Panax ginseng, Rehmannia glutinosa, Angelica polymorpha, Atractylodes macrocephala, Glycyrrhiza uralensis, Ziziphus jujube, and Polygala tenuifolia to boost qi and blood circulation, strengthen the heart, and calm the spirit—skillfully linking heart, spleen, kidney, qi, blood and brain as a whole to treat age-related dementia. The purpose of this review is to outline TCM concepts for the treatment of dementia and illustrated with a historical prescription for the treatment of the condition, with the hope that this description may lead to advances in its management.The Traditional Chinese Medicine (TCM) theory that “kidneys give rise to marrow, and the brain is the sea of marrow” has been a guide for the clinical application of kidney, qi and blood tonics for prevention and treatment of dementia and improvement in memory. As low resistance end-organs, both the brain and the kidneys are subjected to blood flow of high volumes throughout the cardiac cycle. Alzheimer’s disease and vascular dementia are two common causes of dementia, and it is increasingly recognized that many older adults with dementia have both AD and vascular pathologies. The underlying molecular mechanisms are incompletely understood, but may involve atherosclerosis, vascular dysfunction, hypertension, type 2 diabetes, history of cardiac disease and possibly, kidney dysfuntion, leading to reduced erythropoietin production, anemia, brain energy deficit and slow excitotoxicity. During the Ming Dynasty, Zhang Jing-Yue used Qi Fu Yin (seven blessings decoction), comprising  Dementia is a progressive syndrome characterized by memory deficits, cognitive impairment, and deterioration in emotional control and social behavior , 2. The The prevalence of dementia increases exponentially with age, from 3.0% among those aged 65 to 74 years to 18.7% among 75 to 84 years old and 47.2% in individuals over the age of 85 years . An undeAccording to TCM, the brain and bone marrow are the outgrowths of the kidneys. In Lingshu Meridians, it is stated that “at conception, essence is formed.” After essence is formed, the brain and bone marrow are formed. The kidneys contain the essence, the essence sustains the marrow, and the marrow nourishes the brain. According to “Lingshu Discussion on Seas,” humans have a marrow sea, a blood sea, a qi sea and a water/grain sea (stomach). This is the meaning of the four seas. Among the four seas in the human body, the marrow sea refers to the brain. According to the “Category Text” Volume 9: where there is bone, there is marrow, and the brain has the most. Thus, all marrow is related to the brain, and the marrow sea refers to the brain. <《灵枢·经脉》曰:“人始生,先成精,精成而脑髓生。” 肾藏精,精充髓,髓荣脑。《灵枢·海论》歧伯曰:“人有*髓海,有血海,有气海,有水谷之海,凡此四者,以应四海也。”人体四海之一,髓海即指脑。《类经》卷九注:“凡骨之有髓,惟脑为最巨,故诸髓皆属于脑,而脑为髓之海。> , 11. ThiTCM views the body as a single entity and an individual’s being depends on interactions between the different body parts. Qi, blood, and body fluids are essential for life. Qi is interpreted as “life energy” or “life force.” The Chinese character for “qi” (氣) means air or gas and may have some of the characteristics of air or oxygen gas. Original qi originates from the kidneys, the site where “congenital essence,” an essential and vital substance inherited from one’s parents upon conception is stored. Pectoral qi is formed from the combination of inhaled fresh air from the lungs and food essence from the stomach and spleen. It permeates the blood vessels and moves outward during expiration and inward during inspiration. Nutritive qi supplies nourishment to the body. It comes from food and digestive activities and circulates via blood vessels. Protective qi is similar to the immune system and helps in the prevention of illnesses.Phlegm is a pathological substance generated by disturbance of body fluid that blocks qi. Substantial phlegm is visible such as sputum, whereas insubstantial phlegm is invisible. Lipids and lipoprotein metabolic disorders  are believed to contribute to phlegm in the blood, whereas platelet activation, thrombosis, endothelial injury, and atheromatous plaques contribute to blood stasis . Small-mThe TCM theory of “kidneys give rise to marrow, and the brain is the sea of marrow” implies a close relationship between the kidneys, the marrow, and the brain. Chronic renal failure is asymptomatic at first, until kidney function has decreased to less than 25% of normal. Patients then present with nocturia and anorexia, and raised serum levels of nitrogenous compounds such as urea and creatinine. Advanced renal failure causes significant impairment of all renal function and affects virtually all body systems, and causes change in urea, electrolytes, and other blood constituents. End-stage renal disease is the term used when more than 90% of renal function is lost and may be complicated by anemia, bleeding, bone disease, hypertension, congestive heart failure, digestive tract problems, and dementia. It is proposed that part of the renal-cerebral connection may be due to small vessel disease in both the kidneys and brain. There are many hemodynamic similarities between the vascular beds of these organs —both kidAnemia is a common finding in patients with CKD. Analyses of baseline data from adults with chronic kidney disease indicate that the prevalence of cognitive impairment is higher among those with lower eGFR, and this association is independent of traditional vascular risk factors, but rather, related to anemia . The kidL [Epo and EpoR are also expressed in neurons of the CNS, including the neocortex, hippocampus, and hypothalamus, as well as dorsal root ganglia, and Schwann cells . Epo impL . In addiL . CollectL .CKD patients also show raised serum levels of nitrogenous wastes. Decreased cognitive function has been associated with the retention of uremic toxins, with performance improving with more intensive dialysis or kidney transplantation . The enz+/K+ ATPases that maintain resting potential in neurons. This results in depolarization, excessive calcium entry into neurons, activation of calcium dependent enzymes such as proteases, lipases, and