we have focused on the photocycle recovery reaction of PYP, i.e. the dark transition from pB to pG. For this we have recorded spectra from 250 to 600 nm with a time resolution of 100 ms. The broad wavelength range allows us to include the absorption changes of the absorption peak around 280 nm, a part of the spectrum that is generally not included in the analysis of PYP recovery. Because of the rather low time resolution, we ignore formation of pB and focus only on the PYP recovery reaction (see “Discussion”). In most of the recorded time trajectories the sample was illuminated for 5 s with actinic light, 2 s after the start of the measurement. This is enough time to reach a light-induced steady state in the sample.
The rate of photocycle recovery is strongly dependent on environmental conditions such as pH (
• Genick U.K.
• Devanathan S.
• Meyer T.E.
• Canestrelli I.L.
• Williams E.
• Cusanovich M.A.
• Tollin G.
• Getzoff E.D.
,
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
), temperature (
• van Brederode M.E.
• Hoff W.D.
• Van Stokkum I.H.
• Groot M.L.
• Hellingwerf K.J.
), ionic strength (
• Hoersch D.
• Otto H.
• Joshi C.P.
• Borucki B.
• Cusanovich M.A.
• Heyn M.P.
), and buffer composition (
• Shimizu N.
• Kamikubo H.
• Mihara K.
• Imamoto Y.
• Kataoka M.
). Here we have opted to vary the pH over a wide range (pH 4.5 to 10.5), as we expect that this will provide us with the most relevant mechanistic information. The other environmental conditions were kept constant as much as possible. All our measurements were therefore performed at 25 °C (±0.1), with an ionic strength of 0.25 m (adjusted with NaCl), and a buffer mixture that contained 20 mm acetic acid, phosphate, Trizma® base, and boric acid.
The data obtained from our measurements was analyzed in three stages. Note that our analysis was performed on absorption spectra and not absorption difference spectra. First, the data were explored to determine the number of kinetically and spectrally distinguishable states that can be reasonably extracted from the data. Next, for each measured pH value, a kinetic analysis was performed. From this analysis also spectral information was extracted. The results of the analyses at the individual pH values were subsequently analyzed to extract pH dependences. To support the interpretation of these measurements, a light titration of the H108F mutant of PYP was performed to demonstrate that Equation 3 can be used for the analysis in stage 2. Also a pH titration of the dark state was performed, which is necessary for normalization of the data in stage 2 of the analysis.
Analysis Stage 1, Data Exploration—To determine the number of kinetically and spectrally distinguishable states, the data were explored using singular value decomposition (SVD) (
• Hendler R.W.
• Shrager R.I.
). With SVD a matrix containing the time-resolved spectra (A) is deconvoluted into three separate matrices such that A = U·S·sVT (the superscript T indicates the matrix is transposed). Here the U matrix contains columns representing spectral contributions to the A matrix. The V matrix contains columns representing temporal contributions to the A matrix. The S matrix is a diagonal matrix containing scaling factors that indicate the significance of each column in the U and V matrices (columns are sorted with respect to their significance). In Fig. 2 examples of two SVD explorations are shown. Above pH 7, the third component of both the U and V matrix consists mostly of noise (see Fig. 2, B and D, for an example at pH 9.0), which indicates there are only two kinetically and spectrally distinguishable states, i.e. the dark and a light-induced state. Below pH 7, the third component does contain significant structure (see Fig. 2, A and C, for an example at pH 5.5), and the fourth component of both the U and V matrix consists mostly of noise (data not shown). This indicates that there are three kinetically and spectrally distinguishable states below pH 7. To determine the significance of this third state, we took a closer look at the S matrix. For pH 5.5 and 9.0, the logarithm of the scaling values of the first nine ranks from the S matrix are shown in Fig. 2, E and G. It is clear that the scaling value for increasing ranks drops quickly and then stabilizes. To be able to compare the scaling values from the different data sets, the scaling values in each S matrix were normalized by the sum of all scaling values in that S matrix, i.e. effectively expressing the scaling values as fractions. It then becomes clear that the first component of the SVD contributes around 94% to the dataset for pH values above 6.0, and it gradually drops to 86% below this pH value (see Fig. 2F). The third component of the SVD only contributes around 0.04% for pH values above 7.0 and gradually increases to 0.4% below this pH. As such, it is reasonable to assume that this third component can be neglected, which then allows the assumption that only two kinetically/spectrally distinguishable states (i.e. the dark and a light-induced state) are relevant for the entire measured pH range. This is further corroborated by comparing the original data with data reconstructed from the U, V, and S matrices, using either the first 2 or 3 ranks (see supplemental Fig. S1 through S3). When 3 ranks are used, the difference with the original data results in unstructured noise with a maximum level of 1 mOD units at all measured pH values. Therefore, for subsequent kinetic analyses, datasets reconstructed using the first 3 ranks were used. Note, when only 2 ranks are used, similar unstructured noise of the same level is obtained above pH 7. Below pH 7 some structure is observed in the noise, the maximum deviation of which gradually increases going to lower pH values. However, deviations are still lower than 3 mOD units at pH 5, whereas noise levels remain at most 1 mOD units (data not shown).
Analysis Stage 2, Kinetic Analysis—For the kinetic analysis we make use of the fact that the A matrix, which represents the data, can be split into two matrices such that A = D·FT. Here the matrix D contains the spectrum of each relevant species, and the matrix F contains the temporal profile of the contribution of each of the species in matrix D. As such, by modeling the F matrix, we can obtain the spectrum of each species via D = A·FT+, where FT+ is the pseudo-inverse of FT, and therefore we can also determine a simulated A matrix (
• Hendler R.W.
• Shrager R.I.
). In short, we fit a temporal profile to our data, from which we obtain both kinetic and spectral information.
From our SVD exploration, we concluded a two-state model should sufficiently describe our kinetic data. In our experiment the recovery can be seen as the transition from a light-induced steady state mixture of pG and pB back to a dark state with only pG present. The simplest way to model the F matrix in this two-state situation is by having the two-states be interconverted via first-order kinetics (i.e. using exponentials). Here the decay of the fraction of pB is mirrored by the growth in the fraction of pG. To obtain a good fit of the data (i.e. minimal structure in the residuals) two exponentials are required (see Equation 1), as was also shown previously (
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
