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The parents of Evan Ebel, who signed his name "Evil Evan" and is a suspect in two murders, had a "bad streak" that his parents "tried desperately to correct," Colorado Gov. John Hickenlooper said.
Hickenlooper is a friend of Ebel's father Jack Ebel, whose son is the prime suspect in the killing of a pizza delivery man and Colorado's prisons chief Tom Clements, a murder that brought Hickenlooper close to tears this week.
A shootout with Texas cops that killed Ebel Thursday came despite efforts by his parents that literally spanned the globe to curb an increasingly paranoid and violent son.
A blog maintained by Ebel's mother, Jody Mangue, documented a son troubled from youth who was sent to behavioral programs in Jamaica, Samoa, Mexico and Utah.
Even when he was prison, his parents tried to rescue him. Jack Ebel testified in March 2011 before a committee of the Colorado Legislature regarding a proposal that would require inmates to spend time outside of solitary confinement before leaving prison.
"What I've seen over six years is he has become increasingly ... he has a high level of paranoia and [is] extremely anxious. So when he gets out to visit me, and he gets out of his cell to talk to me, I mean he is so agitated that it will take an hour to an hour-and-half before we can actually talk," Jack Ebel told lawmakers.
The governor on Friday issued a statement that sympathized with his friend Jack Ebel whose son may have killed his other friend, Clements.
"Jack is one of the most kind and generous people I know. His son had a bad streak that I know he tried desperately to correct," Hickenlooper said.
"Although Jack loved his son, he never asked me to intervene on his behalf and I never asked for any special treatment for his son," the governor said.
Ebel dropped out of school, where he had been in a special education program for "severely impacted" students. Friends said he "lost it" when his sister, Marin Ebel, was killed in a car crash as a teenager in 2004. The death seemed to set off a string of criminal behaviors and jail stints for Ebel.
The parents haven't spoken publicly since their son's death, but in an undated post on her blog when Ebel was still in prison his mother hinted at the responsibility the parents felt for his criminal life.
"Some people may blame us for what has happened to Evan. I can only say that his dad and I had to make hard decisions when he was younger hoping to avoid where he is now," she wrote.
While the parents remain silent, Hickenlooper told the Denver Post his old friend was devastated.
Evan Ebel, 28, was paroled on Jan. 28 after serving his full prison term. His most recent sentence had been four years for punching a prison guard in 2008, according to state records obtained by the AP.
Authorities suspect Ebel shot and killed Clements at his Monument, Colo., home.
During a traffic stop on Thursday, authorities in Texas said Ebel, who was driving a black Cadillac with Colorado license plates, fired several times at an office and took off, leading police on a chase that at times exceeded 100 mph.
Ebel wasn't stopped until his vehicle slammed into an 18-wheel tractor trailer. He came out of the car with a gun and began firing again at police who cut him down with return fire, police said.
"He wasn't planning on being taken alive. He was planning on hurting somebody," an officer said after the shooting.
Gov. Hickenlooper Was Friends With Suspected Killer's Dad
Shell casings collected after the chase matched casings found in Clements' home, according to a Texas police search warrant affidavit obtained by ABC News.
Authorities said there is no clear motive for the Texas shootout, but they believed the Cadillac was pulled over as part of a drug stop. They were looking into Ebel's alleged affiliation with the prison gang 211 Crew to help explain why he was in Texas.
Ebel, who had the word "hopeless" tattooed on his body and was known to sign his name, "Evil Evan," died Friday of wounds.
Texas authorities said evidence found in the suspect's car -- including a Domino's pizza uniform jacket and a cardboard pizza box -- may also link him to the unsolved murder of Nathan Leon, 27, who was killed delivering pizza near Golden, Colo., last Sunday.
Friends of Ebel, who grew up in Wheat Ridge, Colo., told ABC News that he had been depressed and on edge for years. He had been in prison on an assortment of assault, robbery and menacing charges dating back to 2005, according to jail records.
"He was depressed a lot," Ryan Arici, a friend of Ebel's from Wheat Ridge, told ABC News. "And he was a dark person. His walls were painted black and his windows were blacked out."
ABC News' Colleen Curry and Clayton Sandell contributed to this report


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Timing Belts, Timing Chains & Components at 1A Auto
What is a timing chain or belt and where is it located?
The timing chain is the most important part of an automotive's engine timing system. Typically located on the front of the engine, the timing chain helps the crankshaft rotate in sync with the camshaft. While most older automobiles required one timing chain since they only had one camshaft inside the engine block, today's more complex overhead cam motors can have as many as four cams and multiple chains.
Much like the timing chain, timing belts connect the camshaft to the crankshaft. However, instead of being a metal link design, timing belts are made of reinforced rubber. In order to fit in tighter and more intricate engine designs, the timing belt system may incorporate the use of multiple hydraulic or mechanical timing belt tensioners, and belt tensioner or idler pulleys.
For an even more in-depth explanation of mechanical timing, belts, chains, how they each work, why they fail, maintenance and replacement tips and more, check out our timing belts Vs. timing chains guide.
How do I know the timing belt or chain needs to be replaced?
Since timing belts are rubber, they do require scheduled maintenance. The typical service life of an engine timing belt, at which point a replacement is required, is between 50,000 and 70,000 miles, but you should refer to manufacturer's specifications for your specific application. A worn engine timing belt can break, causing a complete failure of the engine to operate. In vehicles equipped with Interference Engines, this could mean severe damage to the motor since the pistons and valves are only kept from colliding with each other by the timing belt's movement. If the timing belt has been damaged prior to the recommended interval, a timing belt replacement must be obtained and installed immediately.
As for the timing chain, it's rare for them to break. They can often last as long as the engine itself, but this makes them prone to stretch out over time. To compensate, some automotives use timing chain tensioners to eliminate slack in the chain. But if the stretch is too far out of specification, or if the tensioner fails, the camshafts and crank will not be properly aligned with each other. As a result, the engine will lose power and typically trigger the "Check Engine" light on the dashboard. If there is something wrong with the timing chain and a replacement is required, this will need to be taken care of as soon as possible in order to get the car back on the road again.
Can I replace the timing belt or chain myself?
Timing belt and chain replacement can vary from vehicle to vehicle. It's important to seek out the proper information regarding your engine to see what tools you may need. These can range from a three-jaw gear puller to a harmonic balancer puller to a torque wrench. Some vehicles will require these types of tools in order to hold the camshafts and tensioner in place while the timing components are swapped out. We would recommend the advanced do-it-yourselfer who most likely has these tools readily available for this job, otherwise it's a good idea to leave it to the professionals to prevent any potential engine damage.
If you are tackling this job yourself, it's a good idea to have a service guide specific to your vehicle handy. The steps are not the same for every vehicle, but the process usually requires a disconnection of the negative battery cable, raising the vehicle on jack stands, draining the coolant, removing the air intake, disconnecting a few radiator and heater hoses, removing the radiator fan, loosening the serpentine belt, removing the water pump, removing the pulleys, removing the front timing cover, removing the harmonic balancer pulley, and, finally, pulling off the timing belt/chain. To prevent yourself from having to repeat much of the repair, we recommend replacing the water pump as well.


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COOKEVILLE, Tenn. --- When sophomore Allison May stepped in to serve for Tennessee Tech late in the fourth set it looked as though the fifth set was looming. Murray State had built a 24-20 lead and had kept the advantage for the majority of the set.
May had other plans as she tallied two service aces, to go along with kills from junior Kelsey Gray and sophomore Natalie Penrod, to knot up the score at 24. A tech service error gave the Racers the lead again at 25-24, but a pair of kills from Penrod and a Murray State attack error thwarted the Racers comeback as Tech secured the victory.
Along with her solid performance in the closing moments of the match, Penrod was just as effective throughout the match, notching a match-high 24 kills, which included at least four kills in every set.
Freshman Ellen Conti also made an impact on the Tech offense, chipping in with a total of 13 kills, all of which came in the final three sets.