endonucleases, and slow excitotoxicity [The TCM theory that memory and cognition are disordered if channels that connect the heart and brain are blocked by phlegm implies a close relationship between the cardiovascular system and the brain, a connection which is also highly emphasized in Western medicine. Heart disease is any condition that impairs cardiac function regardless of the specific modality that is affected . The funtoxicity . Reductitoxicity , but aretoxicity . They matoxicity , 52, buttoxicity –55. Furttoxicity , and redtoxicity . Dementitoxicity .Increasing evidence indicates that consumption of bioactive compounds or phytochemicals in herbs produces antioxidant, anti-inflammatory, and anti-carcinogenic effects fig. . Unlike Multi-herb formulas rather than single herbs are common in TCM medication. Each herb in a formula has a specific role—sovereign (君), minister (臣), assistant (佐), and courier (使). Sovereign and minister herbs treat the main symptoms and have a major role in the formula. Assistant herbs assist the sovereign and minister herbs to treat the accompanying symptoms, or reduce the side effects of the major herbs. Courier herbs help to lead the other components to the affected area. Interactions between the herbs, such as mutual reinforcement, antagonism, or detoxification, help to determine the formula’s therapeutic efficacy. The nature of the herbs, including the four properties  and the five tastes  as well as characteristics such as meridian-tropism, are taken into consideration by TCM physicians when formulating a prescription .Zhang Jing-Yue 张景岳 (c.1563–1640) had a great influence on the development of TCM, towards the end of the Ming Dynasty in China. He was born in Shaoxing County of Zhejiang Province, the birthplace of many of the country’s most renowned scholars and writers. Zhang traveled with his father to Beijing, where he studied medicine, and became an imperial physician working for Emperor Wanli. Because he made good use of processed Rehmannia glutinosa, he was also called “Zhang Shudi.” He was an outstanding medical scientist, the representative of the ancient Chinese medicine practice of “warm tonification,” and his academic thought has a great influence on future generations. A chapter on dementia (痴呆) in the Ming dynasty book 《Jingyue Quanshu》  describes how the collapse of original qi, together with the presence of impure qi in the meridians and heart orifices, can lead to the problem of dementia.In this chapter, Zhang Jing-Yue was of the opinion that “dementia syndrome is characterized by lack of sputum, or stagnation, or failure, or anxiety, or suspicion, or panic, gradually leading to dementia. Speech and words are in reverse order and movement is sluggish. There is excessive sweating and depression. The symptoms may be extremely unusual and bizarre. The pulse is stringy or increased in frequency, large or small, and often changes. Persons have pathological qi in the heart and liver and biliary systems. The qi is unclear. If the body is strong and there is no reduction in food intake or other weaknesses, it is appropriate to treat using Fu Man Jian. It is the most stable and wonderful prescription. Nevertheless, some cases will recover and some will not, depending on the strength or weakness of stomach qi and original qi. Time is needed for recovery, which cannot be speeded up. Those who have this syndrome may be exacerbated by anxiety or depression leading to absent-mindedness and confusion. In this case, it is important to quickly support the good qi, mostly through the use of Qi Fu Yin or Da Bu Yan”. < 明代《景岳全书》张景岳认为:“痴呆证,凡平素无痰,而或以郁结,或以不遂,或以思虑,或以疑贰,或以惊恐,而渐致痴呆。言辞颠倒,举动不经,或多汗,或善愁,其证则千奇万怪,无所不至。脉必或弦或数,或大或小,变易不常。此其逆气在心或肝胆二经,气有不清而然。但察其形体强壮,饮食不减,别无虚脱等证。则悉宜服蛮煎治之。最稳最妙。然此证有可愈者,有不可愈者,亦在乎胃气元气之强弱,待时而复,非可急也。凡此诸证,若以大惊猝恐,一时偶伤心胆,而致失神昏乱者。此当以速扶正气为主,宜七福饮,或大补元煎主之。> Panax ginseng, Rehmannia glutinosa, Angelica polymorpha, Atractylodes macrocephala, Glycyrrhiza uralensis, Ziziphus jujube, and Polygala tenuifolia  is a mixture of seven herbs, ia Table .No.OrderAccording to one of three foundation books for TCM, Shennong Materia Medica, which is a compilation of oral traditions between 200 and 250 A.D., and attributed to a “divine farmer” Shennong, said to have lived around 2800 B.C.: ginseng has a sweet taste, and mildly cold in nature. It is mainly used for supplementing the five internal organs. It calms the spirit, stabilizes the soul, reduces fear, dispels impure qi, brightens the eyes, increases happiness, and is beneficial for thinking and wisdom. Long-term consumption leads to lightening of the body and extension of years. Another name is Human Title or Ghost Cover. It grows in valleys. < 神农本草经:人参,味甘微寒,主补五脏,安精神,定魂魄,止惊悸,除邪气,明目,开心益智。久服,轻身延年。一名人衔,一名鬼盖。生山谷。> Bencao Gangmu or Compendium of Materia Medica, a Chinese materia medica written by Shi-Zhen Li during the Ming dynasty in 1578 AD: ginseng supplements the five internal organs. It calms the spirit, stabilizes the soul, reduces fear, dispels impure qi, brightens the eyes, increases happiness, and is beneficial for thinking and wisdom. Long-term consumption leads to lightening of the body and extension of years. < 本草纲目:补五脏,安精神,定魂魄,止惊悸,除邪气,明目开心益智。久服轻身延年。> [According to 久服轻身延年。> . As can 久服轻身延年。> .Usage and dosage: for oral consumption, as a decoction 3–10 g. If using a high dose of 10–30 g, should be prepared first as a single decoction and poured into the final prescription herbs. It can also be ground as a powder 1–2 g; or paste; or soaked in liquor/wine, or made into pills. < 用法用量:内服:汤剂,3-10克,大剂量10-30克,宜另煎兑入;或研末,1-2克;或敷膏;或泡酒;或入丸散。>Panax ginseng: not to be consumed by persons with no deficiency, heat syndrome, or no weakness in qi. < 实证、热证而正气不虚者忌服。>Contraindications for According to Shen Nong’s Herbal Classic: Rehmannia has a sweet and cold taste and is non-toxic. It is mainly used for treatment of fractures and muscle injury, internal injury, dispelling blood stasis, filling of the bone marrow, and enhancing muscle growth. As a decoction, it helps to remove the accumulation of cold and heat, and remove paralysis. The raw herb is better. Long-term consumption leads to lightening of the body and anti-aging. < 神农本草经:地黄,甘、寒,无毒。