). The results from this analysis are shown in Fig. 3. For the majority of the measured pH range, the faster of the two exponentials contributes more than 95% to the data. Only at the extremes of pH (pH < 5 and >10), where the two rate constants become more similar, does the contribution of the faster rate constant drop below 95%. Furthermore, the pH dependence of the fast and slow component appears to be slightly different (see Fig. 3). Most notably, the pH at which the highest rate is achieved is slightly shifted toward the lower pH for the slow component. Note that varying the steady state fraction of pB in Equation 1 does not have an effect on the obtained rate constants nor on the quality of the fit. However, there is a large effect on the obtained spectrum for pB. This is illustrated in Fig. 4A, in which the steady state fraction of pB was varied from 0.1 to 1.0 for the dataset at pH 7. It is clear that the resulting spectra are composed of various mixtures of the two-states, pG and pB. The composition of these mixed spectra is described by Equation 2, from which can be deduced that under- or overestimation of the fraction pB leads to either a negative or positive contribution of the pG spectrum.
It is not possible to determine the actual steady state fraction of pB (fpB,ss) without making assumptions. Matters are further complicated by the fact that at high pH a pB species with an absorption maximum around 430 nm appears (
• Hendriks J.
• Hoff W.D.
• Crielaard W.
• Hellingwerf K.J.
), making it very difficult to visually correct for the contribution by pG. But even at neutral pH values there appears to be a small contribution from a species that absorbs around 450 nm (see Fig. 4B). A similar contribution was observed previously (
• Joshi C.P.
• Borucki B.
• Otto H.
• Meyer T.E.
• Cusanovich M.A.
• Heyn M.P.
), but was ignored because of experimental error. This species cannot be explained either by pG or by the high pH pB species. One of the assumptions that is generally made to obtain the pB spectrum is that pB does not absorb above ∼420 nm (
• Joshi C.P.
• Borucki B.
• Otto H.
• Meyer T.E.
• Cusanovich M.A.
• Heyn M.P.
,
• Hendriks J.
• van Stokkum I.H.
• Crielaard W.
• Hellingwerf K.J.
). Fig. 4B indicates that this assumption does not hold. Furthermore, this assumption is not applicable at alkaline pH values. Therefore, we adopted another approach. To obtain the true pB spectra at the various pH values from the kinetic analysis described above, we have used Equation 2 to subtract the contribution of pG from the mixed spectrum obtained in the analysis. A simple photocycle model was used to estimate fpB,ss (see Equation 3), where the rate constant that contributes most to pG recovery was used for kre. The value for fpB,ss was then fine-tuned visually based on spectral shape (e.g. no negative absorption allowed), shape of the first four ranks of an SVD analysis on the pH dependence of the pB spectra (e.g. curve smoothness), pH dependence of the fpB,ss parameter, and analysis of the pH dependence of the resulting pB spectra (described below). The set of pB spectra obtained after this optimization is shown in Fig. 5A.
Light Titration of H108F PYP—To make sure Equation 3 is able to estimate the values for fpB,ss accurately enough, a light titration was performed on a PYP sample, i.e. with the H108F mutant of PYP, as the recovery rate of this mutant protein at pH 8 is slow enough to allow for a nearly complete titration, from 0 to 100% bleach of the sample, with the light source available to us. The recovery rate at each measured light intensity was determined as described above. Again two exponentials were required to fit the data. The major recovery component, which contributes 94% and was the same in all measurements (as expected), was found to be 0.35 s-1. In Fig. 6, the fraction of the absorption at 446 nm that is left after reaching the light-induced steady state is plotted as a function of light intensity. The fitted line was made by scaling Equation 3, with all parameters fixed to measured values (Qy = 0.35; kpe = 0.35 s-1; ∊ dependent on lamp setting). Except for the four highest light intensities in the plot, all data points were obtained using the same shutter setup to illuminate the sample, as the one that was also used for the pH titration. To obtain the four highest light intensities, the shutter had to be removed. At these four light intensities the sample temperature increased significantly, which may explain their sub-optimal fit to the curve. Nonetheless, it is clear that Equation 3 is able to provide a reasonable estimate of fpB,ss, especially under the conditions used for the pH titration.
Analysis Stage 3, pH Dependence of the pB Spectrum—From the pH dependence of the pB spectra we were able to confidently extract two pH transitions, resulting in three distinct spectral species for pB. These are shown in Fig. 5, B and C. Note that the species dominant at neutral pH appear to be a combination of two separate species, as indicated by its bi-lobed absorption spectrum. These two species apparently cannot be separated based on pH, at least not in our experimental approach. As noted above, the species with an absorption maximum around 450 nm cannot be explained either by pG or by high pH pB.
pH Titration of pG—Note that small variations in the amount of protein in the samples (caused by pipetting errors) made it necessary to normalize the spectra obtained from the kinetic analysis above. We selected the absorption of pG at 446 nm to normalize the spectra that were used to determine the true pB spectra. As the spectrum of pG also varies with pH, the pH dependence of the UV/visible absorption spectrum of the dark state of PYP (i.e. pG) was determined for the pH range from pH 0.7 to 13.2. A more thorough analysis of this dataset, using the complete spectra, showed that seven pH transitions were required to describe all pH-induced spectral transitions for this pH range (see Fig. 7). At low pH, transitions with a pKa of 2.0, 3.1, and 4.1 were found. The corresponding spectra in Fig. 7A reveal that the transition with a pKa of 4.1 is accompanied by a minor decrease in absorption around 446 nm and a slight increase around 350 nm. The transition with a pKa of 3.1 results inapBdark species that significantly absorbs around 450 nm. This latter contribution is absent in the most acidic species, which is formed with a pKa of 2.0. At high pH, transitions with apKa of 10.0, 11.5, and 12.2 were found, as well as an apparent pKa of 13.0 for chromophore dissociation (because of the hydrolysis of the thiol ester linkage (
• Hoff W.D.
• Devreese B.
• Fokkens R.
• Nugteren-Roodzant I.M.
• Van Beeumen J.
• Nibbering N.
• Hellingwerf K.J.
)). Note that although we could fit the transition associated with chromophore dissociation as a pH transition, it actually is the result of hydrolysis of the thiol ester bond, which occurs with a rate that is pH-dependent. The corresponding spectra in Fig. 7C reveal that the transition with a pKa of 10.0 is accompanied by a minor change in the shape of the peak at 446 nm and a broadening of the peak around 280 nm that is typical for deprotonated tyrosine residues (
• Hoff W.D.
• Devreese B.
• Fokkens R.
• Nugteren-Roodzant I.M.
• Van Beeumen J.
• Nibbering N.
• Hellingwerf K.J.
,
• Holiday E.R.
). The transition with a pKa of 11.5 results in a blue shift of the main absorption peak to 427 nm. The transition with a pKa of 12.2 results in an additional blue shift of the main absorption peak to 409 nm. Note that for the pH range that we used for the kinetic analysis (pH 4.5-10.5), only three of these seven PYP ground state species contribute significantly (see Fig. 7B). Furthermore, the transition with a pKa of 4.1 appears to coincide with the appearance of the third component in the SVD exploration below pH 7 (data not shown), indicating that these two characteristics are possibly related. Also note that the pH dependences in Fig. 3 and Fig. 5B do not coincide with the pH dependence observed in the SVD exploration.