While Penrod and Conti led the Tech offense, it was Gray who led the Golden Eagles on defense as she recorded a total of nine blocks, including a career-high seven solo tallies.
Gray was joined by three other Golden Eagles that put up a consistent wall against the Murray State attackers. Freshman Madison Keyes matched her career-high with five total blocks, while junior Katrina Dvorken and Penrod posted four total blocks each.
Junior Madison Comperry also made her presence felt as she dished out a match-high 42 assists. Her second set performance was the strongest, as she tallied 12 assists, while the Tech hitters combined for a .257 hitting percentage.
On the other side of the net the Racers were led by Lia Havili who tallied a team-high 39 assists to go along with 11 digs. Cassy Woolverton notched a Racer-high 13 kills, while Kayleigh Sauer added a match-high 22 digs.
The Golden Eagles (4-20, 3-11 OVC) start a four-match road trip Friday, beginning with the Austin Peay Lady Govs, at 7 p.m.


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Discussion in 'Boating - Anything to do with Boating' started by Rich M, Jul 30, 2018.


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Re: Star Wars: The Clone Wars series
I still don't think SW should be reduced to killing, meyhem, and destruction for the pleasure of bloodthirsty little boys.
I don't think it's an overreaction, but sure, I'll wait till it's out on tv then say "I told you so" again.
let us not forget that is actually called "star wars" and not "star romance"...etc.
but one thing that lucas is always adamant about(from interviews ive heard), he will not let it
become sensless violence. althought the prequels were plagued with shoddy acting and too much cgi
there is always a drama and friendships grown and lost etc.
never has it been just violence.
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Gola Classics introduces Varsity, a classic lace up plimsoll. This navy trainer features a canvas upper and a tonal wingflash. The style is completed with white Gola branding to the wingflash and heel counter This relaxed style looks great when work with jeans and is a definite wardrobe essential for the warmer months.
All footwear styles listed on the site are US sizes. We do not currently offer half sizes or varying width fittings. Due to the nature of footwear production fit can vary from style to style, we can not guarantee a style will fit an individual.
Delivery Information
Orders are processed and delivered Monday-Friday (excluding national holidays), we do not despatch or deliver on weekends. Orders placed after 2pm (EST) will be processed the following working day. Unfortunately we can not be held responsible for deliveries that are late. We now offer FREE delivery on all US orders (where service is available). A single order can include up to 10 items for which the shipping prices are fixed. Delivery times are a guide only and are not guaranteed.
Returns Policy
To arrange a return please contact us on email@example.com, within 14 days of receiving your order, stating the reason for return and your order reference. Items must be returned unused and in their original packaging or we will not be able to accept the item. You will be liable for any charges in relation to returning any unwanted items unless the item is faulty. Please do not return items to our warehouse without contacting us first as it may not be accepted and we can NOT reimburse any carriage costs you have incurred as a result of this.


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Our accommodation is beautifully furnished, well appointed and has a warm family atmosphere.
Linda Sugars has been welcoming visitors to the North York Moors for 25 years and has upgraded the facilities constantly to ensure that Sevenford House remains among the best Bed and Breakfast in North Yorkshire.
Rosedale Abbey lies at the centre of the North York Moors National Park, 10 miles from Pickering, allowing easy access to many of the region's most famous attractions and historic sites. The beautiful and tranquil scenery of the moors can be explored on the numerous walking routes and from nearby Pickering or Goathland stations you can catch the North Yorkshire Moors Steam Railway for a unique view of the countryside and its villages.
Bed and breakfast accommodation includes three beautifully decorated guest rooms, a guest lounge with library and open coal fire, a dining room with grand piano that visitors are welcome to use and an information point with tips on things to see and do in North Yorkshire. The nearly four acre landscaped gardens are open to visitors, include a private summer house and offer beautiful views of Rosedale Abbey.
The Coach House is a self catered 3 bedroom holiday cottage with it's own private garden and patio area and offers a more independent option for large families or groups of friends to spend a week or two delving deeper into the Yorkshire Moors.
The menu links above have detailed information on our facilities, photos of the house, gardens and rooms. Visit Ryedale – North Yorkshire Tourism Website has great information on things to do in Ryedale and North Yorkshire, and RosedaleAbbey.com is another excellent reference for the many great local restaurants, pubs and things to see and do in and around Rosedale Abbey.
+Linda Sugars


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Dolphins owner Stephen Ross hopes to renovate SunLife Stadium via a project that relies in part on taxpayer dollars. But he lacks the leverage of an express or implied threat of relocation, which has helped plenty of other teams shake taxpayer dollars out of the tree.
Ross says he won't be using that leverage.
Specifically, Ross told Monday's Pro Football Talk that he won't be expressly or implicitly threatening relocation in order to get a public contribution toward the renovation effort.
"Having been brought up in South Florida and loving the Dolphins I would find that very hard to do," Ross said. "I mean, some times when you're dealing with adversity and people are giving you a hard time you say, 'Hey I'm not bound to stay here.' But really I'm not threatening anybody. The Dolphins belong in South Florida and they are going to stay in South Florida?"
That's good news for Dolphins fans, but perhaps bad news for the effort to get Dolphins and non-Dolphins fans alike to support the effort to raise local hotel taxes by one percent. Ross obviously believes they should.
"We are the only stadium in the country I think, especially in the NFL, that doesn't have one penny of public funds and the only stadium that pays full real estate taxes," Ross explained. "In fact, I think all the rest of them pay no real estate taxes at all, and we're paying full taxes. So that sets us behind and makes us less competitive in that standpoint.
"I'm prepared to put a lot of money into it," Ross said of the renovation effort. "But I think everywhere you see a new stadium you see the public supporting it. And that is one thing Miami has not been supported in Miami. They did the Marlins stadium and that's probably one of the major reasons we're having problems."
Ross is referring to the fact that the local MLB team finagled public funds in part with a promise to make and keep the team competitive — and then the Marlins embarked on a major fire sale only one year in the new building. So how can the Dolphins overcome that?
"It's about jobs, it's about economic development, and you can see all the great events that would otherwise come to Miami," Ross said. "The NFL wants the Super Bowl to come to Miami. It's the warmest weather at that time of the year. You know, with a great stadium they are going to want to come there more and more. The financial impact of that whole community of South Florida with a Super Bowl is enormous. Then you add to that the BCS Bowls and we're going to be announcing several other types of events, soccer and other types of events this year that we can bring if we had a first-class stadium."
Meanwhile, Ross is working toward putting a first-class team into his current second-class stadium, via several big-money free agency signings.
"I bought the team because I wanted to win," Ross said. "I've always listened. I don't go out there and say I want this guy I want that guy. You listen to your football people. They thought this was the time, and they put a great plan together and I've encouraged them. I said, 'Guys let's make some moves, it's time to win.' I think the results you've seen. You could see from the interviews, from the free agents that we have kept and signed and the new ones of how they felt about Miami. It's kind of like a new beginning if you will and it makes it somewhat exciting."
Excitement is the key to getting people to come to the current stadium and to support the efforts to renovate it. Ultimately, putting a consistent winner on the field is the best way to finagle long-term excitement.This video is no longer available. Click here to watch more NBC Sports videos!


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The Municipal Association of South Carolina represents and serves the state's 269 incorporated municipalities. It was formed in 1939 to offer the services, programs and tools that will give municipal officials the knowledge, experience and tools for enabling the most efficient and effective operation of their municipalities in the complex world of municipal government.
This association provides staff support for 11 affiliated associations. It is through these affiliated associations that the association accomplishes its goal of providing training opportunities to municipal employees. The following affiliations are a few examples that are applicable to our town:
• Association of South Carolina Mayors: Provides opportunities for South Carolina mayors to engage in advocating for issues that affect our town; to network, share ideas and best practices with other mayors; and to take part in educational activities. I am fortunate to have been elected to the board of this association and will continue to bring the issues of our town to the forefront.
• The Municipal Court Administration Association: Focuses on fostering/developing interest in sound court administration, encouraging the most advanced technologies in the field of court administration, disseminating information and exchanging ideas among its members. The association offers training and access to a listserve to court administrators, clerks of court, municipal judges, and other city or town employees involved in court administration.