主折跌绝筋,伤中,逐血痹,填骨髓,长肌肉,作汤除寒热积聚,除痹,生者尤良。久服轻身不老。> According to Compendium of Materia Medica: it fills the bone marrow, enhances muscle growth, increases essence and blood, supplements the five internal organs after internal injury, unblocks the blood circulation, is beneficial to the ears and eyes, and blackens the beard and hair. < 本草纲目:填骨髓,长肌肉,生精血,补五脏内伤不足,通血脉,利耳目,黑须发。> Usage and dosage: for oral consumption as a decoction 10–30 g. Or make into pills, pastes, or soak in wine/liquor < 用法用量:内服:煎汤,10-30克,或入丸散;或敷膏;或浸酒。>Contraindications for Rehmannia glutinosa: persons with weakness in stomach, qi stagnation, abundant sputum, abdominal distension, and loose stools < 胃虚弱,气滞痰多,腹满便溏者忌服。>According to Shen Nong’s Materia Medica: Angelica has a bitter taste. It is warm in nature and non-toxic. It is mainly used for treatment of cough, reversed flow of qi, warm malaria, cold and hot contacts in the skin, women with uterine bleeding and infertility, all sore ulcers; and sores. Boil the herb to drink. <神农本草经:当归:苦、温,无毒。主咳逆上气,温瘧,寒热洗洗在皮肤中,妇人漏中绝子,诸恶疮疡,金疮。煮汁饮之 。> Usage and dosage: fried black, grind fine powder, each time use of 9 g, add a cup of water, a little wine, and decoction. 6~12 g. < 用法用量:炒黑,共研细末,每用9克,水一杯,酒少许,煎服。6~12克。>Angelica polymorpha: excessive use causes tiredness, drowsiness, and other reactions. Stopping the medicine leads to disappearance of these symptoms. Allergic reaction: not suitable for persons with menorrhagia, bleeding tendency, yin deficiency, internal heat, and loose stools or diarrhea. Not to be consumed by persons with hot bleeding tendencies. To be consumed with caution by persons with wet fullness and bloating. < 用量过大有疲倦嗜睡等反应,停药消失。过敏反应月经过多,有出血倾向,阴虚内热,大便溏泄者均不宜服用。热盛出血患者禁服。湿盛中满慎服。>Contraindications for There is a saying that “ginseng in the north and Atractylodes in the south,” suggesting that Atractylodes is almost as highly regarded as ginseng in TCM. According to Shen Nong’s Herbal Classic: Atractylodes macrocephala has a bitter taste and warm nature. It is mainly used for treatment of rheumatism, joint ache caused by wind and dampness, muscle paralysis, muscle spasm, and jaundice. It stops perspiration, dispels heat, helps digestion, and is used in decoctions. Long-term consumption leads to lightening of the body, extension of years, and satiety. Another name is mountain thistle, it grows in valleys. < 神农本草经:白术:味苦温。主风寒湿痹死肌,痉疸,止汗,除热,消食,作煎饵。久服轻身延年,不饥。一名山蓟,生山谷。> .Usage and dosage: 6~12 g.< 用法用量:6~12克。>Contraindications for Atractylodes macrocephala: not to be consumed by persons with Yin deficiency, qi stagnation, and nausea < 阴虚燥渴,气滞胀闷者忌服。>Glycyrrhiza uralensis has a sweet taste and a neutral nature. It is mainly used to dispel cold and hot, impure qi in the five viscera  and six hollow organs , toughen tendons and bones, increase muscle mass, multiply strength, facilitate recovery after traumatic injury, and detoxification. Long-term consumption leads to lightening of the body and extension of years. < 神农本草经:炙甘草:味甘,平。主五脏六腑寒热邪气,坚筋骨,长肌肉,倍力,金疮尰,解毒。久服轻身延年。> [According to Shen Nong’s Herbal Classic: 久服轻身延年。> Usage and dosage: 1.5~9 g. < 用法用量:1.5~9克。>Glycyrrhiza uralensis: not appropriate to combine with Jing Da Ji (Radix Euphorbiae), Yuan Hua (Lilac Daphne Flower Bud), and Gan Sui (Radix KanSui) together. < 不宜与京大戟,芫花,甘遂同用。>Contraindications for According to Shen Nong’s Materia Medica, it is mainly used to treat heart and abdominal cold and heat, and aggregation of impure qi; aching, pain; and dampness arthralgia. Long-term consumption leads to calming of the five viscera , lightening of the body, and extension of years. < 神农本草经:主心腹寒热,邪结气聚,四肢酸痛湿痹,久服安五脏,轻身延年。> .According to Compendium of Materia Medica, the seed has a sweet taste and a neutral nature. The processed herb is used for treatment of biliary system weakness, insomnia, polydipsia, and sweating. Its raw form is used for treatment of heaty biliary system and accompanying drowsiness. It is also the drug for treatment of the diseased accupoints: liver meridian of foot—Jueyin, and gallbladder meridian of foot—Shaoyang. < 本草纲目:“其仁甘而润,故熟用疗胆虚不得眠,烦渴虚汗之证;生用疗胆热好眠,皆足厥阴、少阳药也。” > Usage and dosage: put into the decoction, 9~15 g. < 用法用量:入汤剂,9~15克。>.Contraindications for Ziziphus jujube: to be consumed with caution by persons with impure smoldering fire and those with chronic diarrhea. Not to be used with Fang Ji (Radix Stephaniae tetrandrae). < 凡有实邪郁火及患有滑泄症者慎服。恶防己。>.According to Shen Nong’s Herbal Classic: Polygala has a bitter taste and a warm nature. It is mainly used to treat cough with dyspnea, internal injury, make up for deficiency, dispel impure qi, and relieve obstruction in the nine orifices . It is beneficial for wisdom, and makes one clever and not forgetful. It strengthens ambition and resolve, and multiples strength. Long-term consumption leads to lightening of the body and anti-aging. The name of the name of the leaf is Small Grass. Another name is Spine Wan (Lu De Ming Er Ya Yin Sheng cited amaranth). Another name is Spine Around . Another name is Fine Grass. It grows in valleys. < 神农本草经:远志:味苦温。主咳逆,伤中,补不足,除邪气,利九窍,益智慧,使人聪明,不忘,强志倍力。久服,轻身不老。叶名小草,一名棘菀(陆德明尔雅音义引作苋),一名棘绕(御览作要绕),一名细草。生川谷。> According to compendium of Materia Medica: it has a bitter taste and warm nature and non-toxic. It is mainly used to treat forgetfulness. Take Polygala at the end and boil with water to consume < 本草纲目:苦、温、无毒。「主治」善忘症。取远志为末,冲服。> Usage and dosage: boil and consume: 3~9 g. < 用法用量:煎服,3~9克。>Contraindications for Polygala tenuifolia: persons with gastritis and gastric ulcer should use with caution. < 有胃炎及胃溃疡者慎用。>Panax ginseng , tonifying qi and lifting yang, is beneficial for happiness, thinking and wisdom, strengthens the spleen and nourishes the stomach < 人参,补气补阳,开心益智,健脾养胃 > [Atractylodes macrocephala , strengthens the spleen, reduces dampness, benefits qi, and helps in the circulation < 白术,健脾燥湿,加强益气助运之力 > [Rehmannia glutinosa  , Angelica polymorpha , Ziziphus jujube (assistant), and Polygala tenuifolia (assistant) boost the blood, calms the mind, and calms the spirit. < 熟地、当归、枣仁(佐)、远志(佐),补血宁心安神。