## DISCUSSION

The photocycle of PYP has been studied extensively, especially with respect to signaling state formation. This has led to detailed models of the events that lead to the formation of the pB intermediate, which involves chromophore isomerization, proton transfer from the chromophore to Glu-46, and conformational transitions of the protein. However, with respect to the dark recovery reaction of the ground state, much less is known. It has been suggested that this reaction occurs via an intermediate with a specific fold, and an absorption spectrum that is similar to that of the ground state (
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
), but no solid proof could be provided at that point. Here, we provide this proof by applying an improved data analysis method on an extensive kinetic dataset that covers both a large pH (4.5-10.5) and wavelength (250-600 nm) range.
Photocycle Models—Photocycle models are often used to analyze light-induced events in PYP, but rarely do these models explicitly take into account the illumination conditions of the experiment. For this study we have derived a photocycle model that does take this aspect into account (see supplemental material). Initial analyses of the data using a basic model with a one-step recovery step from pB to pG resulted in fits comparable in quality to mono-exponential analyses of the recovery reaction. This is not surprising as mono-exponential recovery is inherent to the basic model. However, the major advantage of using the model is that not only recovery from the light-induced steady state can be fitted but also the formation of the steady state pB/pG mixture. Furthermore, the use of this type of model has the added advantage that it is straightforward to incorporate the effect of sample illumination by the probe lamp, which becomes significant at the pH extremes where photocycle recovery is very slow. Use of this basic model is, however, not sufficient as it has already been shown that mono-exponential decay is not sufficient to accurately describe the data (
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
).
Even though the use of more complicated photocycle models was initially promising, a few complications emerged. Our analysis of the recovery reaction thus far indicates that two-states are involved with a bi-exponential transition. To explain such bi-exponential behavior of the recovery reaction with any photocycle model, a third state needs to be introduced (apart from the electronically excited state of pG). However, spectrally this species could not be distinguished from either pB or pG. Without a clear spectral constraint for this additional state, a dependence between several of the rate constants for the models is introduced that does not allow for a confident estimation of these rate constants. The assumption that the additional species has the same spectrum as either pB or pG also did not improve this situation. Another complication is that during the experiment a small amount of sample is lost. Most likely this is because of aggregation, as spectra of the sample between the start and end of the measurement show a small increase in scattering (data not shown). Our model does not take this process into account, which negatively influences analysis of the data using all extended photocycle models we tested. We therefore decided to analyze our data with a bi-exponential analysis of the photocycle recovery reaction from a steady state situation, as described under “Results.” This latter method is less sensitive to measurement artifacts, such as aggregation, although it still allows us to extract information relevant to our study.
Two-state Model Analysis—Basically our data confirm previous measurements on PYP; however, our data analysis approach allowed us to get more detailed information from the dataset. As explained above, initially it was our intention to use photocycle models that incorporate illumination conditions of both the probe and actinic light to analyze our data. The idea was that such a model would act as a constraint/assumption that would allow accurate determination of the absorption spectra of photocycle intermediates. Furthermore it would describe artifacts introduced by the probe light. However, we soon found that the complexity of the models needed made such an approach unpractical. In part this is caused by the presence of several fast equilibria between spectrally distinct species in the photocycle. This makes it impossible to distinguish between these species, based on kinetic information alone. Therefore, we limited our analysis to dark-state recovery from a light-induced steady state. Furthermore, based on SVD exploration of the data, we assumed that only two kinetically and spectrally distinguishable states are present. The following question then arises. To what extent are photocycle intermediates other than the dark and the signaling state present in a light-induced steady state? To get an answer to this question, we simulated the experiment using a photocycle model that, besides pG and pB, includes the pR and pB′ intermediate. pH-dependent rate constants were taken from our previous work (
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
). We compared these results with a simulation of a two-state model that only has the pG and pB species. The simulations revealed that neither pR nor pB′ accumulates significantly under our measurement conditions. Table 1 lists the simulated steady state amounts of pR and pB′ for the illumination conditions used in our experiment. Only at the extremes of pH enough of either pR or pB′ may accumulate for them to show up in our data. Temporal comparison of the two models shows them to be as good as identical within the time resolution of our experiment (data not shown). Only at the pH extremes, where pR or pB′ start to accumulate in higher amounts, do significant differences become visible. Furthermore, when illumination of the sample is ended, the small amounts of pR and pB′ that do accumulate decay instantly with respect to the time resolution of our measurement (data not shown). These simulations further validate our assumption of a two-state situation for our measurements. The final validation for the two-state assumption is the analysis itself. Recovery from the light-induced steady state could be described by just two spectral species. However, at the pH extremes the quality of the fits was somewhat diminished (see supplemental Fig. S4 through S6). With all the signs pointing to a two-state model for the recovery reaction, it is rather surprising that two exponents were required to satisfactorily describe the transition from the light-induced steady state to the dark state. One would expect mono-exponential behavior from a two-state model. This suggests that most likely a third state is present that cannot be distinguished spectrally from the other two states, presumably because it differs from the other states only in protein conformation. The equilibrium between these structural forms should then be relatively slow to observe the bi-exponential behavior of the recovery. In fact, in a recent publication it was suggested that the bi-exponential behavior of the recovery is caused by trans/cis isomerization of Pro-54 (
• Lee B.C.
• Hoff W.D.
), i.e. presumably a spectrally silent transition between different structural states. Based on the slight difference in the pH dependence of the two obtained rate constants (see Fig. 3), one can conclude that the equilibrium between such structural forms is pH-dependent.
TABLE 1Simulation of steady state percentages of photocycle intermediates
pHkpR→pB′kpB′→pRkpB′→pBkpB→pGPR ⇔ pB′ → pB → pGpB→pG pB
pGpRpB′pB
s−1s−1s−1s−1%%%%%
5.053550109800.3427.350.020.3172.3272.56
5.92232089770.89549.580.040.5849.8050.11
7.06221011408801.562.450.060.0637.4337.47
7.472170170011701.3660.140.060.0539.7539.80
7.862090223017501.3860.500.060.0339.4139.45
8.251690201020201.2558.110.060.0341.8041.83
8.82122073014901.2257.520.060.0342.3942.43
9.1510703208901.257.100.070.0642.7842.83
9.77930505000.7244.410.050.0855.4655.53
10.101070403000.535.690.030.1164.1764.26
10.6217200140.48234.130.022.1963.6665.10
Obtaining the pB Spectrum—The spectral information obtained from our two-state analysis consisted of a spectrum for pG and a mixed spectrum of pB and pG. To obtain the pB spectrum from the mixed spectrum requires a good estimation of the amount of pB present in the light-induced steady state. Typically, one would have used spectral constraints to achieve a good estimation. As we were not able to use any spectral constraints other than that the spectrum cannot have a negative component, we had to use other constraints. Here we have made use of a photocycle model again. As the two-state model is able to describe the data reasonably well, we derived an equation from it for the light-induced steady state fraction of pB (Equation 3). This fraction is dependent on the illumination conditions, which we measured, and on the recovery rate. Of the two rates we obtained from our analysis, the dominant one contributes more than 95% to the signal for most pH values. Therefore, we used this dominant rate to estimate the amount of pB in the light-induced steady state. The obtained values were subsequently fine-tuned visually, based on spectral and pH dependence characteristics. We are aware that this is a weak point in the analysis, as it introduces a human bias into the analysis. However, typical changes in the calculated fraction of pB were less than 0.01; they never exceeded 0.04 (see also supplemental Fig. S7). As expected, the largest corrections were necessary at the extremes of pH. Furthermore, a test with the PYP mutant H108F, in which the amount of light was varied, showed that using Equation 3 allows one to make a reasonable estimate of the amount of pB in the light-induced steady state (see Fig. 6).
Identification of a New Intermediate (pG)—From previous work it was already clear that the pB spectrum is pH-dependent. From our data we were able to separate three spectral species based on pH (see Fig. 5). Basically, these results confirm previous work (
• Shimizu N.
• Imamoto Y.
• Harigai M.
• Kamikubo H.
• Yamazaki Y.
• Kataoka M.
,
• Hendriks J.
• Hoff W.D.
• Crielaard W.
• Hellingwerf K.J.
). At low pH pB has an absorption maximum of 368 nm, at medium pH an absorption maximum of 355 nm, and at high pH an absorption maximum of 430 nm, with corresponding pKa values at 5.9 and 9.9, respectively. However, the spectral species at medium pH appears to be a mixture of at least two species, with distinct absorption maxima at 355 and 450 nm. The peak at 450 nm has not been observed before and can be explained by neither the pG nor the high pH pB species. It must therefore be a distinct species related to the 355 nm pB species. The goal of our experiment was to find the intermediate through which recovery to pG occurs. We find it likely that the 450 nm species represents this intermediate. The suggested characteristics of this intermediate, a specific fold and an absorption spectrum that is likely very similar to that of the ground state (
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
), seem to fit. As this intermediate is spectrally similar to pG, and also is a precursor to pG, we have decided to name it pG′. Note that in a previous publication (
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
) we suggested the rather confusing name pBdeprot, which we will no longer use. As the function of pG′ is to catalyze the recovery reaction, it would be logical for the pH dependence of PYP recovery to be governed by the relative presence of this intermediate. We were therefore surprised that we could not separate pG′ from the related pB intermediate on the basis of pH dependence. The only explanation can be that pG′ and the 355 nm pB species are involved in a pH-independent equilibrium, as is illustrated in the inset of Fig. 5C. Furthermore, if pG′ indeed catalyzes the recovery reaction, it is likely that the rate of recovery via pG′ is much faster than the rate of direct recovery via any one of the pB species. The pH dependence of the rate of recovery via pG′ should therefore become visible by division of the most prominent rate constant obtained with the bi-exponential fit (see Fig. 3, circles) by the fraction that the pG′-containing pB spectrum contributes (see Fig. 5B, solid line). Indeed such a treatment results in a curve with a clear, single pH transition. Analysis with Equation 4 revealed a pKa of 7.07 (see Fig. 8). Deviations at the pH extremes are likely because of the low abundance of pG′ at these pH extremes, allowing direct recovery from pB to become significant. The result further shows that at low pH, recovery via pG′ is more efficient with an apparent rate constant of 2.70 s-1 compared with high pH for which an apparent rate constant of 1.59 s-1 is found.