• The S.C. Municipal Finance Officers, Clerks & Treasurers Association: Offers training programs throughout the year covering the wide range of responsibilities of finance officers, clerks and treasurers. MFOCTA offers a listserve to its members and co-sponsor of the Municipal Clerks and Treasurers Institute.
• The Municipal Technology Association of S.C.: Promotes the effective use of information technology by municipalities. The affiliate offers members access to a listserve and training on how to provide better services and achieve greater efficiencies using the latest technological innovations for municipal government.
• The S.C. Association of Stormwater Managers: Offers a listserve and training sessions about stormwater management policies and best practices. It also provides members a forum for sharing ideas and keeping current on state and federal laws
• The S.C. Community Development Association: Provides a forum for learning about the community development process. Its members include municipal, county, regional and state community development professionals; employees of private companies with an interest in community development; elected officials and volunteers.


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- Publish date:
- Updated on
- Year
- 1978
OSU fires coach Woody Hayes for attacking an opposing player
On December 30, 1978, Ohio State University (OSU) makes the decision to fire its 65-year-old football coach, Woody Hayes, one day after Hayes punched a player on the opposing team near the end of the Gator Bowl.
Born in Clifton, Ohio, in 1913, Hayes played college football at Denison University before serving in the U.S. Navy during World War II, rising to the rank of lieutenant commander. He returned to his alma mater as a coach upon his discharge in 1946. After three seasons with Denison and two with the Redskins of Miami University of Ohio, Hayes took the head coaching position at OSU in 1951. In his 28 seasons with the Buckeyes, Hayes compiled an overall record of 238-72-10, including 13 Big Ten titles, four national championships, and four appearances in the Rose Bowl. His 238 wins placed him ninth on the all-time list of top NCAA Division I coaching victories (as of 2007), and he was inducted into the College Football Hall of Fame in 1983.
Despite his prodigious coaching ability, Hayes (who died in 1987) is also remembered for his volatile temper and violent outbursts, which sometimes threatened to overshadow his teams' performance on the field. The most egregious example came on December 29, 1978, during the Buckeyes' 15-17 loss to the Clemson Tigers in the Gator Bowl. With OSU down by two points in the closing seconds of the game, Clemson linebacker Charlie Bauman intercepted a pass and was knocked out of bounds on the Buckeyes' sidelines. As Bauman was getting up, Hayes punched him in the throat, after which he was restrained by several OSU players. On December 30, an embarrassed OSU administration fired Hayes, who would never coach again.


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- Research article
- Open Access
The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions
© Thomas et al.; licensee BioMed Central Ltd. 2012
- Received: 28 November 2011
- Accepted: 13 March 2012
- Published: 14 May 2012
It is well known that the deterministic dynamics of biochemical reaction networks can be more easily studied if timescale separation conditions are invoked (the quasi-steady-state assumption). In this case the deterministic dynamics of a large network of elementary reactions are well described by the dynamics of a smaller network of effective reactions. Each of the latter represents a group of elementary reactions in the large network and has associated with it an effective macroscopic rate law. A popular method to achieve model reduction in the presence of intrinsic noise consists of using the effective macroscopic rate laws to heuristically deduce effective probabilities for the effective reactions which then enables simulation via the stochastic simulation algorithm (SSA). The validity of this heuristic SSA method is a priori doubtful because the reaction probabilities for the SSA have only been rigorously derived from microscopic physics arguments for elementary reactions.
We here obtain, by rigorous means and in closed-form, a reduced linear Langevin equation description of the stochastic dynamics of monostable biochemical networks in conditions characterized by small intrinsic noise and timescale separation. The slow-scale linear noise approximation (ssLNA), as the new method is called, is used to calculate the intrinsic noise statistics of enzyme and gene networks. The results agree very well with SSA simulations of the non-reduced network of elementary reactions. In contrast the conventional heuristic SSA is shown to overestimate the size of noise for Michaelis-Menten kinetics, considerably under-estimate the size of noise for Hill-type kinetics and in some cases even miss the prediction of noise-induced oscillations.
A new general method, the ssLNA, is derived and shown to correctly describe the statistics of intrinsic noise about the macroscopic concentrations under timescale separation conditions. The ssLNA provides a simple and accurate means of performing stochastic model reduction and hence it is expected to be of widespread utility in studying the dynamics of large noisy reaction networks, as is common in computational and systems biology.
- Langevin Equation
- Intrinsic Noise
- Diffusion Matrix
- Fano Factor
- Stochastic Simulation Algorithm
Biochemical pathways or networks are typically very large. A well-characterized example is the protein-protein interaction network of the yeast Saccharomyces cerevisiae with approximately a thousand putative interactions involving an approximate equal number of proteins[1]. It is also a fact that a significant number of species are found in low copy numbers in both prokaryotic and eukaryotic cells[2, 3]. Recent mass spectrometry-based studies have, for example, shown that 75% of the proteins in the cytosol of the bacterium Escherichia coli appear in copy numbers below 250 and the median copy number of all identified proteins is approximately 500[3]. This means that simulation methods intended to realistically capture the inner workings of a cell have to (i) be stochastic to take into account the significant intrinsic noise associated with low copy number conditions; (ii) be able to simulate fairly large networks in a reasonable amount of time. The stochastic simulation algorithm (SSA)[4] has been and still is the algorithm of choice for a large number of studies exploring the role of noise in biology. The advantage of the algorithm is that it is exact, i.e., it exactly samples the trajectories of the stochastic process described by the chemical master equation (CME), the accepted mesoscopic description of chemical kinetics. Its disadvantage is that it simulates every reaction event and hence is not particularly suited for the study of large networks[5]. This problem is an outstanding challenge in the fields of computational and systems biology.
A common way of circumventing the problem is to simulate a network of species which is much smaller than the size of the full network but which nevertheless captures the essential dynamics. For example, the three elementary (unimolecular or bimolecular) reactions which describe the enzyme-assisted catalysis of substrate S into product P via the Michaelis-Menten reaction,, can be replaced by a single effective reaction. Note that here E and C denote the free enzyme species and the enzyme-substrate complex species, respectively. The latter first-order reaction is non-elementary, i.e., it can be broken down into a set of fundamental elementary reactions. The implicit assumption in this lumping or coarse-graining method is that the transients in the average concentrations of some species decay over much longer timescales than those of the rest of the species. Hence, one can argue that the relevant network to be simulated is that involving the slowly varying species only. In the Michaelis-Menten example, the fast species were the enzyme and the complex and the slow species are the substrate and product. The dynamics of this reduced network are of course only a faithful approximation of those of the full network, the Michaelis-Menten reaction, whenever the rate constants guarantee reasonable timescale separation.
On the macroscopic level, where molecule numbers are so large that intrinsic noise can be ignored, there is a well-known practical recipe for obtaining this reduced or coarse-grained network from the full network of elementary reactions. One writes down the rate equations (REs) for each species, decides which species are fast and slow, sets the time derivative of the concentration of the fast species to zero, solves for the steady-state concentrations of the fast species and finally substitutes these concentrations into the equations for the slow species. This procedure is the deterministic quasi-steady-state assumption (QSSA). The result is a set of new REs for the slow species only; corresponding to these reduced equations is the coarse-grained network, i.e., the network of reactions between slow species whose macroscopic rate laws are dictated by the new REs. Generally, all coarse-grained networks will have at least one reaction which is non-elementary; however those reactions involving the interaction of only slow species in the full network will naturally also remain elementary in the coarse-grained network. The deterministic QSSA presents a rigorous method of achieving a coarse-grained macroscopic description based on the deterministic REs[6]. Its major shortcoming is that it ignores the inherent stochasticity of the system.