> [Glycyrrhiza uralensis  is beneficial to qi, nourishes the heart, and harmonizes the different medicines, thus leading to filling of the qi and blood and calming of the mind and spirit, resulting in recovery from disease. < 炙甘草,益气和中养心,调和诸药,从而使气血充、心神安则病愈。> [Qi Fu Yin is used for treatment of heart and qi deficiency, treatment of neurasthenia, and treatment of age-related dementia. It has effects on benefitting qi, supplementing the blood, nourishing the heart, and calming the spirit. < 治疗心气虚,治疗神经衰弱,治疗老年性痴呆。有益气补血,养心宁神的功效。> , 67. Pan智,健脾养胃 > , 67 . At益气助运之力 > , 67. Reh补血宁心安神。> , 67 Glyc心神安则病愈。> , 67Persons who have heat syndrome and no weakness in qi should not consume. Women should stop using it during menstruation < 实证、热证而正气不虚者忌服。妇女经期停用。> , 67.None recorded to our knowledge.The following herbs are boiled in 400 cc of water and left to simmer until the volume reduces to 280 cc. This can be divided into 2 servings per day, to be taken on an empty stomach, when warm. <上药用水400毫升,煎取280毫升,空腹时温服。> . Review Panax ginseng, Rehmannia glutinosa, Angelica polymorpha, Atractylodes macrocephala, Glycyrrhiza uralensis, Ziziphus jujube, and Polygala tenuifolia to boost qi and blood circulation, strengthen the heart, and calm the spirit—skillfully linking heart, spleen, kidney, qi, blood and brain as a whole to treat age-related dementia.Alzheimer’s disease and vascular dementia are two common causes of dementia, and it is increasingly recognized that many older adults with dementia have both AD and vascular pathologies. The underlying molecular mechanisms are incompletely understood, but may involve atherosclerosis, vascular dysfunction, hypertension, type 2 diabetes, history of cardiac disease, and total homocysteine, and possibly, lack of erythropoietin formation by the kidney. During the Ming Dynasty, Zhang Jing-Yue used Qi Fu Yin (seven blessings decoction), comprising


Although bone age plays a special role in determining the child’s age, there are some variations in skeletal growth of different people. The aim of this study was to compare the bone age with chronological age of children aged 2–18 years old in order to recognize whether Greulich-Pyle (GP) method could be reliable for Iranian children? The standard radiograph of Left hand was taken in 40 healthy subjects, then the bone age was determined according to GP. Mean ± SD bone ages were delayed 1.12 ± 0.65, 0.82 ± 1.34 and 0.10 ± 0.51 years than the mean chronological ages in 2.99–5.99, 10–13.99 and 14–17.99 age group, respectively; and advanced −0.33 ± 3.12 years in the 6–9.99 age group. In BMI levels <18.5, 18.5–24.9, 25–29.9 and ≥30, Mean ± SD bone ages in males were delayed 2.25 ± 0.21, 0.14 ± 0.55, 0.87 ± 0.41 and 4.05 ± 0.70 years than the mean chronological ages, respectively. In BMI range of 18.5–24.9 and BMI ≥ 30, Mean ± SD bone age in females was delayed 0.50 ± 0.49 and 0.45 ± 0.63 years than the mean chronological ages, respectively. For BMI < 18.5, Mean ± SD bone age in females were advanced −0.40 ± 2.69 years than mean chronological ages. Considering these differences, Iranian boys may have a different pattern of bone growth from GP standards. Specifications TableValue of the Protocol•The data demonstrate that variations in skeletal growth in children aged 6–9.99 and 10–13.99 years old are greater than those evaluated by the Greulich and Pyle atlas, and that there are significant differences between bone and chronological ages.•The categorizing groups based on BMD shows a great differences among our Mean ± SD chronological and bone ages in those male subjects who are underweight (BMD < 18.5) and obese (BMI ≥ 30). The Mean ± SD of difference of bone and chronological ages in male normal children (18.5 ≤ BMI ≤ 24.9) and overweight children (25 ≤ BMI ≤ 29.9) were 0.14 ± 0.55 and 0.87 ± 0.41 years, respectively.•The Mean ± SD difference of chronological and bone ages in female subjects that are underweight, normal, and obese is −0.40 ± 2.69, 0.50 ± 0.49 and 0.45 ± 0.63, respectively.•The variation in skeletal growth of Iranian children and children’s group that the GP Atlas was taken could be related to environmental, economic, social, cultural and racial differences. Although this discrepancy is greater in boys than girls, so it may be affected by gender in Iran. Thus, GP Atlas could be more accurate to determine bone age in Iranian girls than boys.The Mean ± SD bone and chronological ages, weight, height and BMI of subjects based on gender are presented. Although the Mean ± SD BMI in boys was more than girls, the difference between Mean ± SD chronological and bone age in boys was lower than that of girls. The difference of Mean ± SD of chronological and bone ages in group 1 was more than other aged grouped , Fig. 1.The aim of this methodology is to determine the differences between bone age and chronological age in Iranian children which could lead more accurate estimation of bone age. To compare bone age with chronological age, 40 children were randomly selected. Subjects aged between 2 and 18 years living in Tehran City, Iran. Exclusion criteria were history of systemic diseases more than 1 month, history of chronic systemic diseases, history of hospitalization for more than a week, and people with height or weight above the percentile of 97% or below 3 percent percentile. After receiving informed consent from patients and their parents, radiograph of the left wrist and hand was taken. Based on GP , all radT-Test by SPSS.The normality of data was checked out by Shapiro–Wilk Test. For each group, mean ± SD of bone age, chronological age, weight, height and BMI (Body Mass Index) were determined; then data were analyzed using One-Sample Although GP method is reliable and known method, there are some variations in skeletal growth of the Iranian children and children’s group that the GP Atlas was taken that could be related to environmental, economic, social, cultural and racial differences. Since this discrepancy is greater in boys than girls, it may be affected by gender in Iran. Therefore, GP Atlas could be more accurate to determine bone age in Iranian girls than boys.


