Estimation of Free-Energy Differences from Computed Work Distributions: An Application of Jarzynski’s Equality

Equilibrium free-energy differences can be computed from nonequilibrium molecular dynamics (MD) simulations using Jarzynski’s equality (Jarzynski, C. <i>Phys. Rev. Lett.</i> <b>1997</b>,<i> 78</i>, 2690) by combining a large set of independent trajectories (path ensemble). Here we present the multistep trajectory combination (MSTC) method to compute free-energy differences, which by combining trajectories significantly reduces the number of trajectories necessary to generate a representative path ensemble. This method generates well-sampled work distributions, even for large systems, by combining parts of a relatively small number of trajectories carried out in steps. To assess the efficiency of the MSTC method, we derived analytical expressions and used them to compute the bias and the variance of the free-energy estimates along with numerically calculated values. We show that the MSTC method significantly reduces both the bias and variance of the free-energy estimates compared to the estimates obtained using single-step trajectories. In addition, because in the MSTC method the process is divided into steps, it is feasible to compute the reverse transition. By combining the forward and reverse processes, the free-energy difference can be computed using the Crooks' fluctuation theorem (Crooks, G. E. <i>J. Stat. Phys.</i> <b>1998</b>,<i> 90</i>, 1481 and Crooks, G. E. <i>Phys. Rev. E</i> <b>2000</b>,<i> 61</i>, 2361) or Bennett’s acceptance ratio (Bennett, C. H. <i>J. Comput. Phys</i>. <b>1976</b>,<i> 22</i>, 245), which further reduces the bias and variance of the estimates

A Redox Regulatory System Critical for Mycobacterial Survival in Macrophages and Biofilm Development

<div><p>Survival of <i>M</i>. <i>tuberculosis</i> in host macrophages requires the eukaryotic-type protein kinase G, PknG, but the underlying mechanism has remained unknown. Here, we show that PknG is an integral component of a novel <i>r</i>edox <i>ho</i>meostati<i>c</i><i>s</i>ystem, RHOCS, which includes the ribosomal protein L13 and RenU, a Nudix hydrolase encoded by a gene adjacent to <i>pknG</i>. Studies in <i>M</i>. <i>smegmatis</i> showed that PknG expression is uniquely induced by NADH, which plays a key role in metabolism and redox homeostasis. <i>In vitro</i>, RenU hydrolyses FAD, ADP-ribose and NADH, but not NAD+. Absence of RHOCS activities <i>in vivo</i> causes NADH and FAD accumulation, and increased susceptibility to oxidative stress. We show that PknG phosphorylates L13 and promotes its cytoplasmic association with RenU, and the phosphorylated L13 accelerates the RenU-catalyzed NADH hydrolysis. Importantly, interruption of RHOCS leads to impaired mycobacterial biofilms and reduced survival of <i>M</i>. <i>tuberculosis</i> in macrophages. Thus, RHOCS represents a checkpoint in the developmental program required for mycobacterial growth in these environments.</p></div

L13, a ribosomal protein associated with RenU, is phosphorylated by PknG.

<p>(<b>A</b>) Representative <i>in vitro</i> phosphorylation of RenU.6H preparations purified from <i>M</i>. <i>smegmatis</i> (left) or <i>E</i>. <i>coli</i> (right) by purified PknG. PI, phosphatase inhibitors. (<b>B</b>) <i>In vitro</i> phosphorylation of corresponding fractions eluted from ion exchange columns by PknG. Numbers indicate the NaCl concentrations used in elution buffer. Samples loaded to the ion exchange columns were obtained from an immobilized Cobalt affinity chromatography of <i>M</i>. <i>smegmatis</i> RenU.6H (+) cell lysates or control lysates (-). (<b>C</b>) <i>In vitro</i> phosphorylation of purified 6H.L13 or 6H.SmpB by PknG. (<b>D</b>) Co-purification of L13 from <i>M</i>. <i>smegmatis</i> lysates by exogenous RenU.6H. Another recombinant 6H-tagged protein (6H.SHMT) was used as a control. Blots were detected by Anti-L13 or Anti-6H antibodies. (<b>E</b>) <i>In vitro</i> phosphorylation of recombinant or native L13 protein associated with RenU by PknG kinase activity. (<b>F</b>) <i>In vitro</i> phosphorylation of purified 6H.L13 by <i>M</i>. <i>smegmatis</i> cell lysates, followed by pull-down using Nickel-agarose beads.</p

PknG-catalyzed phosphorylation of L13 at a mycobacterial specific site, T11, is required for mycobacterial biofilm growth.