Based on the analysis so far, it would seem that a multitude of pB species are in a fast equilibrium of which there is one, pG′, through which accelerated recovery of pG is achieved. If this hypothesis would completely describe the photocycle events, our analysis should only have needed a single exponent to describe the recovery from pB to pG. However, it turns out that a minor second exponent is required for a proper fit of the data. One might explain this via the equilibrium reaction of pG′ with pB. However, if so, it should be possible to distinguish between pG′ and the other pB species based on kinetics, which is not the case. Still the spectral similarity of pG and pG′ may be a complicating factor here. However, we feel that the explanation given above is more likely, i.e. that the bi-exponential behavior is caused by a spectrally silent difference in structure that is involved in a relatively slow equilibrium.
If the amount of pG′ that is present indeed governs the observed recovery rate, then logic would dictate that this should be reflected in the pB spectrum of the H108F mutant used in the light titration test. At pH 8 the recovery reaction of H108F is slower than that of wild type PYP. When we compare the ratio between the absorption at 358 and 446 nm for wild type (2.9) and H108F (9.4), we indeed see that the amount of pG′ is significantly less in H108F in comparison with wild type PYP (see supplemental Fig. S14).
pH Dependence of pG—The minor third component, detected below pH 7 by the SVD exploration (see Fig. 2), might be explained by heterogeneity in pG. The pH titration on the dark state of PYP revealed this heterogeneous nature of pG. It appears that one of the pH transitions (with a pKa of 4.1) between different ground state species has a pH dependence, comparable with the contribution of the third component in the SVD exploration. The pH dependence of the dark state of PYP further revealed that formation of pBdark appears to have two pH transitions. One, with a pKa of 3.1, results in a pBdark species that also has a small contribution of a species that absorbs around 452 nm. A pH-independent equilibrium between these two species could explain this. The second transition with a pKa of 2.0 then results in a pure pBdark spectrum. Furthermore, the pH titration of the dark state of PYP revealed two transitions at alkaline pH, with pKa values of 11.5 and 12.2, each of which results in a species with a distinct absorption maximum. The species with an absorption maximum of 409 nm is most likely an (partially) unfolded PYP (
• Hoff W.D.
• Devreese B.
• Fokkens R.
• Nugteren-Roodzant I.M.
• Van Beeumen J.
• Nibbering N.
• Hellingwerf K.J.
). Also, an additional transition at alkaline pH with a pKa of 10.0 was found. Unlike the former transitions that were accompanied by a major change of the absorption spectrum (from 446 to 427 to 409 nm), the transition with a pKa of 10.0 is more subtle; it causes only a small change in the main absorption peak. The pKa of 10.0 suggests that the transition is because of deprotonation of tyrosine residues. This is corroborated by a broadening of the absorption peak around 280 nm, typical for the presence of tyrosinates (
• Hoff W.D.
• Devreese B.
• Fokkens R.
• Nugteren-Roodzant I.M.
• Van Beeumen J.
• Nibbering N.
• Hellingwerf K.J.
,
• Holiday E.R.
). It is therefore likely that a similar transition also exists for pB. However, for pB we observed a pKa of 9.9, which is accompanied by a major spectral change. A separate transition due to deprotonation of tyrosine residues, like for the dark state of PYP, may therefore not be visible for pB. The quality of the pH-dependent fit of the pB spectra was relatively poor at very alkaline pH. It is therefore likely that an additional high pH pB species exists. However, as we do not have spectral data for pB above pH 10.5, it was not possible to confidently determine its characteristics.
PYP Photocycle Tree—Based on this study we have built a model for the photocycle of PYP (see Fig. 9) that includes all the different pH transitions described above. At first sight, this photocycle model looks rather complex. With a total of 18 different species, not including electronically excited states, naming all these species starts to become a problem. Therefore, we have grouped these species into photocycle functional groups. Essentially, we stay with the naming convention that we have used before, only now names such as pG, pR, and pB represent a group of species instead of a single species. This is meaningful as proteins are very dynamic, both structurally and with regard to the protonation state of their amino acid residues. Three basic states of PYP can be discriminated, i.e. pG, pR, and pB (also known in other nomenclatures as P, I1, and I2 (or I2′) or PYP, PYPL, and PYPM). The pG state is the dark state of PYP. Absorption of blue light by this state can lead to the formation of the pR state. Upon formation of pR the chromophore is isomerized from trans to cis, while maintaining a structural state highly similar to that of pG. Adaptation of the protein to the new isomerization state of the chromophore results in the formation of pB. Subsequently, pG is recovered from pB, completing/closing the photocycle. Besides these basic states of PYP, several others need to be defined as well. The pBdark state can be considered a chemically/pH-induced nonfunctional state of PYP. Although this state may be spectrally similar to pB, the chromophore is still in the trans state. As the protein structure is perturbed in this state, the protein has become nonfunctional. Although this state may be irrelevant biologically, it has to be taken into account in certain in vitro experiments. Though generally pBdark is considered the low pH form of pG, the high pH form of pG is a pBdark form as well. At very low and very high pH values one also has to take into account chromophore hydrolysis (
• Hoff W.D.
• Devreese B.
• Fokkens R.
• Nugteren-Roodzant I.M.
• Van Beeumen J.
• Nibbering N.
• Hellingwerf K.J.
,
• Meyer T.E.
• Devanathan S.
• Woo T.
• Getzoff E.D.
• Tollin G.
• Cusanovich M.A.
).
In its simplest form, the photocycle of PYP has three states as follows: pG, pR, and pB. In between these basic states of PYP, additional intermediates can be detected. In Fig. 9 only those intermediates have been depicted that occur in the pR to pB and in the pB to pG transition. Interestingly, these additional intermediates, i.e. pB′ and pG′, only seem to occur in the biologically relevant path of the photocycle (indicated by boxes with thick lines in Fig. 9). The pB′ state is characterized by the protonation of the chromophore by the carboxylic acid group of Glu-46 (
• Xie A.
• Kelemen L.
• Hendriks J.
• White B.J.
• Hellingwerf K.J.
• Hoff W.D.
). This protonation is reversible, but pB′ can also be converted to pB (
• Hendriks J.
• van Stokkum I.H.M.
• Hellingwerf K.J.
). At low pH the chromophore may well be protonated directly via the solvent, thereby bypassing pB′.
Note that in the model the reaction between pB′ and pB is unidirectional. It has been suggested that this reaction actually is an equilibrium. In that case we should have observed pB′-like absorption characteristics mixed with our pB spectra. This is not the case, except for the low pH pB form, which is spectrally similar to pB′. Therefore, if pB′ and pB are in equilibrium, this equilibrium must lie far toward pB, in which case it is more convenient to consider it as being a unidirectional reaction.
The pG′ state has been difficult to identify and is characterized by its ability to catalyze photocycle recovery. It appears that pG′ is in a fast equilibrium with only one of the pB states, i.e. the pB state that dominates around pH 8. Strikingly, pG′ appears to be in a fast pH-independent equilibrium with this pB state, making it impossible to separate these two intermediates based on either kinetics, pH, or both, at least for the conditions used in this study. Combined with the apparent spectral similarity of pG′ to pG, it is easy to see how the pG′ intermediate remained undetected for so long. Recovery via pG′ appears to have a pH dependence with only a single pH transition (see Fig. 8). It is unclear what causes the pKa of 7.07 for recovery via pG′. The more complex pH dependence of the PYP recovery rate (see Fig. 3) can be explained by the fact that the amount of pG′ is pH-dependent as well. Direct recovery of pB to pG seems to be possible as well, although it is much slower than recovery via pG′. Direct recovery from pB to pG is only significant at the extremes of pH, where the amount of pG′ is minimal.
The absorption characteristics of pG′ not only indicate that the chromophore is deprotonated but also that this deprotonated chromophore has specific interactions with the protein. Interestingly, a recent molecular dynamics study on PYP indicates that helix formation is a dynamic bottleneck in the recovery reaction of PYP (
• Vreede J.
• Hellingwerf K.J.
• Bolhuis P.G.
). It might be that this helix formation is one of the key elements involved in the equilibrium between pB and pG′.
Concluding Remarks—The model depicted in Fig. 9 can easily be extended to take into account other aspects of PYP. Here we have limited ourselves to the following: the isomerization state of the chromophore, the protonation state of the chromophore, Glu-46 and tyrosine residues, the folding state of the protein, and its visible absorption maximum. Other aspects that could be considered are as follows: the protonation state of other amino acid residues, the presence/absence of salt bridges (
• Hoersch D.
• Otto H.
• Joshi C.P.
• Borucki B.
• Cusanovich M.A.
• Heyn M.P.
), and the secondary structure of a specific part of the protein (
• Vreede J.
• Hellingwerf K.J.
• Bolhuis P.G.
). Furthermore, we have limited ourselves to pathways induced by blue light excitation of pG. Other paths such as light-induced branching reactions from pR or pB, for example, or illumination of pBdark species can of course also be added. One thing is very clear from this model, whenever PYP is used for experiments chances are that mixtures of intermediates are formed that cannot always be distinguished based on kinetic information alone. On the practical side, the complexity of the model makes it very difficult to apply it directly to the analysis of datasets. However, it is a powerful tool that can help with the choice of experimental conditions. Furthermore, it can serve as a guide for the analysis of data.
The use of a light-induced steady state pB/pG mixture has enabled us to identify/visualize the photocycle intermediate pG′. Although this approach can be used in many other cases as well, it does have its limitations. In order for a significant amount of signaling state to accumulate, the recovery rate has to be relatively slow. For example, for the E46Q mutant of PYP, significant amounts of pB cannot be accumulated in part of the relevant pH range (
• Hendriks J.
• Hoff W.D.
• Crielaard W.
• Hellingwerf K.J.
). The advantage of using a light-induced steady state is that it is fairly easy to estimate the amount of signaling state that is accumulated in that steady state, using measurable parameters such as light intensity and recovery rate. This is valuable information that can be used in the data analysis process. In our case it was a key part of the identification of pG′. This intermediate may play a crucial role in further studies of thermal chromophore isomerization in PYP.