These transitions do not require for their realization the energy-consumable anisotropic rotation of the amino group of A around the exocyclic C6-N6 bond. They are controlled by the non-planar transition states with quasi-orthogonal geometry (symmetry C1) joined by the single intermolecular (Т)N3H···N6(А) H-bond . The Gibbs free energies of activation for these non-dissociative, dipole-active conformational transitions consist 7.33 and 7.81 kcal∙mol−1, accordingly. Quantum-mechanical (QM) calculations in combination with Bader’s quantum theory of “Atoms in Molecules” (QTAIM) have been performed at the MP2/aug-cc-pVDZ//B3LYP/6-311++G level of QM theory in the continuum with ε = 4 under normal conditions.In this study it was theoretically shown that discovered by us recently (Brovarets’  It was shown by NMR methods5 that Watson-Crick ↔ Hoogsteen breathing in DNA duplex containing A∙T rich region occurs via the switching of the Watson-Crick DNA base pair (bp) from the anti- to syn-conformation with the probability ~10−2 and represents one of the pathways for the reaction of formaldehyde with DNA10. Thorough calculations by the method of molecular dynamics indicate that А·Т(WC) ↔ А·Т(Н) transitions of actually bps and anti ↔ syn transitions of the A around the glycosidic bond are closely correlated processes, for which Gibbs free energy of activation is 10–11 kcal∙mol−1 under normal conditions8.Spontaneous transition of the DNA base pairs from the Watson-Crick (WC) to Hoogsteen (H) configuration and 13. Comprehensive analysis of the current literature data showed that the nature of these biologically-important processes has not been investigated at all. Currently in the literature there is only one single theoretical work devoted to the study of the anti ↔ syn non-dissociative transitions in irregular pairs of nucleotide bases that do not have an exocyclic amino group in its composition14.Based on analysis of the microstructural nature of these transitions, it is quite logical to connect it with the analogical properties of the isolated DNA bps1), short-lived wobbled (w) conformers – А·Т(wWC), А·Т(wrWC), А·Т(wН) and А·Т(wrН) for each of the four classical А·Т(WC) DNA bps – Watson-Crick А·Т(WC), reverse Watson-Crick А·Т(rWC), Hoogsteen А·Т(Н) and reverse Hoogsteen А·Т(rН)11. It is known from the literature data, that these bps joined by different H-bonds are formed due to the rotation of the DNA base relative to the other on 180° around: the (A)N1–N3(T) axis for the reverse Watson-Crick А·Т(rWC) or in other terms Donohue DNA bp23; the (A)C9′-N9 axis for the Hoogsteen A·T(H) bp30 and the (A)N7–N3(T) axis for the reverse Hoogsteen A·T(rH) or in other terms Haschemeyer–Sobell bp34.Recently, we have theoretically revealed novel high-energetic, significantly non-planar (symmetry CWC), А·Т(wrWC), А·Т(wН) and А·Т(wrН) conformers have essentially non-planar structure joined by the two anti-parallel N6H/N6H′···O4/O2 and N3H···N6 H-bonds . These specific intermolecular contacts involve pyramidalized A amino group, acting simultaneously as an acceptor and a donor of the H-bonding. The transition states (TSs) – TSА·Т(WC)↔А·Т(wWC), TSА·Т(rWC)↔А·Т(wrWC), TSА·Т(Н)↔А·Т(wН) and TSА·Т(rН)↔А·Т(wrН) – of the dipole-active conformational transformations of the basic, plane-symmetric state of the classical А·Т DNA bps into the high-energetic, essentially non-planar wobbled bps and vice versa possess wobble structures (symmetry C1) and are joined by the N6H/N6H′···O4/O2 and N3H···N6 H-bonds. The А·Т(wWC), А·Т(wrWC), А·Т(wН) and А·Т(wrН) conformers was found to be dynamically stable structures with short lifetime τ = (1.4–3.9) ps. It was assumed that these conformational transitions are directly related to the thermally-driven fluctuational behavior of DNA – pre-melting and breathing7.It was found that revealed А·Т(wWC), А·Т(wrWC), А·Т(wН) and А·Т(wrН) control the А·Т(wWC)/А·Т(wrWC) ↔ А·Т(wН)/А·Т(wrН) conformational transitions. Moreover, in view of the recently discovered conformational transitions for the classical A·T DNA bps - А·Т(WC) ↔ А·Т(wWC), А·Т(rWC) ↔ А·Т(wrWC), А·Т(Н) ↔ А·Т(wН) and А·Т(rН) ↔ А·Т(wrН)11, they are also intermediates of the biologically-important А·Т(WC)/А·Т(rWC) ↔ А·Т(Н)/А·Т(rН) conformational transitions.In this work it was established for the first time that just-mentioned novel conformers А·Т(wWC) ↔ А∙Т(wH) and А∙Т(wrWC) ↔ А∙Т(wrH), and together with them conformational transition of the А∙Т DNA bps – А∙Т(WC)/А∙Т(rWC) ↔ А∙Т(H)/А∙Т(rH), does not require for their realization the rotation of the amino group of A around the exocyclic C6N6 bond35.Energetically favorable mechanism of the conformational pairwise transformation of the intermediates А∙ТN3H···N6(А) H-bond between the imino group of T and pyramidilized amino group of A. The Gibbs free energies of activation for these non-dissociative, dipole-active conformational transitions consist 7.33 and 7.81 kcal∙molWC) ↔ А∙Т(wH) and А∙Т(wrWC) ↔ А∙Т(wrH) – are realized via the anisotropic rotation of the amino group of A N6H/N6H′···O4/O2(T) and (T)N3H···N6(A) H-bonds) around the exocyclic C6N6 bond. In TSs of these conformational transitions the pyramidality of the amino group of A significantly