<p>(<b>A</b>) Sequence alignment of the N-terminal 20 amino acids of L13 proteins from different bacteria. Residues marked with asterisks (T11, T12, and S14) are potential targets of phosphorylation by PknG. (<b>B</b>) Phosphorylation of L13 and its mutants by PknG. In L13(3A), all three residues (T11, T12, and S14) were mutated to alanine. Inhibition was achieved by pre-incubation of PknG in 1 mM AX20017 (AX). (<b>C</b>) Chromatograms confirming wild type and mutant alleles of <i>rplM</i> on <i>M</i>. <i>smegmatis</i> chromosomes. The chromosomal loci were amplified from genomic DNA of <i>M</i>. <i>smegmatis</i> strains by primers that anneal to DNA sequences outside the regions homologous to the allelic exchange substrates, followed by cloning and sequencing. (<b>D</b>) Biofilm of <i>M</i>. <i>smegmatis</i> strains. Similar to <i>Ms</i>Δ<i>pknG</i> and <i>Ms</i>Δ<i>renU</i>, <i>Ms</i>.<i>L13(T11A)</i> exhibited defective biofilm growth, while biofilm of <i>Ms</i>.<i>L13(T11E)</i> was largely identical to wild type. <i>In trans</i> expression of an allele encoding L13(T11E) restored biofilm growth to <i>Ms</i>.<i>L13(T11A)</i> strain. (<b>E</b>) Quantitation of biofilm growth of <i>M</i>. <i>smegmatis</i> strains. The biofilm biomass was harvested and quantified by determining total protein per plate. Error bars represent standard deviations of biological triplicates. Statistical significances of differences were analyzed using Students <i>t</i>-test; ns, not significant difference. (<b>F</b>) Biofilm of <i>Mtb</i> strains. Similar to <i>Mtb</i>Δ<i>pknG</i> and <i>Mtb</i>Δ<i>renU</i>, <i>Mtb</i>.<i>L13(T11A)</i> exhibited defective biofilm growth while <i>in trans</i> expression of an allele encoding L13(T11E) restored its biofilm growth. Addition of PknG inhibitor AX20017 (+AX) had no effect on the biofilm of the complemented strain. (<b>G</b>) Quantitation of biofilm growth of <i>Mtb</i> strains. The biofilm biomass was harvested and quantified by determining total protein per plate. Error bars represent standard deviations of biological triplicates. Statistical significances of differences were analyzed using Students <i>t</i>-test; ns, not significant difference.</p

Both <i>pknG</i> and its adjacent gene <i>renU</i> are each required for oxidative stress resistance.

<p>(<b>A</b>) Alignment of the <i>pknG</i> loci from <i>Mtb</i> and <i>M</i>. <i>smegmatis</i>. <i>renU</i> (previously annotated as <i>mutT3</i>) shares the same intergenic region with the operon encoding <i>pknG</i>. Bar, 1kb. (<b>B</b>) Both <i>pknG</i> and <i>renU</i> are each required for <i>M</i>. <i>smegmatis</i> resistance to H₂O₂ (left) and diamide (right). Wild type <i>M</i>. <i>smegmatis</i> (circles), <i>Ms</i>Δ<i>pknG</i> (triangles) and <i>Ms</i>Δ<i>renU</i> (squares) were grown in 7H9 medium. At the indicated times (arrows), 10mM H₂O₂ or 15mM diamide was added. Growth was estimated through optical absorbance at 600 nm (OD_600nm). Error bars represent standard deviation of biological triplicates. (<b>C</b>) <i>pknG</i> and <i>renU</i> are each required for <i>Mtb</i> resistance to H₂O₂ (left) and diamide (right). Wild type <i>Mtb</i> (circles or striped bars), <i>Mtb</i>Δ<i>pknG</i> (triangles or black filled bars) and <i>Mtb</i>Δ<i>renU</i> (squares or grey filled bars) of were grown in 7H9-OADC medium. At the indicated times (arrows), 20 mM H₂O₂ or 10 mM diamide was added. Growth was estimated through measuring optical absorbance at 600 nm (OD_600nm, top) or determining colony forming units (CFU, bottom) by serial dilution plating. Error bars represent standard deviation of biological triplicates.</p