## Acknowledgments

We thank Dr. I. H. M. van Stokkum and Dr. T. Gensch for providing valuable feedback regarding the manuscript.

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# Measurement of the top-quark mass in the fully hadronic decay channel from ATLAS data at $$\sqrt{s}=7\mathrm{\,TeV}$$ s = 7 TeV

Aad, G. et al. (2015) Measurement of the top-quark mass in the fully hadronic decay channel from ATLAS data at $$\sqrt{s}=7\mathrm{\,TeV}$$ s = 7 TeV. European Physical Journal C, 75(4), 158.

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# Let's make one big, happy "Congratulations!" thread.

Update: The proposal in question has been implemented.

We recently had a discussion about the tag that raised some important issues, but did not result in a consensus. I'd like to pitch the following solution to the community.

Let's consolidate all celebration posts into one bumpable, meta question, similar to the thread for reopen requests.

These are several reasons that I want to propose this.

• These posts are not questions, and they are all very similar. I know we allow other unconventional threads to be posted as questions, but this is very clearly a family of such posts. It would make sense to aggregate them into one place to be viewed all at once, like a .

• Celebrations make more sense as answers than questions. Congratulations posts don't need to be answered. Sometimes those being recognized post an answer to say thanks, but it makes just as much sense to express this sentiment as a comment. Since comments can be upvoted, this presents no loss of visibility.

• Bumping an answer list allows for continuous recognition of past achievements. Many users may not think to use the search to read through past celebrations, and even so, the interface is not as good as a list of answers. This way, every time a new celebration occurs, we not only get to celebrate the occasion at hand, but remember the reasons for past celebrations, too.

• Shaking up tradition broadens the format to allow celebration of other cool achievements. Two months ago, Martin Sleziak opened this chat room as an experiment, and I think it's really great. The informality of the chat format has allowed a variety of "quirkier" achievements to be recognized, such as high "meta-rep" and crazy good editing. It's really cool to see these celebrations of individuality alongside those of pure numbers. However, the chat format has its problems. Discussion is intermingled with the list of results, which makes them more difficult to read. The visibility is inarguably lower, there's no voting to show appreciation, it's less accessible to newcomers, and, as demonstrated by the comments in this thread, some users just plain don't like to use chat. A consolidated celebrations thread would be allow us to adopt some of this flexibility, in addition being more unobtrusive to those who consider these posts off-topic. I do think we should still use this room- it's a great place to talk about data explorer, and a great platform to celebrate smaller achievements that are not big enough to warrant a meta post. But we need something else, too.

• The main thing we'd lose would be having the name of the person being celebrated in the title of the question, which would then get less attention on the meta front page. One way we might compensate for this could be to change the accepted answer whenever a new celebration is posted, thus identifying the "incumbent celebration."

What do you think, community?

• Re the last point: Or edit the title, "Congratulations thread, [whoever caused the last bump] edition". Nov 25 '14 at 11:47
• Re chat and "there's no voting to show appreciation" I do not get that at all. One can "star" a message, which seems as good as an upvote for the purpose at hand. The issue of downvotes (that seem somewhat unpleasant in that context) is gone. And starred messages can be found among the rest. To be clear, I do not argue against that thread, just this particular reason seems like a nonreason to me, since in fact for this purpose in my opinion there is a better suited form of voting in chat then here.
– quid Mod
Nov 25 '14 at 13:01
• Let's just not have user specific congratulatory threads instead.
– Asaf Karagila Mod
Nov 25 '14 at 13:49
• Agreeing with @AsafKaragila: let's just not. Simply downvote, close, and delete off-topic threads as they appear -- business as usual.
– user147263
Nov 25 '14 at 15:13
• @Asaf The issue is that not everyone thinks they are off topic. I do, but others' opinions matter. To me, this seems like a good compromise.
– Alexander Gruber Mod
Nov 25 '14 at 18:49
• Alexander, of course that I don't want to have another divisive issue on meta, there are plenty of serious issues to go around. And I do understand this sort of activity and why people do it. But just like we don't congratulate people on earning gold badges (or a particular number thereof), we shouldn't congratulate them on some reputation barrier. This community is very large as it is, and growing larger with each day. This means that eventually more and more people get to 100k in faster rates, which means that at some point the congratulation thread will be bumped all too often too.
– Asaf Karagila Mod
Nov 25 '14 at 19:01
• Just rip this band-aid (or adhesive strip, if you prefer) now, and get it over with.
– Asaf Karagila Mod
Nov 25 '14 at 19:01
• We might see how good a compromise this idea is once we will have had the first discussion whether it was not a bad idea to replace that really noteworthy achievement in the title by a much less worthy one after so little time (and perhaps if this was not done for sinister motives).
– quid Mod
Nov 25 '14 at 19:46
• @quid Well, we could just agree that the accepted answer denotes the most recent post, not the most important. Or we could just forget about the accepted answer / title change thing. That would be fine by me.
– Alexander Gruber Mod
Nov 26 '14 at 0:18
• My comment is based on "title/accept is most recent." My point is: some things might be "headline" for a long time and some others only for 20 minutes. This could cause problems. Especially since everybody can generate "news" essentially as they see fit. So somebody could come along and "push away" somebody they do not like from the "headline" via posting something. Or somebody could suspect it happened. If at all, yes, I would strongly prefer no title changes or accept (except perhaps at a later point some overview answer could be generated and accept, but this could also be in the post).
– quid Mod
Nov 26 '14 at 7:18
• Do these posts come up so often as to be actively cluttering up meta? It seems as though they're on the order of, say, one per month if that - as opposed to something like the reopen thread, which gets a new note every few days. I just don't see what problem this is trying to solve. Nov 29 '14 at 8:38
• I think it is a nice tradition for the first $n$ people to hit some barrier. Then around $n$ threads need to be kept open and the barrier number updated as appropriate (assuming that, for the most part, the same $n$ people will be the ones hitting the thresholds). I don't find the threads bothersome in any way. Dec 1 '14 at 7:02

Personally, I find these posts not bad. However, if a somewhat complicated procedure should be needed to fit them in (edit the title, change the accpeted answer, and what not), then I would say the thing has run its course and it is better to stop that tradition, as oppose to continuing it in this rigid form.

Perhaps this is a pity for some (including me), but the alternative seems a lot worse to me, be it either the constant quibble about the posts or this rigid form that gets proposed here.

Update: The proposal in question has been implemented.

This is a good compromise and I think its score (presently 31) indicates a clear positive community response.

But I think the thread should be simple rather than complicated. In particular,

• Don't try to have an accepted answer mean anything. This is not what accepting an answer means and this would be confusing.

• Don't try to merge old congratulations threads with this new thread. Just use it for future congratulations.

• Don't edit the title of the question. Leave it as something generic.

• Don't argue about the details. Just make the thread and allow people to make answers congratulating specific users.

The point is that we don't want a rigid and complicated procedure; this is what quid's answer complains about:

if a somewhat complicated procedure should be needed to fit them in (edit the title, change the accpeted answer, and what not), then I would say the thing has run its course and it is better to stop that tradition, as oppose to continuing it in this rigid form.

The main problem with the current congratulations threads is that they are recurring, uninteresting, and a much bigger deal than they really should be. I would find it a bit sad to ban the sentiment of such posts altogether, but on the other hand, I think it is silly to keep congratulating people on this level (and the Hot Meta Posts list should be reserved for more important things). This proposal solves all these issues effectively.