On the mesoscopic level, or, in other words, whenever the size of intrinsic noise becomes comparable with the average molecule numbers, the description of chemical kinetics is given by the CME. One would hope that under conditions of timescale separation, just as one can write effective REs for a coarse-grained network starting from the REs of the full network, in a similar manner one can obtain an effective (or reduced) CME for the coarse-grained network starting from the CME of the full network. The effective REs have information about the macroscopic concentrations of the slow species only, while the effective CME has information about the fluctuations of the slow species only. This line of reasoning has led to a stochastic formulation of the QSSA which is in widespread use. In what follows we concisely review the CME formulation of stochastic kinetics and point out compelling reasons which cast doubt on the validity of the popular stochastic QSSA.
If we are modeling the full network, then the constituent reactions have to be all elementary. For such reactions, the propensity and microscopic rate functions have been derived from molecular physics[7] and hence the CME for the full network is fundamentally correct. Now say that we are modeling a coarse-grained network in which case some reactions are non-elementary. Microscopic considerations do not tell us anything about the form of the propensity functions for such reactions. Rather the propensities and the microscopic rate functions are a heuristic extrapolation of the macroscopic reaction rates at the heart of the effective REs for the non-elementary reactions: is obtained by performing the substitution on f .
Hence it follows that the CME for the coarse-grained network is not rigorously derived from that of the full network under conditions of timescale separation but rather is heuristic and hence its validity is a priori doubtful. Janssen was the first to investigate this question by means of an analytical approach applied to a simple chemical example, the dissociation of N2O5; he showed that "the master equation for a complex chemical reaction cannot always be reduced to a simpler master equation, even if there are fast and slow individual reaction steps"[8]. This suggests that even if the molecules numbers are quite large, the conditions for timescale separation required for the validity of the deterministic QSSA are not generally enough to guarantee the validity of the heuristic CME, a hypothesis which has been recently verified in the context of the Michaelis-Menten reaction with substrate input[9]. In other words, the heuristic CME is not the legitimate stochastic equivalent of the deterministic QSSA, in the sense that it does not correctly describe the statistics of the intrinsic noise about the macroscopic concentrations as given by the reduced REs of the coarse-grained network.
Notwithstanding the fundamental objections of Janssen, and frequently in the name of pragmatism, many studies[10–13] have employed the heuristic CME to obtain a coarse-grained stochastic description of various complex networks. A number of studies[14–17] have reported good agreement between the results of stochastic simulations of the full and coarse-grained networks for enzyme reactions and circadian oscillators which has enhanced faith in the heuristic approach of stochastic modeling of networks with non-elementary reactions and given it the status of a mainstream methodology.
In this article we seek to derive a rigorous alternative to the heuristic approach. Given the CME of the full network of elementary reactions, we derive a reduced linear Fokker-Planck equation (FPE) which describes the noise statistics of the same network when the molecule numbers are not too small and under the same conditions of timescale separation imposed by the deterministic QSSA. This new FPE is the legitimate mesoscopic description of intrinsic noise about the macroscopic concentrations of the coarse-grained network as obtained by the deterministic QSSA. The noise statistics from this approach are compared with stochastic simulations of the full network and with simulations of the coarse-grained network using the conventional heuristic approach. In all cases our approach agrees very well with the full network results. In contrast, we show how the size of intrinsic noise as predicted by the conventional approach can be different by more than an order of magnitude than the actual value and how in some instances this approach even misses the existence of noise-induced oscillations. We also show using our method how one can obtain the regions of parameter space where the conventional approach qualitatively fails and where it fares well.
The article is organized as follows. In the Results section, we discuss in general terms the procedure of obtaining a rigorous mesoscopic description under conditions of timescale separation akin to those of the deterministic QSSA. We then apply this novel method to two different examples: an enzyme mechanism capable of displaying both Michaelis-Menten and Hill-type kinetics and a gene network with a single negative feedback loop. The results from our method are contrasted and compared with stochastic simulations of the full network and with those of the coarse-grained network using the conventional heuristic method. We finish by a discussion. Detailed derivations concerning results and applications can be found in the Methods section.
where is the macroscopic concentration of species i and ε i is proportional to the noise about this concentration. This substitution leads to an infinite expansion of the master equation. The first term, that proportional to Ω1/2, leads to the deterministic equations for the mean concentrations as predicted by the CME in the macroscopic limit of large volumes (or equivalently large molecule numbers). The rest of the terms give a time-evolution equation for the probability density function of the fluctuations,. This partial differential equation is an infinite series in powers of the inverse square root of the volume (see[22] for the general form of this equation). Truncating this series to include only the first term, i.e., that which is proportional to Ω0, leads to a second-order partial differential equation, also called the linear Fokker-Planck equation or the linear noise approximation (LNA)[21, 23, 24]. The solution of the latter equation is a multivariate Gaussian probability distribution and hence expressions for the statistics of intrinsic noise about the macroscopic concentrations, e.g., the variance of fluctuations, can be obtained straightforwardly from this formalism, a distinctive advantage over the CME. The restrictions which must be kept in mind are that this method only provides a reliable approximation to the CME if the molecule numbers are sufficiently large (small intrinsic noise) and the chemical network is monostable (see also the Discussion and conclusion section).
Hence we can now formulate two questions to precisely determine the validity of the heuristic CME in timescale separation conditions: (i) in the macroscopic limit, are the mean concentrations of the heuristic CME exactly given by the reduced REs obtained from the deterministic QSSA? (ii) are the noise statistics about these mean concentrations, as given by the LNA applied on the heuristic CME, equal to the noise statistics obtained from applying the LNA on the CME of the full network? If the heuristic CME is correct then the answer to both these questions should be yes.
The first question can be answered straightforwardly. The deterministic equations for the mean concentrations of the heuristic CME, in the macroscopic limit of infinite volumes, necessarily only depend on the macroscopic limit of the heuristic microscopic rate functions in the heuristic CME. More specifically, consideration of the first term of the system-size expansion leads to a deterministic set of equations of the form, where S is the stoichiometric matrix with elements S ij =r ij −s ij [23]. As discussed in the Introduction, the vector of heuristic microscopic rate functions for the heuristic CME,, is constructed from the macroscopic rate function vector,, of the reduced REs by performing the substitution on. Given the ansatz, equation (3), we can see that the heuristic method guarantees, by construction, that. This implies that the first term of the system-size expansion applied to the heuristic CME leads to a deterministic set of equations of the form, which indeed are the reduced REs obtained from the deterministic QSSA. Hence, we can conclusively state that in the macroscopic limit, the heuristic CME does reproduce the correct mean concentrations for timescale separation conditions.
The second question, regarding agreement in noise statistics not simply in the means, has not been considered before and presents a considerably more difficult challenge. In what follows we briefly review the LNA applied to the heuristic CME of the coarse-grained network which we shall call the hLNA and we derive the LNA applied to the full network under conditions of timescale separation, a novel method which we refer to as the slow-scale LNA (ssLNA).
The LNA applied to the heuristic CME
The application of the LNA to the heuristic CME has been the subject of a number of studies[23, 25–30]. For a step-by-step guide to implementing the LNA we refer the reader to the supplementary material of Ref.[9]. Here we simply state the well known results. We shall use the underline notation to denote a matrix throughout the rest of the article.
Given the coarse-grained network, reaction scheme (1), one can construct the stoichiometric matrix S with elements S ij =r ij −s ij and the macroscopic rate vector with entries. Note that the latter, as discussed in the Introduction, encapsulates the macroscopic rate law for each individual reaction composing the network. Note also that the k's for a coarse-grained network are generally functions of the macroscopic concentrations and not constants as for a full network (of elementary reactions). The reduced REs are then given by and consequently the Jacobian matrix J has elements, where ∂ j denotes the partial derivative with respect to, the concentration of species j.
where F is a diagonal matrix whose non-zero diagonal entries are the elements of the macroscopic rate function vector, i.e.,.
where I is the identity matrix, i is the imaginary unit number and ω is the frequency.