<i>renU</i> encodes a Nudix hydrolase required for biofilm growth.

<p>(<b>A</b>) Relative Nudix hydrolase activity of RenU on a substrate panel (left). Nucleoside diphosphate derivatives (NDPX) are preferred substrates compared to nucleoside triphosphates (NTP). A catalytically-inactive mutant of RenU (RenU^DEAD) protein, in which 3 glutamate residues (E74, E77, and E78) in the Nudix box were mutated to alanines, exhibits no phosphatase activity towards the preferred substrates (right). (<b>B</b>) Kinetics studies of Nudix hydrolase activity of RenU on the three NDPXs as preferred substrates ADP-ribose, FAD, and NADH. (<b>C</b>) Rate of RenU catalytic activity on NADH compared to its oxidative form NAD⁺. Fit curve is shown for NADH. (<b>D</b>) The Nudix hydrolase activity of RenU is required for <i>M</i>. <i>smegmatis</i> biofilm growth. Wild type <i>M</i>. <i>smegmatis</i>, <i>Ms</i>Δ<i>renU</i>, and the mutant strains completed with wild type RenU or RenU^DEAD were assayed for biofilm growth. Whereas <i>renU</i> fully restored biofilm growth to <i>Ms</i>Δ<i>renU</i>, <i>renUDEAD</i> failed to complement the mutant. Shown images are representatives of biological triplicates. (<b>E</b>) The Nudix hydrolase activity of RenU is required for <i>Mtb</i> biofilm growth. Wild type <i>Mtb</i> H37Rv, <i>Mtb</i>Δ<i>renU</i>, and the mutant strains completed with wild type RenU or RenU^DEAD were was assayed for biofilm growth. Whereas <i>renU</i> fully restored biofilm growth to <i>Mtb</i>Δ<i>renU</i>, <i>renUDEAD</i> failed to complement the mutant. Shown images are representatives of biological duplicates. (<b>F</b>) Quantitation of biofilm growth of <i>Mtb</i> strains. The biofilm biomass was harvested and estimated by determining total protein per plate. Error bars represent standard deviation of biological triplicates. Statistical significances of differences were analyzed using Students <i>t</i>-test; ns, not significant difference.</p

A UDP-X Diphosphatase from <i>Streptococcus pneumoniae</i> Hydrolyzes Precursors of Peptidoglycan Biosynthesis

<div><p>The gene for a Nudix enzyme (<i>SP_1669</i>) was found to code for a UDP-X diphosphatase. The <i>SP_1669</i> gene is localized among genes encoding proteins that participate in cell division in <i>Streptococcus pneumoniae.</i> One of these genes, <i>MurF</i>, encodes an enzyme that catalyzes the last step of the Mur pathway of peptidoglycan biosynthesis. Mur pathway substrates are all derived from UDP-glucosamine and all are potential Nudix substrates. We showed that UDP-X diphosphatase can hydrolyze the Mur pathway substrates UDP-N-acetylmuramic acid and UDP-N-acetylmuramoyl-L-alanine. The 1.39 Å resolution crystal structure of this enzyme shows that it folds as an asymmetric homodimer with two distinct active sites, each containing elements of the conserved Nudix box sequence. In addition to its Nudix catalytic activity, the enzyme has a 3′5′ RNA exonuclease activity. We propose that the structural asymmetry in UDP-X diphosphatase facilitates the recognition of these two distinct classes of substrates, Nudix substrates and RNA. UDP-X diphosphatase is a prototype of a new family of Nudix enzymes with unique structural characteristics: two monomers, each consisting of an N-terminal helix bundle domain and a C-terminal Nudix domain, form an asymmetric dimer with two distinct active sites. These enzymes function to hydrolyze bacterial cell wall precursors and degrade RNA.</p></div

Correlation of NADH and RHOCS, and role of L13 phosphorylation by PknG.