increases: this causes increase of the energy of the N3H···N6 H-bond and decrease of the energy of the intermolecular N6H/N6H′···O4/O2 H-bond. The transitions states of these reactions – TScysА·Т(wWC)↔А·Т(wН), TStransА·Т(wWC)↔А·Т(wН) and TScysА·Т(wrWC)↔А·Т(wrН), TStransА·Т(wrWC)↔А·Т(wrН) – have close energy in corresponding conformational transformations . Thus, these TSs of the mutual conformational transformation of the wobble intermediates – А∙Т(wWC) ↔ А∙Т(wH) and А∙Т(wrWC) ↔ А∙Т(wrH) of the classical А∙Т DNA bps – А∙Т(WC)/А∙Т(rWC) ↔ А∙Т(H)/А∙Т(rH) – determine their conformational transformations.Two other mechanisms – the А∙Т of their mutual conformational transformations together with their harmonic vibrational frequencies at the B3LYP/6–311++G level of theoryel – electronic energy, while Ecorr – thermal correction.The Gibbs free energy G for all structures has been received at the MP2/6-311++G level of theory by the formula:int have been obtained at the MP2/6-311++G level of theory as a difference between the BSSE-corrected72 electronic energy of the bp and electronic energies of the isolated bases.The electronic energies of interaction ∆E78 has been applied for the analysis of the electron density distribution by AIMAll program package79, using wave functions calculated at the B3LYP/6-311++G level of theory. We considered the presence of the  bond critical point (BCP), a bond path between the donor and acceptor of the intermolecular contact and positive value of the Laplacian at this BCP (Δρ > 0) as criteria for the existence of the H-bond or attractive van der Waals contact formation84.Bader’s quantum theory of Atoms in Molecules (QTAIM)86 in TSs of the conformational transitions have been estimated by the Espinosa-Molins-Lecomte (EML) formula88:The energies of the attractive van der Waals contacts89:91.The energies of the conventional AH···B H-bonds have been calculated by the Iogansen’s formula92.In this study the numeration for the DNA bases is generally accepted95, in particular in the base-pair recognition pocket of the high-fidelity DNA-polymerase46. At this, we have relied on the experience received in the previous works98 on the related topic and systems, in which the negligibly small impact of the stacking and sugar-phosphate backbone on the tautomerisation processes has been shown.In this study we have provided investigations at the basic, but sufficient level of the isolated H-bonded pairs of nucleotide bases, that adequately simulates the processes in real biological systems11 we have succeed to establish in the classical А∙Т DNA bps with Cs symmetry – Watson-Crick (WC), reverse Watson-Crick А·Т(rWC), Hoogsteen А·Т(Н) and reverse Hoogsteen А·Т(rН) DNA bps – novel high-energetic, dynamically-stable, mirror-symmetrical А∙Т(wWC)R,L, А∙Т(wH)R,L, А∙Т(wrWC)R,L and А∙Т(wrH)R,L conformational states. Their distinguished feature independently of the pair, in which they are realized, is significantly non-planar structure (С1 symmetry), caused by the pyramidal structure of the ≥C6N6H2 amino fragment of the A DNA base, in which the amino group acts simultaneously as a donor and an acceptor of the specific intermolecular interaction with T through the two (Т)N3H···N6(A) and (A)N6H/N6H′···O4/O2(T) H-bonds. Each of the four А∙Т Watson-Crick DNA bps transfers into the aforementioned conformer via two mirror-symmetric pathways through the TSА∙Т(WC)↔А∙Т(wWC)R,L, TSА∙Т(rWC)↔А∙Т(wrWC)R,L, TSА∙Т(H)↔А∙Т(wrH)R,L and TSА∙Т(rH)↔А∙Т(wrH)R,L (C1 symmetry). At this, the structures, which names differ from each other only by the subscripts R and L, are mirror-symmetrical, that is enantiomers. It is well known that enantiomers have identical scalar physico-chemical characteristics and differ only by the direction of the dipole moment.In our previous paperLet analyze the biological significance of these non-usual conformers of the classical А∙Т DNA bps.In this context it was fixed important result – these conformers are responsible for the two different WC/rWC ↔ H/rH mechanisms of the non-dissociative conformational transformation of the А∙Т DNA bps Fig. –3.FigureА∙Т(wWC)R,L↔А∙Т(wH)R,L and TSА∙Т(wrWC)R,L↔А∙Т(wrH)R,L (C1 symmetry) with low values of imaginary frequency . Both of them are joined by the one-single intermolecular (T)N3H···N6(A) H-bond  between the imino group of T and pyramidilized amino group of A. In this case, conformational transformations of the А∙Т DNA bps are realized by the following non-dissociative scenario (each of them – by the mirror-symmetric pathways): А∙Т(WC) (0.00) ↔ TSА∙Т(WC)↔А∙Т(wWC)R,L (7.13) ↔ А∙Т(wWC)R,L (5.36)11 ↔ TSА∙Т(wWC)R,L↔А∙Т(wH)R,L (7.33) ↔ А∙Т(wH)R,L (5.35) ↔ TSА∙Т(wH)R,L↔А∙Т(H) (7.24) ↔ А∙Т(H) (−0.44)11 and А∙Т(rWC) (0.00) ↔ TSА∙Т(rWC)↔А∙Т(wrWC)R,L (7.26) ↔ А∙Т(wrWC)R,L (5.97)11 ↔ TSА∙Т(wrWC)R,L↔А∙Т(wrH)R,L (7.81) ↔ А∙Т(wrH)R,L (5.79) ↔ TSА∙Т(wrH)R,L↔А∙Т(rH) (7.41) ↔ А∙Т(rH) (−0.03)11. Notably, obtained energetic barriers are in good coincidence with the molecular-dynamic data for the А∙Т(WC) ↔ А∙Т(H) transition .First of these conformational transformations, which are the most energetically favorable mechanisms, are controlled by the soft TSА∙Т(wWC)R,L↔А∙Т(wH)R,L and TSА∙Т(wrWC)R,L↔А∙Т(wrH)R,L, which pairwise link