Note that we have chosen to compute the variance and power spectrum as our noise statistics for the following reasons. The variance can be used to calculate the Fano factor (variance of fluctuations divided by the mean concentration) and the coefficient of variation (standard deviation of fluctuations divided by the mean concentration)[32]. The coefficient of variation provides a non-dimensional measure of the size of intrinsic noise, and is a particularly natural measure when the probability distribution solution of the CME is approximately Gaussian. The Fano factor multiplied by the volume provides another non-dimensional measure of the noise level: it gives the size of the fluctuations in the molecule numbers relative to that of a Poissonian distribution with the same mean number of molecules. Generally these measures provide different but complementary information and both have been reported in recent experiments[33, 34]. Hence in this article we calculate both measures. We also calculate the power spectrum which gives the intensity of fluctuations at a given frequency; a peak in the spectrum indicates noise-induced oscillations[35], a phenomenon which is of importance in biochemical networks responsible for biological rhythms such as circadian clocks[36].
The LNA applied to the full network under conditions of timescale separation
where and and are, respectively, the vectors of concentration fluctuations about the macroscopic concentrations of the slow and fast species. The operator ∇ denotes the vector of derivatives with respect to components of. The matrix J F is the Jacobian of the REs of the full network, while the diffusion matrix D F is given by equation (5) with S and F now equal to the stoichiometric matrix and the diagonal matrix of the macroscopic rate function vector for the full network, respectively.
Note that while equation (4) is based on the heuristic CME and therefore inherits all its problems, equation (8) has no such problems: it is derived from the CME of the full network of elementary reactions, which is fundamentally correct. Hence, ideally, we would obtain the multivariate Gaussian solution of the two linear FPEs, compare and then decide upon the validity of the heuristic CME. Unfortunately, this direct comparison is impossible because equation (4) gives a joint probability distribution function for the slow species only, whereas equation (8) leads to a joint probability density function for both slow and fast species.
where S and J F are the stoichiometric and Jacobian matrices of the full network and F is the diagonal matrix of the macroscopic rate function vector of the full network with the macroscopic concentrations of the fast species expressed in terms of the macroscopic concentrations of the slow ones. The matrices S and J F are partitioned into sub-matrices of the following sizes: S s is an N s ×R matrix, S f is an N f ×R matrix, J s is an N s ×N s matrix, J sf is an N s ×N f matrix, J fs is an N f ×N s matrix and J f is an N f ×N f matrix. Note that N s and N f are the number of slow and fast species respectively. Note also that the fact that the new diffusion matrix can be written in the form of equation (10) immediately implies that it is symmetric and positive semi-definite, two crucial properties of the diffusion matrices for all FPEs[21]. It also follows that the variance and power spectrum of the slow species according to the ssLNA can be calculated from equations (6) and (7) with Dh replaced by Dss.
Such a reaction scheme is compatible with the reduced REs: defining as the macroscopic concentration vector of the slow species, we have since as required by the deterministic QSSA. Note that while S' is not the only stoichiometric matrix which is compatible with the reduced REs (any matrix of the form where A is some general matrix will do), it is the matrix which is uniquely selected by adiabatic elimination of the concentration fluctuations of the fast species. Of course this reduced reaction scheme characterized by S' is only physically meaningful if its entries are time-independent and integer-valued. Under timescale separation, this condition is not generally fulfilled. Rather this constitutes an additional, stronger condition. Hence it follows that generally a reduced CME description does not exist under timescale separation conditions, though a reduced Langevin equation description, i.e., the ssLNA, always exists.
In the rest of this article, we apply the systematic comparison method developed in the Results section to two examples of biological importance: enzyme-facilitated catalysis of substrate into product by cooperative and non-cooperative mechanisms and a genetic network with a negative feedback loop. For each of these, we shall obtain the noise properties of the coarse-grained versions of the circuits in the limit of large molecule numbers using the ssLNA and the hLNA. Because the expressions for the noise statistics from these two are quite simple, we shall be able to readily identify the regions of parameter space where the hLNA, and hence the heuristic CME, is correct and where it gives misleading results. The theoretical results are confirmed by stochastic simulations based on the CME of the full network and on the heuristic CME of the coarse-grained network.
Application I: Cooperative and non-cooperative catalytic mechanisms
Substrate S is input into the compartment where the reaction is occurring, it reversibly binds with an enzyme EE which has two free binding sites to form the first complex EES with one binding site occupied by a substrate molecule. This complex either decays into the original enzyme EE and a product molecule or else it can reversibly bind to another substrate molecule leading to a second complex SEES with both binding sites occupied by substrate molecules. Finally, this last complex decays into the first complex and product P. Note that reaction scheme (14) is the considered full network, since only elementary reactions are involved.
Deterministic analysis and network coarse-graining
Note that throughout the rest of this article, the notation [X] will generally denote the steady-state concentration of species X, unless it appears in the context of a differential equation as in equation (15) where then it necessarily refers to the instantaneous concentration.
Stochastic analysis of the coarse-grained network: ssLNA and hLNA methods
A comparison of equations (21) and (24) leads one to the observation that the latter can be obtained from the former by setting q1=q2=0. Substituting these conditions in the Langevin equation, equation (18), we obtain physical insight into the shortcomings of the conventional heuristic method. This method rests on the incorrect implicit assumption that under conditions of timescale separation, the reversible elementary reactions involving the fast species do not contribute to the intrinsic noise in the substrate concentration.
Stochastic Michaelis-Menten and Hill-type kinetics
Hence, the first case leads to Michaelis-Menten (MM) kinetics (non-cooperative kinetics) and the second to Hill-type kinetics with a Hill coefficient of two (cooperative kinetics).
Comparison of equations (27) and (28) shows that the heuristic CME description of the coarse-grained network overestimates the size of intrinsic noise whenever the deterministic kinetics are Michaelis-Menten. Interestingly, comparison of equations (29) and (30) shows the opposite for Hill-type kinetics: the size of noise predicted by the heuristic CME underestimates the true value. Note also that for both types of kinetics, the heuristic CME predicts the correct noise statistics in the limit of very small and very large substrate concentrations (which correspond to very large and very small free enzyme concentrations in steady-state conditions, respectively). The predictions for the Michaelis-Menten case agree with those reported by a recent simulation-based study[17] and a study using the LNA applied to the full network[9]. Indeed, this agreement is an important benchmark for the ssLNA. To our knowledge, the results for the Hill-type kinetics have not been obtained before.
The following observations can be made from Figures4a and4b. The ssLNA quantitatively agrees with the results of stochastic simulations of the full network for a large enough volume Ω. In contrast, the heuristic approach, hLNA, and stochastic simulations based on the corresponding heuristic CME, are generally in quantitative disagreement with the results of the ssLNA and of stochastic simulations of the full network, even if the volume is very large. For example, for the case k1=5×10−7 and Θ=1/2, the CVS and FFS from the hLNA are approximately 11 and 112 times smaller, respectively, than the prediction of the ssLNA. In Figure4c we also illustrate the large differences which these statistics imply, by showing sample paths (trajectories) of the CME of the full network, of the heuristic CME and of the Langevin equation given by the ssLNA, equation (18). This confirms that: (i) the hLNA and, hence, the heuristic CME on which it is based, predicts the correct mean concentrations but incorrect noise statistics even if the molecule numbers are considerably large; (ii) the Langevin equation obtained from the ssLNA is a viable accurate simulation alternative to SSA simulations based on the heuristic CME.
Besides quantitative disagreement we also note that the qualitative dependence of the FFS and the CVS with Θ as predicted by the heuristic approach is also very different than the predictions of the ssLNA and stochastic simulations with the full network. For example, for the case k1=5×10−7, according to stochastic simulations of the full network and the ssLNA, the FFS reaches a maximum at Θ=1/2, whereas the heuristic approach predicts a monotonic increase of the FFSwith Θ. The case Θ<0.5is particularly interesting because the ssLNA and stochastic simulations of the full network lead to ΩFFSwhich is much greater than 1, whereas the heuristic approach predicts ΩFFS which is below 1. Hence, for Θ<0.5, the ssLNA correctly predicts the fluctuations to be super-Poissonian, i.e., the size of the fluctuations is larger than those of a Poissonian with the same mean number of substrate molecules, whereas the hLNA incorrectly predicts the opposite case of sub-Poissonian fluctuations.