• Votes have many rationals, it is not easy to tell what they represent unless there is a clear statement in the post about how votes will be interpreted. I upvoted the question because I think it worth discussing, but I am leaning more towards what quid wrote. Jan 14 '15 at 0:40
• @Kaveh Upvotes on meta mean "I agree". Downvotes mean "I disagree".
– 6005
Jan 14 '15 at 1:33
• No, they don't. People use votes for various reasons, e.g. I here upvoted because I think it is a useful discussion, not that I agree with the proposed solution. Jan 14 '15 at 1:46
• @Kaveh They generally do. I'm not sure how to convince you of this, maybe you simply haven't spent much time on meta. Here is another post; notice nobody argues with the premise of the question.
– 6005
Jan 14 '15 at 2:06
• I have seen countless times new users on meta get upset by downvotes, before it is explained by more experienced users that downvotes mean disagreement and upvotes mean agree. I didn't really know this was subject to debate. Of course, some users (e.g. you) are not voting this way, and I will contend this might be a significant portion of the meta population, but if you ask anyone here how voting works on meta, they will explain what I have said.
– 6005
Jan 14 '15 at 2:08
• @Goos: Checking the highest voted answer to the first link you gave, you will see a couple of reasons to downvote that don't mean "I disagree." I notice that I long ago downvoted the answer you linked directly to, because I disagree with it; it is sometimes right, but overgeneralized. Jan 14 '15 at 2:52
• I have been visiting this meta (and other metas) for years now and I can tell you that it is not the case, people may vote for usefulness/suitability of a topic, for agreement with a proposal, or even because of general behavior of the user among other things. If someone wants to have a poll on a specific issue it should be stated as so in the post like Asaf does in his answer here. Jan 14 '15 at 4:42
• @Kaveh I wasn't really arguing about how people actually vote, only about how votes are to be interpreted. I should concede that "I agree" is a vague way of saying what I mean; for example, "I agree" can mean that you agree with the usefulness/suitability of a topic.
– 6005
Jan 14 '15 at 5:00
• @Kaveh I don't agree that a separate poll is needed unless the voting looks close. I stand by that 41 upvotes, 5 downvotes (or whatever it was at the time) indicates a "clear positive community response". If you think my interpreting this as clear positive support for the proposal and you think it important enough I suggest you take it up on another thread. I would not see, for instance, 10 upvotes and 7 votes, as a clear positive response, or even as a positive response. But 41 upvotes is clear approval in my mind.
– 6005
Jan 14 '15 at 5:03
• More concretely: If only 5 people downvote a proposal, then only 5 people expressly disagreed with any aspect of it. In particular at most 5 people disagreed with the proposal. And 41 upvotes I would assume includes at least 20 (being safe) votes in favor of the proposal itself.
– 6005
Jan 14 '15 at 5:04

This is a poll answer.

Please UPVOTE this if you think there shouldn't be any more personal congratulatory meta posts.

Please DOWNVOTE this if you think there should be some solution (regardless to a single thread, or continuing in this fashion) for personal congratulatory posts on meta.

If you don't care, please don't vote.

(I will occasionally update the current score for the benefit of those that did not reach 1,000 points yet.)

Update: Current score, +6, -8 (12 Jan 2015 09:51:10 PM UTC)

• Whatever the score is, please add +1 from me on the upvote side.
– Asaf Karagila Mod
Dec 8 '14 at 8:47
• I've voted on your behalf! Dec 8 '14 at 10:11
• @Integrator: Does that mean that you're abstaining?
– Asaf Karagila Mod
Dec 8 '14 at 10:12
• I'm unable to choose sides. Dec 8 '14 at 10:17
• @Integrator: Is this because you don't want to assume the axiom of choice?
– Asaf Karagila Mod
Dec 8 '14 at 10:17
• @quid: It might take another couple of days. I don't expect this to be similar to the "Let's kill the [homework] tag" or "Let's require registration for asking questions" or "Let's have silly hats", but I'm not in a hurry.
– Asaf Karagila Mod
Dec 8 '14 at 18:02
• Let us see. What I am worried about is that it does not get enough visibility anymore. After all 40+ voted OP. Maybe let us make it an "issue" in the election? This is only semi-serious. [Deleted the first comment to avoid excessive clutter.] Current count +3/-3
– quid Mod
Dec 9 '14 at 7:03
• I think the number of votes matches the importance of the issue. Those who want congratulatory posts can always create them. Those who don't want them can vote accordingly. Whichever side has more support will determine the outcome. The system works.
– user147263
Dec 9 '14 at 7:07
• @quid, it is a possibility, yes.
– Asaf Karagila Mod
Dec 9 '14 at 7:20
• @Beh, no this is when it doesn't work. Since meta is not active enough to drown these things in the noise, it becomes annoying when this gets done, closed and reopened and deleted and undeleted. This is where the system needs fixing, and we need to decide on some result and empower the moderators to act, if need be, without hindrance.
– Asaf Karagila Mod
Dec 9 '14 at 7:23
• @AsafKaragila You know there is the (celebration) tag for hiding those posts, if they are annoying.
– user147263
Dec 9 '14 at 7:33


text:
CCC '98 S4 - Lottery

View as PDF

Points: 10
Time limit: 2.0s
Memory limit: 16M

Problem types

You have just won the lottery. All that separates you from your multi-million dollar prize is your correct answer to the following skill-testing question:

In your twenty seconds you see your fortune slipping away because you don't know whether the answer is

or

Finally you guess 63811, but that answer is incorrect. Your math teacher set the question and expected you to remember that multiplication is done before addition. The correct answer is 51471.

Your job is to write a program to insert parentheses into lottery questions such as the above so as to make clear the order of operations.

Input Specification

The input to your program consists of a line containing an integer, , followed by lines of input. Each of the lines contains an expression consisting of integers, and the operators +, -, and X denoting addition, subtraction, and multiplication respectively. Adjacent integers are separated by one operator. There is a single space to the left and to the right of each operator and no input line contains more than 80 characters.

Output Specification

Your output should consist of the same lines, with a blank line between them, with parentheses inserted in the lines so as to indicate the order of operations. Multiplication should be done first, from left to right, and addition and subtraction should then be done from left to right. Spaces surrounding operators should be preserved.

Sample Input

3
10 + 20 X 30
1 + 2 + 3 - 4
123 + 456 X 789 - 876

Sample Output

10 + (20 X 30)

((1 + 2) + 3) - 4

(123 + (456 X 789)) - 876

• Dingledooper  commented on July 19, 2019, 7:00 p.m.

(1 + 2) - (3 X 4)

• 4fecta  commented on July 18, 2019, 5:58 p.m.

Should "1 + 2 - 3 X 4" return "(1 + 2) - (3 X 4)" or "1 + 2 - (3 X 4)"?


text:
# Tag:Douroux

• ## We sit in our ivory towers and think

I’m on vacation, and re-reading two ‘metabiographies’: Philippe Douroux : Alexandre Grothendieck : Sur les traces du dernier génie des mathématiques and Siobhan Roberts : Genius At Play: The Curious Mind of John Horton Conway . Siobhan Roberts’ book is absolutely brilliant! I’m reading it for the n-th time, first on Kindle, then hardcopy, and […]

• ## Grothendieck’s gribouillis (4)

Fortunately, there are a few certainties left in life: In spring, you might expect the next instalment of Connes’ and Consani’s quest for Gabriel’s topos. Here’s the latest: $\overline{\mathbf{Spec}(\mathbb{Z})}$ and the Gromov norm. Every half year or so, Mochizuki’s circle-of-friends tries to create some buzz announcing the next IUTeich-workshop. I’ll spare you the link, if…


text:
# Anyone ever work with a DE like this?

1. Nov 29, 2009

### Mosis

This has come up in my research, and my supervisor and I don't really know how to proceed. It reads

$$r^2\frac{d^2Y}{dr^2} + r\frac{dY}{dr} - \left(\frac{3}{2}\right)^2Y = Cr^3Y^{1/3}$$

I know the RHS is an equidimensional DE which has the nice solution $$Y = r^{\pm 3/2}$$, but I have no idea how this could help with that nasty $$Y^{1/3}$$ term sitting on the RHS.