<p>(<b>A</b>) Induction of PknG expression in <i>M</i>. <i>smegmatis</i>. Western analysis was used to detect PknG expression following the exposure of wild type <i>M</i>. <i>smegmatis</i> cultures (OD₆₀₀ of 2) to various oxidative stimuli including NADH (upper) and FAD (lower) for 30 minutes. All chemicals were used at 10 mM except for bleomycin, which was used at 10 μg/ml. Samples were separated on SDS-PAGE, followed by immunodetection using an anti-PknG antibody or an anti-DivIVA antibody, as a control. Non-induced lysates from wild type <i>M</i>. <i>smegmatis</i> and <i>Ms</i>Δ<i>pknG</i> were used as controls. NADH uniquely induced expression of PknG. (<b>B</b>) Titration of the induced PknG expression by increasing NADH concentrations (0–30 mM) for 30 minutes, followed by Western analysis using anti-PknG antibody. (<b>C</b>) Time course of PknG expression (0–60 minutes) following cell exposure to 10 mM NADH. Detection of PknG was similar to (A) and (B). (<b>D</b>) Quantitation of cellular NADH (top), NAD⁺ (middle), and FAD (bottom) levels following oxidative stress induced by H₂O₂. <i>M</i>. <i>smegmatis</i> cells were exposed to 1 mM H₂O₂ for 1 hour. Bars show means with standard deviations from 3–6 biological repeats. *, p < 0.0001; ns, not significant relative to wild type <i>M</i>. <i>smegmatis</i>). (<b>E</b>) Effect of PknG-catalyzed phosphorylation of L13 on its association with RenU in the cytoplasm. Expression of PknG in <i>M</i>. <i>smegmatis</i> strains was induced by NADH. Cells were disintegrated by French Press, followed by ultracentrifugation to remove ribosomes. RenU.6H was added to the non-ribosomal fraction, followed by pull-down using Cobalt-agarose beads. The presence of L13 in the pulled down materials was detected by Western analysis using anti-L13 antibody. (<b>F</b>) Effect of L13(T11E), a phosphorylation-mimic form of L13, on <i>in vitro</i> NADH hydrolytic activity of RenU. Initial rates from a continuous fluorescence excitation assay were fit by nonlinear least squares to the Michaelis-Menten equation to determine K_m and V_max values for RenU. Reaction was performed at 37°C. Error bars represent standard deviations of triplicates. The extent of the uncatalyzed reaction was ~10% of the RenU catalyzed reaction. (<b>G</b>) Effect of L13(T11E) on the catalytic activity of RenU. In the presence of L13(T11E), a 20.6% increase in V_max was observed (p < 0.05x10^-3), whereas K_m, reflecting the binding affinity of RenU to NADH, was not affected by L13(T11E).</p

Intracellular trafficking and survival of <i>M</i>. <i>tuberculosis</i> RHOCS mutants.

<p>(<b>A</b>) Intracellular survival of <i>Mtb</i> strains. Macrophages were infected with <i>Mtb</i> strains for 3 hours, followed by 0 or 72-hour chase. CFUs were counted after 4–5 weeks of growth at 37°C. Bars represent percentages of CFUs remaining at 72-hour compared to 0-hour time point. Error bars represent standard deviations from 3–6 repeats. *, p < 0.001; ns, not significant relative to <i>Mtb</i> H37Rv; #ns, not significant between the two indicated groups. Order of strains is as in 7B. (<b>B</b>) Quantitative analysis of lysosomal delivery following phagocytosis of <i>Mtb</i> strains by macrophages. Macrophages were infected with FLUOS-stained <i>Mtb</i> strains for 1 hour, followed by 16-hour chase. Infected macrophages (see C below) were used for quantitation. Biological triplicates of 50 events were counted for each <i>Mtb</i> strain. Error bars represent standard deviations. Error bars represent standard deviations from 3–6 repeats. *, p < 0.0001; ns, not significant relative to <i>Mtb</i> H37Rv; ** ns, not significant between the two indicated groups. (<b>C</b>) A model depicting activity and function of RHOCS in mycobacteria. PknG was previously shown to de-repress the TCA cycle through its phosphorylation of GarA, an inhibitor of α-ketoglutarate decarboxylase and glutamate dehydrogenase. Increased TCA cycle activities, hypoxia, or impaired oxidative phosphorylation (OXPHOS), lead to elevated NADH levels. To protect mycobacterial cells against the change in redox status, PknG expression is up-regulated, leading to the signaling cascade including L13 and RenU, which degrades NADH and FAD and restores their optimal level. AMP, adenosine monophosphate, FAD, flavin adenine dinucleotide; FMN, flavin mononucleotide; NMNH, nicotinamide mononucleotide.</p