The power spectrum for the substrate fluctuations has also been calculated (see the Methods section). Although there is some quantitative disagreement between the predictions of the ssLNA and hLNA both are in qualitative agreement: the spectrum is monotonic in the frequency and hence no noise-induced oscillations are possible by this mechanism. More generally, it can be shown that the spectra of the hLNA and ssLNA are in qualitative agreement for all full networks with at most one slow species because as can be deduced from equation (7), for such networks, the spectrum for a single species chemical system is invariably monotonic in the frequency.
Application II: A gene network with negative feedback
Finally, we study an example of a gene network with autoregulatory negative feedback. Such a feedback mechanism is ubiquitous in biology appearing in such diverse contexts as metabolism[40], signaling[41], somitogenesis[42] and circadian clocks[43]. Two reasons hypothesized for its widespread occurance are that (i) it supresses the size of intrinsic noise[44, 45] thereby providing enhanced stability and (ii) it can lead to concentration oscillations or rhythms which are crucial to the control of several aspects of cell physiology[36].
Note that the gene with two bound proteins is inactive, in the sense that it does not lead to mRNA production. This implies that sudden increases in protein concentration lead to a decrease in mRNA transcription which eventually results in a lowered protein concentration; this is the negative feedback or auto-inhibitory mechanism. The reaction network as given by reaction schemes (31) and (32) is our full network for this example. Note that the first two reactions in reaction scheme (31) are not in reality elementary chemical reactions but they are the simplest accepted forms of modeling the complex processes of transcription and translation and hence it is in this spirit that we include them in our full network description.
Deterministic analysis and coarse-grained network
Stochastic analysis of the coarse-grained network: ssLNA and hLNA methods
where K3=k−3/k3. Note that the coupled Langevin equations (34) imply that the fluctuations in the mRNA and protein concentrations are affected by noise from all of the 11 constituent reactions of the full network (reaction schemes (31) and (32)) except from those of the reversible reaction P + GP⇌G P2. As shown in the Methods section, the noise from this reaction becomes zero due to the imposition of cooperative behavior in the feedback loop.
A comparison of equation (37) and equation (38) shows that the ssLNA and hLNA are generally different except in the limits of p1→0 and q→1. From the Langevin equations (34) we see that setting p1=0 implies ignoring the noise due to the reversible reaction P + G⇌GP, while setting q=1 is equivalent to ignoring the noise from the reversible reaction P + E⇌EP. Hence, as for the previous example of enzyme kinetics, we can state that the hLNA and the heuristic CME upon which it rests, implicitly (and incorrectly) assume that under conditions of timescale separation, the reversible reactions involving the fast species do not contribute to the intrinsic noise in the slow species.
Furthermore, by the comparison of equations (37) and (38), one can also deduce that the heuristic CME provides a statistically correct description when the protein concentration [P] is either very small or very large, in other words the case of very weak or very strong transcriptional repression (and corresponding non-saturated and saturated degrading enzyme conditions).
Detailed comparison of the noise statistics from the ssLNA and hLNA
We emphasize that the main message brought by our analysis is that there are significantly large regions of parameter space (blue region in Figure9a), where the hLNA (and the heuristic CME) does not predict noise-induced oscillations but such oscillations are obtained from stochastic simulations of the full network as well as being captured by the ssLNA.
Qualitative discrepancies in the prediction of noise-induced oscillations arise because the hLNA does not correctly take into account the fluctuations stemming from the rate limiting step of the cooperative binding mechanism. The latter involves the slow binding reaction between a protein molecule P and a gene G leading to a complex GP. The reason for this is the hLNA's tacit assumption that the fast species are not involved in slow reactions. This rate limiting reaction is at the heart of the negative feedback loop that is responsible for concentration oscillations in many biological networks such as circadian clocks[36] and hence why we speculate that the hLNA misses the occurrence of noise-induced oscillations in certain regions of parameter space.
Discussion and conclusion
Concluding, in this article we have rigorously derived in closed-form, linear Langevin equations which describe the noise statistics of the fluctuations about the deterministic concentrations as predicted by the reduced REs obtained from the deterministic QSSA. Equivalently, the ssLNA, as the method was called, is the statistically correct description of biochemical networks under conditions of timescale separation and sufficiently large molecule numbers. We note that our method provides an accurate means of performing stochastic simulation in such conditions. This is particularly relevant since it has been proven that there is generally no reduced CME description in such cases[8]. Another advantage of the ssLNA is that it enables quick computation of noise statistics through the solution of a set of simultaneous, deterministic linear algebraic equations. By applying the ssLNA to two biologically relevant networks, we showed that this procedure can lead to particularly simple and compact expressions for the noise statistics, which are in very good numerical agreement with stochastic simulations of the CME of the full network under the above conditions. This is in contrast to the heuristic CME, which generally performed with poor accuracy and in some instances even missed the prediction of noise-induced oscillations.
The limitations of the ssLNA are precisely those of the conventional LNA on which it is based. Namely, if the system is composed of at least one bimolecular reaction, then it is valid for large enough molecule numbers (or, equivalently, large volumes) and provided the biochemical network is monostable. If the system is purely composed of first-order reactions and if one is only interested in variance and power spectra, then the only requirement is that of monostability. This is since in such a case it is well known that the first and second moments are exactly given by the LNA. For monostable systems with bimolecular reactions, the finite-volume corrections to the LNA can be considerable when the network has implicit conservation laws, when bursty phenomena are at play and when steady-states are characterized by few tens or hundreds of molecules[24, 47–49]. These problems probably become exacerbated when the network is bistable or possesses absorbing states[50]. Hence, it is clear that although the ssLNA presented in this article is valid for a considerable number of biologically interesting cases, it cannot be homogeneously applied to all intracellular reaction networks of interest. These require the development of methods beyond those presented in this article and hence present an interesting research challenge for the future.
A necessary and sufficient condition for timescale separation is that the timescales governing the decay of the transients in the average concentrations are well separated. Fast species are those whose transients decay on fast timescales while the slow species are those whose transients decay on slow timescales. At the microscopic level, there are several different scenarios which can lead to timescale separation. Grouping chemical reactions as fast or slow according to the relative size of their associated timescales, Pahlajani et al.[51] obtain timescale separation by defining fast species as those which are involved in fast reactions only and slow species as those involved in slow reactions only. Zeron and Santillan[52] use a similar but less restrictive approach whereby the fast species are involved in fast reactions only and the slow species can participate in both fast and slow reactions. Another method is that due to Cao et al.[53] who define slow species as those involved in slow reactions only and fast species as those participating in at least one fast reaction and any number of slow reactions. While the three aforementioned scenarios will lead to timescale separation, it must be emphasized that they only constitute a subset of the possible scenarios leading to such conditions. The derivation behind the ssLNA is not based on any particular microscopic scenario, rather it simply requires that the timescales of the transients in the average macroscopic concentrations are well separated. Hence it follows that in the limit of large molecule numbers, the methods developed in[51–53] cover only a sub-space of the parameter space over which timescale separation is valid. In the Methods section we indeed show that Pahlajani's approach[51] leads to a reduced linear Langevin equation which is a special case of the ssLNA, the case where the matrix B in equation (10) can be neglected. The approaches of Zeron and Santillan[52] and of Cao et al.[53] lead to a reduced CME description. As we have shown in the Results section, under conditions of timescale separation and for small intrinsic noise, there always exists a reduced linear Langevin description of monostable stochastic reaction networks (the ssLNA) but there is generally not a physically meaningful reduced master equation description. The latter is only obtained if one imposes stronger conditions.