the А∙Т(wWC)R,L and А∙Т(wH)R,L, А∙Т(wrWC)R,L and А∙Т(wrH)R,L conformers, are transition states of the WC/rWC ↔ H/rH conformational transformations of the classical А∙Т DNA bps.Herewith, some R structures transform into the other R structures, the same concerns L-structures. Saying in other words, pathways of these dipole-active conformational transformations are mirror-symmetric. In fact, the TS35 and is controlled by the TScysА∙Т(wWC)R,L↔А∙Т(wH)L,R, TStransА∙Т(wWC)R,L↔А∙Т(wH)L,R and TScysА∙Т(wrWC)R,L↔А∙Т(wrH)L,R, TStransА∙Т(wrWC)R,L↔А∙Т(wrH)L,R, that have non-planar structure (С1 symmetry) and quite high values of the imaginary frequencies (~252 i cm−1). These TSs are joined by the two anti-parallel intermolecular (Т)N3H···N6(A) and (A)N6H/N6H′···O4/O2(T) H-bonds; notably, first of them is significantly stronger than the second one. The attractive O2···N7 and O4···N7 van der Waals contacts with weak energy  also participate in the stabilization of the TScysА∙Т(wWC)R,L↔А∙Т(wH)L,R and TScysА∙Т(wrWC)R,L↔А∙Т(wrH)L,R, accordingly.High-energetic mechanism of the WC/rWC ↔ H/rH conformational transitions of the А∙Т DNA bps is connected with anisotropic rotation of the amino group of A around the exocylic С6-N6 bondvice versa and WC/rWC ↔ H/rH conformational transitions of the classical А∙Т DNA bps occur in such a case :In this case, the R structures transform into the L structures and А∙Т(WC)↔А∙Т(wWC)R,L (7.13) ↔ А∙Т(wWC)R,L (5.36)11 ↔ TScysА∙Т(wWC)R,L↔А∙Т(wH)L,R (14.89) ↔ А∙Т(wH)L,R (5.35) ↔ TSА∙Т(wH)L,R↔А∙Т(H) (7.24) ↔ А∙Т(H) (−0.44)11;А∙Т(WC) (0.00) ↔ TSА∙Т(WC)↔А∙Т(wWC)R,L (7.13) ↔ А∙Т(wWC)R,L (5.36)11 ↔ TStransА∙Т(wWC)R,L↔А∙Т(wH)L,R (14.88) ↔ А∙Т(wH)L,R (5.35) ↔ TSА∙Т(wH)L,R↔А∙Т(H) (7.24) ↔ А∙Т(H) (−0.44)11;А∙Т(WC) (0.00) ↔ TSА∙Т(rWC)↔А∙Т(wrWC)R,L (7.26) ↔ А∙Т(wrWC)R,L (5.97)11 ↔ TScysА∙Т(wrWC)R,L↔А∙Т(wrH)L,R (15.01) ↔ А∙Т(wrH)L,R (5.79) ↔ TSА∙Т(wrH)L,R↔А∙Т(rH) (7.41) ↔ А∙Т(rH) (−0.03)11 andА∙Т(rWC) (0.00) ↔ TSА∙Т(rWC)↔А∙Т(wrWC)R,L (7.26) ↔ А∙Т(wrWC)R,L (5.97)11 ↔ TStransА∙Т(wrWC)R,L↔А∙Т(wrH)L,R (15.00) ↔ А∙Т(wrH)L,R (5.79) ↔ TSА∙Т(wrH)L,R↔А∙Т(rH) (7.41) ↔ А∙Т(rH) (−0.03)11  level of QM theory in the continuum with ε = 4 under normal conditions).А∙Т(rWC) (0.00) ↔ TS101, stay planar.It should be noted that the orientation of the methyl group of the T DNA base does not alter in the course of all reactions of conformational transitions. At this, the heterocycles of the DNA bases, capable for the out-of-plane bendingSo, obtained by us results launch the conception of the “mechanics” of the non-dissociative WC/rWC ↔ H/rH conformational transformations of the classical А∙Т DNA bps.anti ↔ syn transition of A around the glycosidic bond  and reorganization of stacking and hydratation8. Simple comparison of the energetics, determining these processes, clearly indicates that the first two of them plays a leading role. This fact gives hope that obtained in this paper data are closely related to the nature of the А∙Т(WC) ↔ А∙Т(H) thermal fluctuation process, which occurs in DNA7. This conclusion can be verified, applying the newest methods of ab initio dynamics for the short fragments of DNA.Of course, in the composition of DNA these conformational transitions represent a self-consistent transformation of the bps, the 11, we offered novel non-dissociative mechanisms of the А∙Т(WC) ↔ А∙Т(H) and А∙Т(rWC) ↔ А∙Т(rH) conformational transitions, that do not require for their realization energy-consuming anisotropic rotation of the amino group of the A DNA base around the C6-N6 exocyclic bond. Figuratively speaking, at the transformation of the A base from the anti- to syn-conformation leading to the formation of the Hoogsteen А∙Т(H) and reverse Hoogsteen А∙Т(rH) bps, it dynamically relies as on the support on the T DNA base through the pyramidilized amino group of A, interacting with it in the TS region by one single (Т)N3H···N6(А) H-bond.By applying developed by us novel ideas according the high-energetic conformers of the classical А∙Т DNA bps7, most likely occurs by the non-dissociative mechanism: A, rotating from the anti- to syn-configuration, interacts with T via the intermolecular H-bonds along the entire process of the conformational transformation.In the light of the obtained by us results, it could be suggested that the А∙Т(WC) ↔ А∙Т(H) conformational transition in DNA duplex, which was registered experimentally


The entry for the column “MSA principal city” and row “Tdap ≥1 dose” should have read “88.8 (87.2 to 90.1). Entries for the column “Difference between non-MSA and MSA principal city” and row “MenACWY, ≥1 dose” should have read -7.4 (-10.0 to -4.7), row “MenACWY, ≥2 dose” should have read -12.0 (-19.5 to -4.6), row “HPV, ≥1dose” should have read -10.8 (-14.0 to -7.6), and row “HPV UTD” should have read -10.0 (-13.3 to -6.6). Entries for the column “Difference between MSA nonprincipal city and principal city” and row “MenACWY, ≥1 dose” should have read 0.1 (-2.0 to 2.2), row “HPV, ≥1dose” should have read -7.0 (‑9.6 to -4.4), and row “HPV UTD” should have read -5.5 (-8.3 to -2.6).In the report “National, Regional, State, and Selected Local Area Vaccination Coverage Among Adolescents Aged 13–17 Years — United States, 2017,” on page 914, in Table 2, the entry for the column “Difference” and row “≥3 Hepatitis B doses” should have read -2.6 (-4.8 to