PknG kinase activity is required for biofilm growth in mycobacteria.

<p>(<b>A</b>) Role of <i>pknG</i> in <i>M</i>. <i>smegmatis</i> planktonic growth. Wild type <i>M</i>. <i>smegmatis</i> mc²155 (red filled circles), its derived <i>Ms</i>Δ<i>pknG</i> mutant (blue filled triangles), and the complemented strain <i>Ms</i>Δ<i>pknG</i>/<i>pknG</i> (green filled squares) were grown in 7H9 medium supplemented with 0.2% glucose with shaking at 200 r.p.m. and 37°C. Growth was assessed by measuring optical absorbance at 600 nm. Error bars represent standard deviation of biological triplicates. Differences between wild type and <i>Ms</i>Δ<i>pknG</i> in stationary phase are not significant. (<b>B</b>) Role of <i>pknG</i> in <i>Mtb</i> planktonic growth. Wild type <i>Mtb</i> H37Rv (open circles) and its derived <i>Mtb</i>Δ<i>pknG</i> mutant (open triangles) were grown in 7H9-OADC medium with 0.2% glucose (blue) or 1% glucose (red). Cultures were shaken at 200 r.p.m. and 37°C. Growth was assessed by measuring optical absorbance at 600 nm. Error bars represent standard deviation of biological triplicates. Differences between wild type and <i>Mtb</i>Δ<i>pknG</i> in stationary phase (5–11 hours) are statistically significant (two-tailed <i>t</i>-test, p<0.05). (<b>C</b>) <i>pknG</i> is required for <i>M</i>. <i>smegmatis</i> biofilm growth. Wild type <i>M</i>. <i>smegmatis</i>, <i>Ms</i>Δ<i>pknG</i>, and the mutant strains complemented with the <i>M</i>. <i>smegmatis</i> (<i>Ms-pknG</i>) or <i>M</i>. <i>tuberculosis</i> (<i>Mtb-pknG</i>) gene. Pictures were taken after 7 days of static growth at 30°C. Shown images are representatives of biological triplicates. (<b>D</b>) <i>pknG</i> is required for <i>Mtb</i> biofilm growth. Wild type <i>Mtb</i> H37Rv, <i>Mtb</i>Δ<i>pknG</i>, and the complemented strain were assayed as previously described [<a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1004839#ppat.1004839.ref010" target="_blank">10</a>]. Pictures were taken after 6 weeks of growth at a static humidified condition of 37°C and 5% CO₂. Shown images are representatives of biological triplicates. (<b>E</b>) Quantitation of biofilm growth of <i>Mtb</i> strains. Biofilms were harvested and quantified as described in Experimental Procedures. Error bars represent standard deviation of biological triplicates (*, p<0.0001; ns, not significant difference between wild type H37Rv and the complemented strain). (<b>F</b>) PknG kinase activity is required for <i>Mtb</i> biofilm growth. Wild type <i>Mtb</i> H37Rv, <i>Mtb</i>Δ<i>pknG</i>, and the complemented strain were assayed in the absence (-) or presence (+) of 1 mM AX20017, a specific inhibitor of PknG. Pictures were taken after 6 weeks of growth at static humidified condition of 37°C and 5% CO₂. Shown images are representatives of biological triplicates.</p