These results are in line with those of Mastny et al.[54] which show that for the Michaelis-Menten reaction without substrate input, the sQSPA method, a rigorous singular-perturbation approach, leads to a reduced master equation whenever the free enzyme or complex concentrations are very small (see Table II of Ref.[54]). This equation has the same form as the heuristic CME. This implies that for such conditions the error in the predictions of the heuristic CME should be zero, a result which is reproduced by the ssLNA (see Application I in the Results section). However, note that though these concentration conditions can be compatible with the deterministic QSSA they are not synonymous with it. Generally, the sQSPA methods do not lead to a reduced stochastic description consistent with the deterministic QSSA over the whole parameter space, whereas the ssLNA does, albeit within the constraints that molecule numbers should not be too small and that the network is monostable.
Finally we consider the approach of Shahrezaei and Swain[55], who derived the probability distribution for a linear two-stage model of gene expression under conditions of timescale separation. Their method rests on an exact solution of the generating function equation corresponding to the CME in the limit that the protein lifetime is much greater than that of the mRNA. In the Methods section we show that the ssLNA applied to their example leads to the same variance as obtained from their reduced probability distribution. The upside of their method over the ssLNA is that they obtain the full probability distribution valid for all molecule numbers. The downside of their method is that the generating function equation can only be solved exactly for networks composed of first-order processes (as in the gene example presented by Shahrezaei and Swain) or very simple bimolecular reactions[19] and hence the method has a restricted range of applicability compared to the ssLNA.
While the stochastic simulation algorithm explicitly simulates every individual reaction event, the Langevin approach yields approximate stochastic differential equations for the molecular populations. This is computationally advantageous whenever the reactant populations are quite large[5]. This reasoning can be deduced from the relationship between the propensities and the microscopic rate functions as given by. It is well known that in the large population number limit, the vector is approximately equal to the vector of macroscopic concentrations and hence the magnitude of the propensities increases with the reaction volume or equivalently with molecule numbers. In particular, this implies that the time between consecutive reaction events, given by where r∈(0,1) is a uniform random number, decreases with increasing reaction volume. This means that the time spent by the stochastic simulation algorithm increases with increasing volume because more reaction events have to be resolved within the same time window. Given this reasoning we can compare the discussed methods in terms of speed and accuracy. The computation time of the Langevin methods, hLNA and ssLNA, is independent of the volume and hence if the molecule numbers are not too small, both methods are much quicker than simulating any reduced CME of the coarse-grained network or the CME of the full network. However the ssLNA enjoys the additional advantage that under conditions of timescale separation, it is as accurate as the CME of the full network. The same argument does not generally hold for the hLNA.
We emphasize that besides deriving the ssLNA method, in this paper we have used it to determine the range of validity of the conventional heuristic CME approach and the size of errors in its predictions. To our knowledge, this is the first study which attempts to answer these important and timely questions via a rigorous, systematic theoretical approach.
Our main message is that, the "conventional wisdom" that the heuristic CME is generally a good approximation to the CME of the full network under conditions of timescale separation is incorrect, if one is interested in intrinsic noise statistics and the prediction of noise-induced oscillations.
Derivation of the ssLNA
The linear FPE describing the full network is given by equation (8). It is well known that with every FPE one can associate a set of Langevin equations (stochastic differential equations)[37]. Note that the Langevin and FPE formalisms are exactly equivalent but as we show now, the Langevin description is ideal for deriving a reduced description in timescale separation conditions.
This Langevin equation is the ssLNA: it is an effective stochastic description of the intrinsic noise in the slow variables in timescale separation conditions. Using standard methods[37] it can be shown that the FPE which is equivalent to this effective Langevin equation is equation (9). The ssLNA can also be derived more rigorously using the projection operator formalism as shown in[57].
A note on the reduced Jacobian of the ssLNA
Hence the Jacobian in the ssLNA equations (9) and (12) is the same as the Jacobian of the reduced REs.
Note that equations (48) are formally the same as obtained by taking the average of the LNA equations (39) and (40) (this general agreement between the LNA and linear stability analysis is discussed in[21]). This implies that the timescales of the fast and slow variables in the ssLNA (and hence of the CME under timescale separation conditions and in the macroscopic limit) is the same as the timescales obtained from the REs.
Details of the derivations for the two-subunit enzyme network
The ssLNA recipe: Langevin equation and noise statistics
Note that the row number of the stoichiometric matrix reflects the species number, while the column number reflects the reaction number. The order of the entries in the macroscopic rate function vector reflects the reaction number.
The power decays monotonically with frequency, which implies no noise-induced oscillation; this statement is generally true for all networks (full or coarse-grained) which have just one slow species.
The hLNA recipe: Langevin equation and noise statistics
In the Results section, it was shown that a reduced CME description becomes possible whenever the effective stoichiometric matrix S' as given by equation (13) evaluates to integer values. For the reaction scheme under consideration, it can be shown that, where q1and q2 are given by equations (19) and (20). The latter two quantities are generally real values and time-dependent and hence a reduced CME description is not generally possible. A simple choice which makes S' integer-valued is the null choice, q1=q2=0, and indeed it is for these values that in the main text we show that the hLNA (and hence the heuristic CME) is a valid description of stochastic kinetics under timescale separation conditions.
Details of the derivations for the gene network example
Reduced rate equations
where is the Michaelis-Menten constant of the enzyme which degrades the protein species.
Derivation of the ssLNA results
We cast the species in the full network (as given by reaction schemes (31) and (32)) into the form required by the convention set in the Introduction. We denote the slow species by X1=M and X2=P and the fast species by X3=GP, X4=G P2 and X5=EP. Note that the form of the gene with no bound protein (G) as well as the free enzyme species (E) do not appear explicitly in our description due to the two inherent conservation laws (same as shown in the previous section for the enzyme example except that here we immediately remove the extra species). The eleven constituent reactions are numbered in the following order: G→G + M, GP→GP + M, P + G→GP, GP→P + G, P + GP→G P2, G P2→P + GP, M→∅, M→M + P, P + E→EP, EP→P + E, and EP→E.
Note that the columns of S reflect the reaction number, while the rows reflect the species number. Similarly, the reaction number is reflected in the entries of the macroscopic rate function vector.
Note that we have drawn the limit of cooperative binding on p1, p2, q and J.
Derivation of the hLNA results
The covariance matrix and the spectra can be obtained as for the ssLNA. The variances and spectra are given by equation (78) and equation (79) with D M replaced by Dh,M, and D P replaced by Dh,P.
In the main text, we show that the hLNA (and hence the heuristic CME) is the correct stochastic description under timescale separation when p1=0and q=1. Indeed one finds that this choice satisfies the condition derived in the Methods section, which is necessary to have a reduced CME description under timescale separation conditions. Namely the choice p1=0 and q=1 forces the effective stoichiometric matrix S' given by equation (13) to assume strictly integer values.
Comparison with other stochastic model reduction methods
In this section, we compare the predictions of the ssLNA with the predictions of other stochastic model reduction techniques in the literature. Specifically, we compare with the recent methods of Pahlajani et al.[51] and of Shahrezaei and Swain[55].
where the parameter b=k s /kdMhas been interpreted as the burst size (the average number of proteins synthesized per mRNA transcript)[56]. This is the deterministic QSSA.
The steady state variance predicted by the above Langevin equation is given by equation (87). The same result has also been previously obtained by Paulsson[32] by applying the LNA to the full network given by (83) and subsequently taking the limit of timescale separation. Hence, the result obtained from the ssLNA agrees with the exact method of Shahrezaei and Swain. The advantage of the ssLNA is that it is generally applicable to arbitrarily complex biochemical networks, whereas the generating function method of solving CMEs is typically restricted to networks composed of at most first-order reactions or very simple bimolecular reactions[18, 19].
The variance of fluctuations predicted by the above Langevin equation is given by equation (87) with b≪1. Hence, it is clear that the method of Pahlajani et al. cannot capture the fluctuations about the steady-state concentrations for all choices of rate constants which are compatible with the deterministic QSSA. Rather their assumption regarding the form of the diffusion matrix limits their analysis to a subset of the parameter space over which the deterministic QSSA and consequently the ssLNA are valid. Indeed, the fact that the method by Pahlajani et al. is generally a special case of the ssLNA can also be seen by direct comparison of the diffusion matrices of the two methods, namely, equations (91) and (10).