Really, we just want to learn SOMETHING about this DE. What do the solutions look like? Does a bounded solution exist? What can we do if C is small? etc.

Any ideas?

2. Nov 29, 2009

### Mosis

wow, 46 views, no reply - is this really that uncommon?

3. Nov 29, 2009

### HallsofIvy

Staff Emeritus
The problem is that that is a seriously non-linear equation. Since solutions of non-linear equations do not add to another solution, Knowing a solution to LHS= 0 does not help. For C small, you might try a perturbation method, writing the solution as a series in powers of C: $y_0+ Cy_1+ C^2y_2+ \cdot\cdot\cdot$. Substituting that into the equation and equating like powers of C will give, for the "C0" term, the homgeneous equation as the equation for $y_0$. Then the "C1 term will gvie a non-homogeneous equation for $y_1$ with the (now known) $y_0$ on the right side, etc.

Last edited: Dec 11, 2009
4. Nov 29, 2009

### Mosis

I was thinking about that kind of perturbation scheme, but the fact that we have Y^1/3 makes that intractable i.e. what to do with $$\left(Y_0 + CY_1\right)^{1/3}$$?

5. Nov 29, 2009

### hamster143

You could express it as $$Y_0^{1/3} (1+C(Y_1/Y_0)+C^2(Y_2/Y_0)+...)^{1/3} = Y_0^{1/3}(1 + \frac{1}{3}(C(Y_1/Y_0)+C^2(Y_2/Y_0)...) - \frac{1}{9}(C(Y_1/Y_0)+C^2(Y_2/Y_0)...)^2 ...$$

because $$(1+x)^a = 1 + xa + x^2 a(a-1)/2! + x^3 a(a-1)(a-2)/3! + ..., x \ll 1$$

It will be incredibly messy beyond the first two terms, but at least you can get some results.

By the way, it would simplify your calculations if you did a substitution $$z = \ln{r}$$.

Last edited: Nov 29, 2009
6. Nov 30, 2009

### hamster143

Further thoughts.

There are two distinct cases, C>0 and C<0.

Once you rewrite your expression as r = ln z, $$Y'' = 9/4 Y + C e^{3z} Y^{1/3}$$,

it's easy to see that, as long as C>0, all nontrivial solutions are unbounded (because the second derivative is always positive, therefore the graph of the function is convex).

Every individual solution will look like a bunch of exponentials a*exp(b*z) stitched together, depending on which of the two terms on the right hand side dominates (most of the time, one of the two terms would be negligible). Draw a plot of the solution of the equation $$9/4 Y = C e^{3z} Y^{1/3}$$. Use log scale on the vertical axis. Think about the behavior of a solution of your DE that passes through a point to the left of the curve. See if you can prove (or at least convince yourself) that almost all solutions either blow up because Y crosses below zero, or diverge as exp(-1.5 z) as $$z\rightarrow -\infty$$ and as exp(4.5 z) as $$z\rightarrow +\infty$$.

If C<0, things are more complicated, and many solutions would blow up at some point because there's nothing to keep Y from going negative.

Last edited: Nov 30, 2009
7. Nov 30, 2009

### Mosis

why are you assuming Y is positive? also, exp(-r) is convex but decays as r goes to infinity. Of course, it blows up in the other direction, but in my case, Y = Y(r), and r > 0, so I'd really need some information about the first derivative to conclude all nontrivial solutions are unbounded.

8. Nov 30, 2009

### Mosis

also, I'm not sure why you suggest plotting the solution to that equation - how can that help me?

9. Nov 30, 2009

### hamster143

Because Y^(1/3) is only defined for positive Y.

Yes, exp(-r) is convex. But notice my variable change: from r that goes 0..infinity, to z that goes from minus infinity to infinity.

If you have a function whose argument goes from minus infinity to infinity and its graph is convex, it must be unbounded on either side.

Because either of two terms on the right hand side would dominate, depending on which side of the graph you are. So, as long as you're sufficiently far to the left, solutions of DE would behave as if there's no pesky cubic root term.

10. Dec 1, 2009

### trambolin

I agree with hamster143, even it does not help you in terms of the details it will give you a lot of information about the vector field. My suggestion is setup your DE in MATLAB or your favorite solver while keeping it nonlinear, and let the solutions from many (I mean many) initial conditions and get a feeling for the differential equation. You can even read off if there is a bounded solution or not.

You never know! It might act like almost linear in the region of interest. (Happened to me also)

11. Dec 2, 2009

### gato_

you can further write that using the fact that $$Y=exp(9z/2)$$ is an exact solution, and write $$Y=exp(9z/2)y(z)$$, with :
$$y''+9y'+18y-Cy^{1/3}$$
which is autonomous, and proceed by local analysis.

12. Dec 2, 2009

### gato_

you can check that the critical points are
$$y = y'=0$$, and $$y=(-C/18)^{3/2}, y'=0$$
the second stationary points leads to a non bounded solution Y(z). The jacobian at (0,0) is not defined, but we have the inequality
$$[y'^{2}/2+9y^{2}-Cy^{4/3}3/4]'=-9y'^{2}$$
So, for C<0, y=0=y' is stable, and you might have bounded solutions
* beware when simulating that in matlab, y^(1/3) does not give the proper solution. try sign(y)*abs(y)^1/3 instead.

13. Dec 11, 2009

### cheky

what do you mean by a convex graph


text:
# Chapter 4 - Polynomials - Study Summary - Practice Exercises: 13

degree = $1$

#### Work Step by Step

The degree of a term is equal to the sum of the exponents of the variables. $5x$ has one variable whose exponent is $1$. Thus, the degree of $5x$ is $1$.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# 500g of water is heated so that its temperature rises from 30C to 72C in...

Question 467
A
9000J
B
12600J
C
21000J
D
25200J
E
88200J
##### Explanation
$$\frac{Q}{t} = \frac{0.5\times4200\times(72-30)}{7}=12600J$$

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text:
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## Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 16th 2020
• (edited Jan 16th 2020)

In order to formalize some physics, I am looking for a suitable mathematical concept. It looks like the putative concept ought to be something like “pro-finite $\mathfrak{su}(2)$-representations”, but I am not sure yet. And once I am sure, I’ll be wondering if there is any decent established theory for such things.

The following is the motivation (taken from here):

Motivation

There is the remarkable observation (MSJVR 02, checked in AIST 17) that in the BMN matrix model supersymmetric M2-M5-brane bound states are identified with “limit sequences” of isomorphism classes of finite-dimensional complex Lie algebra representations of su(2).

Concretely, if

$\mathbf{N}( \{N^{(M2)}_i,N^{(M5)}_i\}_i ) \;\;\coloneqq\;\; \underset{ i }{\oplus} N^{(M2)}_i \cdot \rho_{N^{(M5)}_i} \;\;\in\;\; \mathfrak{su}(2)Rep^{fin}$

denotes the representation containing

of the

• $N^{(M5)}_i$-dimensional irrep

(for $\{N^{(M2)}_i, N^{(M5)}_i\}_{i} \in (\mathbb{N} \times \mathbb{N})^I$ some finitely indexed set of pairs of natural numbers)

with total dimension

$N \;\coloneqq\; dim\big( \mathbf{N}( \{N^{(M2)}_i,N^{(M5)}_i\}_i ) \big)$

then:

1. an M5-brane configuration corresponds to a sequence of such representations for which

$N^{(M2)}_i \to \infty$

for fixed $N^{(M5)}_i$

and fixed ratios $N^{(M2)}_i/N$

2. an M2-brane configuration corresponds to a sequence of such representations for which

$N^{(M5)}_i \to \infty$

for fixed $N^{(M2)}_i$

and fixed ratios $N^{(M5)}_i/N$

for all $i \in I$.

Hence, by extension, any other sequence of finite-dimensional $\mathfrak{su}(2)$-representations is a kind of mixture of these two cases, interpreted as an M2-M5 brane bound state of sorts.

$\,$

Question. I’d like to extract a precise definition of “M2-M5 brane bound state” from the above. It must subsume suitable limits of finite-dimensional $\mathfrak{su}(2)$-representation as above. But taken where? And identified how?