This work was supported by the German Research Foundation (DFG project No. STR 1021/1-2) and by SULSA (Scottish Universities Life Science Alliance), both of which are gratefully acknowledged.
Authors' Affiliations
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My mom and grandma refer to babies as "little bits". I love making these bibs as gifts for new babies and their mamas. It's my Buster Bib pattern in the small size. (Sorry, I'm not currently selling anything right now.)
I've refined my process for making these...and added some fun details, too.
I put them together using a "quilt as you go" method. This bib was made for a mama who didn't find out the gender of her babe. I had fun pulling out green and yellow scraps. :)
Before I get too far ahead of myself, I'll quilt about half way down the fabric and add a monogram to the other side. (The bibs are reversible.)
I particularly loved the happy bias-cut stripey binding. It still makes me happy. :)
These bibs are reversible, so the buttons are sewn back to back on both sides. I felt like I was being pretty clever when I came up with that one!
What about you? Do you have a go-to gift you like to make for new little bits?


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For the past few decades, turtles have not been alone on their tropical nesting beaches around the world. They now share many of these beaches with humans: some who eat turtle eggs (or turtles), some who build hotels, condos and beachfront homes, and others who just want to soak up the sun.
But there is another group of humans spending lots of time on turtle beaches with a different goal in mind: counting turtles.
These turtle counters use any means necessary to perform their censuses. Some use small planes to fly low over beaches and count turtle tracks from the air. Some cruise coastlines in boats. Some even count while on horseback. And some do it the old-fashioned way: boots (or bare feet) in the sand, walking miles of beach.
Although sauntering sandy tropical beaches while the surf rolls underfoot might sound idyllic, I can tell you that it is not for the soft of foot. I worked with many volunteers unaccustomed to the physical toil of beach patrolling whose feet paid the price in blisters and bandages.
Despite the hardships, walking where land meets sea in search of prehistoric reptiles for endless hours under starlit skies is simply magical. We would lie on our backs during breaks, catching our breath, swigging water, and counting shooting stars that would brighten the sky like a glimpse of daytime.
But nothing — I mean nothing — compared to witnessing a turtle emerging from the water. The monotony of the breaking and receding surf would give way subtly to a large, black shape, slowly plodding from the water toward the dry sand ahead. We would wait breathlessly for her to haul her massive body out of the ocean and commence the ancient ritual of reproduction. While each nesting event is unique, the telltale track carved into the wet sand by the flippers of a nesting sea turtle is the mark counted by turtle census workers around the world that together provide enough information to assess turtle nesting populations.
With so many beaches, and so many people around the world trying to count turtles, it's hard to get a global perspective from all of these separate censuses. So just like the population censuses that national governments conduct every decade, we started the State of the World's Sea Turtles (SWOT) initiative — a partnership between Conservation International, Duke University, and the IUCN Marine Turtle Specialist Group — to compile local censuses contributed by hundreds of researchers to provide the first comprehensive global view of sea turtle nesting around the world.
In the most recent SWOT Report, we compiled nesting data from more than 1,000 nesting sites worldwide for the green sea turtle (Chelonia mydas) and published the most detailed map of green turtle nesting ever created. Download the map here (PDF – 8.91 MB)
The map shows how widespread green turtle nesting is, with nesting sites circling the globe's midsection and extending throughout tropical latitudes. Nesting sites range from a handful of turtles a year to the tens of thousands of nesting females annually nesting at places like Raine Island, Australia, and Tortuguero, Costa Rica.
A striking feature of this map is that green turtle nesting is viewable at multiple scales: while the global view shows the dizzying multitude of nesting sites worldwide, three dozen inset maps allow the viewer to zoom in to more local geographies. These close-up views show the incredible diversity of places where green turtles are, from island archipelagoes in the central Pacific Ocean to mainland beaches on major continents. This means that to protect turtles throughout their nesting range, different strategies need to be developed depending on geography.
Also, by presenting available census data from all around the world, the SWOT maps show areas of high coverage (where there's lots of turtle counting going on) and areas towards which more efforts could be directed.
Turtle researchers aren't the only ones who understand the importance of the SWOT maps. Recently, the global green turtle map — and mapper Andrew DiMatteo — won the International Conservation Mapping Award, given by Esri, the world's leading provider of geographic information systems (GIS) software services.
This prestigious award shows not only the high quality of the map products SWOT produces, but also recognizes the importance of data-sharing and collaboration among colleagues from all corners of the world for improving shared knowledge and prioritizing conservation work across many geographies.
As long those turtle-counting humans continue their beach-to-beach censuses, SWOT will continue to put turtles (and their counters) on the map.
Bryan Wallace is scientific director of CI's Marine Flagship Species Program.


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Howdie folks! If you're like me than you love BBQ. I have just recently purchased a little ole cheapie smoker and wanted to share with you how it's working so far.
My 1st attempt Smokin Beef Ribs wasn't too bad but I definitely have some must do's for the next round.
Hope you enjoy the pics and come back for more!


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Protecting our planet's ecosystems and natural resources is a big job, and no one can do it alone. Collaboration between the world's leading scientists is crucial in order to combine valuable data and convince decision-makers to take action. This week, a team made up from staff from CI, the New England Aquarium (NEAq), National Geographic, the Waitt Institute and the Monterey Bay Aquarium Research Institute set out to discover and explore seamounts—largely-unstudied ecosystems—in Indonesia's Raja Ampat archipelago, an integral part of the Bird's Head Seascape. The expedition is led by CI's Chief Scientist for Oceans Dr. Greg Stone. Here is his most recent update from the water—brought to you directly from the NEAq blog.
Our expedition team is now on board and we are sailing towards our first anchorage in Raja Ampat. Here we hope to discover and explore seamounts—the major purpose for our collaborative and interdisciplinary voyage.
Seamounts: Hidden Mountains
The 6,600-ton nuclear submarine USS San Francisco entered a poorly mapped area 400 miles southwest of Guam in the eastern Pacific Ocean (just to the north of where we are now). Gliding at 38 miles per hour 500 feet down, without warning the colossal submarine buckled into the side of an uncharted seamount. It was a tragic event resulted in pandemonium on the sub after the impact, but the thick inner hull protecting the nuclear reactor held and the crippled sub managed to surface. One sailor was killed and 75 injured, but the remaining crew and the sub survived the May 16, 2005, collision.
The San Francisco had slammed into one of the estimated 60,000 seamounts that rise from the seafloor in all oceans of the Earth. Most are uncharted, only a few hundred have ever been visited, fewer than 1000 have names, and only a handful have been intensively studied.
Seamounts are extinct and active underwater volcanoes. These hidden mountains of the sea rival the Rocky Mountains in size and challenge coral reefs for biodiversity. Seamounts contain many new and endemic species and are now recognized as the last frontier in earth geography, ocean science and conservation.
Seamounts of Raja Ampat
The primary purposes of this project is to document and photograph Seamounts and the biology that lives on them for a National Geographic Magazine story written by me and photographed by Brian Skerry, and also to add new and important information about deep sea biodiversity in the Raja Ampat Seascape in order to enhance our understanding and conservation of this reagion. We are diving in one of the most biodiverse places in the ocean, the Raja Ampat seascape, a large marine conservation site where CI and partners works with governmental and non-governmental partners.
Complex networks of deep and shallow currents swirling around, up and over these giant seamounts enrich the water, increasing ocean productivity and encouraging concentrations of ocean life.
Seamounts are important because fish and other forms of marine life and seafood in many shallow parts of the ocean are greatly diminished. Increasingly, commercial fishers are being forced to the deeper recesses of the sea, which has brought seamounts to center stage in the environmental debate about over-fishing because sometimes huge nets are dragged over the seamounts, which can harm or even destroy their unique life forms.
Our plan is to use a helicopter, Remotely Operated Vehicle (ROV) and SCUBA diving to find and survey seamounts of this region. We hope to find and explore previously unknown seamounts and will report back on that soon.
To read all the posts from this expedition, check out the NEAq Global Explorers blog.