Is “profinite $\mathfrak{su}(2)$-reps” a thing in representation theory? (i.e. pro-objects in the category of finite-dimensional representations.) Or maybe ind-objects instead? Or something else?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 16th 2020

Ah, never mind, I see the answer now. It was right in front of me all along.

The space of interest here is not $\mathfrak{su}(2)Rep$ but the image of $Span(\mathfrak{su}(2)Rep)$ in weight systems. Now taking the linear span is what makes sense of expressions like $\frac{1}{N} \cdot \rho_{N}$. And their limit with $N\to \infty$ gotta be taken in the space of weight systems, because that’s precisely what the multi-trace observables of the BMN model observe.

Okay, case closed. Thanks for listening.


text:
# Network firewalls, routing etc

### In which years of study are needed have basic online safety

Usefulness: 🔧 🔧
Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

You need to route network traffic cleverly, e.g. for a vpn, filesync client or ssh, or various server security concerns. Note that you might also want to look at application firewalls which stop specific software from doing things, as opposed to network activity.

Oh sweetie. If you are getting advice from me, you are royally fucked. I’m so sorry to find you in this bind. I’m not the kind of person who knows about network stacks. Otherwise how would I have had time to do the other things for which you would read the blog?

Anyway, let’s discover this together.

## macOS

Certain friendly apps for macOS exist to make this possibly easier.

icefloor

IceFloor 2 is group based, like the old ServerAdmin firewall tool. Control filtering, bandwidth, logs, connections and custom PF configurations.

(No longer current for recent macOS)

murus

macOS features one of the best network firewalls: PF (Packet Filter). It comes in an “under the hood” fashion, installed and disabled by default.

Murus’ purpose is to unleash its potential. With its easy and intuitive icons-based and drag&drop-based interface, visual layers of abstraction and a friendly view of the PF firewall it’s a lean and mean tool to protect your Mac and network.

Designed with ease of use in mind, yet full of advanced options and monitoring tools, Murus is perfect for everybody; from the average user to the experienced UNIX guru/system administrator and even for educational purposes.

Configure and start the PF firewall in one click using built-in presets, use Murus graphical ruleset editor design tool or write fully customized rulesets using the advanced rule editor.

Murus Pro includes Vallum, an application-layer firewall. This allows you to take full control of your Mac at both application and network level.

## Linux

Do you need one? Yeah. Do you have one by default? Usually not.

So what’s the Linux networking routing doo-dad called? iptables. This is how you interact with the netfilter subsytem. How does it work? idk; every time I touch it I break things because there are thousands of moving parts in a modern computer’s networking system.

There are a million firewall configuration apps for Linux, but the one that n00bs like me seem to get recommended is ufw the Uncomplicated FireWall, which is installed on This in turn provides a simpler front end to iptables which, as preaged, is not actually enabled per default.

Here you go, turn it on.

sudo ufw enable

This blocks incoming traffic.

You might need to enable certain traffic if you run servers. The following 2 are equivalent.

sudo ufw allow ssh
sudo ufw allow 22/tcp
sudo ufw status verbose

gufw is a notionally even friendlier GUI for ufw. GuFW AFAICT doesn’t add value in terms of visualising your firewall, which would be the big win. It’s just some knobs and buttons to type the firewall commands for you.

If your needs are a little more professional, you probably want to set up firewalls using basic iptables, which TBH doesn’t seem that complicated. An example of someone getting fancy with that, including cute hand-rolled DDOS protection, is diveyez’ fw.sh.

At this point in seeking help on the issue of firewall configuration, the local office geek will tell you that the real pros use BSD for their network routing. But if you are a real pro, you ain’t reading this, I suspect. Gather your dilletantism about you like a cloak and sneak away.

I suppose if I wanted to prove something with slightly less re-learning I could try a stripped-down [hardened]{filename}hardened_os.md) distro like Alpine.


text:
## Detect ARCH Effects

### Test Autocorrelation of Squared Residuals

This example shows how to inspect a squared residual series for autocorrelation by plotting the sample autocorrelation function (ACF) and partial autocorrelation function (PACF). Then, conduct a Ljung-Box Q-test to more formally assess autocorrelation.

Load the NASDAQ data included with the toolbox. Convert the daily close composite index series to a percentage return series.

```load Data_EquityIdx; y = DataTable.NASDAQ; r = 100*price2ret(y); T = length(r); figure plot(r) xlim([0,T]) title('NASDAQ Daily Returns')```

The returns appear to fluctuate around a constant level, but exhibit volatility clustering. Large changes in the returns tend to cluster together, and small changes tend to cluster together. That is, the series exhibits conditional heteroscedasticity.

The returns are of relatively high frequency. Therefore, the daily changes can be small. For numerical stability, it is good practice to scale such data.

Plot the Sample ACF and PACF.

Plot the sample ACF and PACF for the squared residual series.

```e = r - mean(r); figure subplot(2,1,1) autocorr(e.^2) subplot(2,1,2) parcorr(e.^2)```

The sample ACF and PACF show significant autocorrelation in the squared residual series. This indicates that volatility clustering is present in the residual series.

Conduct a Ljung-Box Q-test.

Conduct a Ljung-Box Q-test on the squared residual series at lags 5 and 10.

`[h,p] = lbqtest(e.^2,'Lags',[5,10])`
```h = 1x2 logical array 1 1 ```
```p = 1×2 0 0 ```

The null hypothesis is rejected for the two tests (`h = 1`). The p values for both tests is `0`. Thus, not all of the autocorrelations up to lag 5 (or 10) are zero, indicating volatility clustering in the residual series.

### Conduct Engle's ARCH Test

This example shows how to conduct Engle's ARCH test for conditional heteroscedasticity.

Load the NASDAQ data included with the toolbox. Convert the daily close composite index series to a percentage return series.

```load Data_EquityIdx; y = DataTable.NASDAQ; r = 100*price2ret(y); T = length(r); figure plot(r) xlim([0,T]) title('NASDAQ Daily Returns')```

The returns appear to fluctuate around a constant level, but exhibit volatility clustering. Large changes in the returns tend to cluster together, and small changes tend to cluster together. That is, the series exhibits conditional heteroscedasticity.

The returns are of relatively high frequency. Therefore, the daily changes can be small. For numerical stability, it is good practice to scale such data.

Conduct Engle's ARCH Test.

Conduct Engle's ARCH test for conditional heteroscedasticity on the residual series, using two lags in the alternative hypothesis.

```e = r - mean(r); [h,p,fStat,crit] = archtest(e,'Lags',2)```
```h = logical 1 ```
```p = 0 ```
```fStat = 399.9693 ```
```crit = 5.9915 ```

The null hypothesis is soundly rejected (`h = 1`, `p = 0`) in favor of the ARCH(2) alternative. The F statistic for the test is `399.97`, much larger than the critical value from the ${\chi }^{2}$ distribution with two degrees of freedom, `5.99`.

The test concludes there is significant volatility clustering in the residual series.


text:
# Numerical Integration in Python

• Python
Mark44
Mentor
I need to evaluate say G(1,1,0), G(2,1,0) .... G(10,1,0). These function are independent of each others. But in for loop these will look something like

Code:
for(int i=1; i <= 10; i++){
cout  <<  G(i, 1,0);
}
which are evaluated sequentially. These can be evaluated in parallel. But how? I am not familiar with multithreading, but I think we can get Kx speed up, where K is the number of cores.
My computer, which I've had for five years, has an Intel Quad Core Duo processor, with each of the four cores having effectively two processors. This means that eight threads can be running. You're not going to have K cores, with K being an arbitrary integer. Your program also can't be using all the cores, as some of them will be used by the OS and other processes.

If I were going to write the code, I wouldn't have it iterate 10 times. Instead, I would iterate 4, or 8, or some power of 2, times, and I wouldn't use a loop. If I had four threads to play with, I would create four thread objects to evaluate G(1, 1.0), G(2, 1.0), G(3, 1.0), and G(4, 1.0). When the last one of these is finished, I would use the threads to evaluate G with a first argument of 5, 6, 7, and 8.

Also, I wouldn't output them in the loop. Instead, I would store the return values from G() in an array, and print them out when the array is full.

I can't say too much more about threads and such, since there's quite a bit more than I have explained. Threads are pretty low level. At a higher level of abstraction are async objects, which are easier to work with.

EngWiPy