Scientific Reports 10.1038/s41598-019-47976-x, published online 08 August 2019Correction to: In the original version of the Article, the full error bars were omitted from Figure 9A due a technical error during publication.In addition, in the Results and Discussion Section under the subheading ‘Direct transfer of a qPCR protocol to the QS3D dPCR system’,E. amylovora31 was transferred to the QS3D dPCR platform, using the same primer and probe concentrations and thermal cycling conditions as a first step to test the QS3D dPCR technology.”“A known qPCR protocol for now reads:E. amylovora31 was transferred to the QS3D dPCR platform, using the same primer and probe concentrations and thermal cycling conditions as a first step to test the QS3D dPCR technology.”“A known qPCR protocol for Finally, in the Legend of Table 2,cLog Copies mL−1 = Log [copies μL−1rxn × 1/Drxn tube x1/EDNA Extr. X1/VolPl.Macerates]; i.e. Log (copies μL−1rxn × 1.79 × 103) for Apple, and Log (copies μL−1rxn × 4.81 × 103) for pear.”“now reads:cLog Copies mL−1 = Log [copies μL−1rxn × 1/Drxn tube × 1/EDNA Extr × 1/VolPl.Macerates]; i.e. Log (copies μL−1rxn × 1.79 × 103) for Apple, and Log (copies μL−1rxn × 4.81 × 103) for pear.”“This has been corrected in both the PDF and HTML versions of the Article.


This work aims to discuss a predator-prey system with distributed delay. Various conditions are presented to ensure the existence and global asymptotic stability of positive periodic solution of the involved model. The method is based on coincidence degree theory and the idea of Lyapunov function. At last, simulation results are presented to show the correctness of theoretical findings. It is well known that the qualitative analysis of predator-prey models is an interesting mathematical problem and has received great attention from both theoretical and mathematical biologists –5. In paf1(u) = au, (II) u(t) represents the prey density at time t, c > 0 is the half-saturation constant, a > 0 denotes the search rate of the predator. Holling type II functional response is most typical of predators that specialized on one or a few prey  stands for the integer part of t, t ∈  and t ≠ 0, 1, 2, ⋯. The solution Following , 43 and t ∈ =r¯2α.It follo))]=r¯1α,∑k=0α-1[))]=r¯2α.It follo(a) If x1(η1) ≥ x2(η2), then it follows from  and and(a) I2r¯1α≔χ1,x1(k)≥x12r¯1α≔χ2.By (23),24)From  and (24)Thus by , 25) an an(a) If2r¯2α≔χ4,x2(k)≥x22r¯2α≔χ5.It follo(b) If x1(η1) < x2(η2), then it follows from  has no relation with ρ ∈ . Let M = max{χ3, χ6, χ9, χ12} + M0, where M0 > 0 which satisfies v = {v(k)} = {(x1(k), x2(k))T} of  and and(b) I2r¯2α≔χ7,x2(k)≥x22r¯2α≔χ8.By (31),32)From , we getThus by , 33) an an(b) Ifr¯2α≔χ10,x1(k)≥x1r¯2α≔χ11.It follo¯2α≔χ11.3It follov = {v(k)} ∈ X: ‖v‖ < M}, then Ω is an open, bounded set in X and (a) of Lemma 1 is satisfied. When v ∈ ∂Ω ∩ KerL, v = {T} with ‖v‖ = max{|x1|, |x2|} = M. Thenϕ = μQNv + (1 − μ)Gv, μ ∈ , whereJ be the identity mapping, we haveLv = Nv has at least one solution in α periodic solution in α positive periodic solution of system  and (H2) are satisfied and furthermore suppose that there exist positive constants ν, σ1and σ2such thatThen the positive ω-periodic solution of system , y*(k))T} of (Define a function given by  and 45)V by V(N(f system  and 41)V by V(N(ondition  that ∃ ϵk))T} of . The proRemark 1In },where r1Example 2 Consider the model as follows:r1(k) = 0.2 + cos kπ, r2(k) = 0.1 + cos kπ, m = 2, k1 = 0.2, k2 = 0.12, k3 = 0.24, k4 = 0.35,. So σ1 = 0.18, σ2 = 0.23, ν = 0.04. Thus the conditions (H1)-(H2) and  and  of Theormx1(k)]}.where r1Based on the previous works and some biological meanings of predators and preys, we propose a new discrete delayed predator-prey system. By using the continuation theorem in coincidence degree theory, we present a set of sufficient conditions to ensure to ensure the existence of positive periodic solution of the discrete delayed predator-prey system. In addition, we also discussed the global asymptotic stability of positive periodic solution for the considered system. The obtained theoretical findings have important significance in biological ecology. Considering the effect of random factor, it is meaningful for us to deal with the dynamics of stochastic predator-prey system. This topic will be our future research direction.S1 Figx1(t) and the red line stands for x2(t).The blue line stands for (DOC)Click here for additional data file.S2 Fig(DOC)Click here for additional data file.S3 Figx1(t) and the red line stands for x2(t).The blue line stands for (DOC)Click here for additional data file.S4 Fig(DOC)Click here for additional data file.


Nature Communicationshttps://www.nature.com/articles/s41467-019-10342-6, published online 24 May 2019.Correction to: 2+/Au → CO2+ A’, ‘CO2+/Au → CO2+ B’ and ‘CO2+/Au → O2– C’ instead of the correct panel a, b, c ‘CO2+/Au → CO2+’, ‘CO2+/Au → O2+’ and ‘CO2+/Au → O2–’, respectively. This has been corrected in the PDF and HTML versions of the Article.The original version of this Article contained errors in Fig. 1. The titles at the top of each panel a, b, c were incorrectly given as ‘CO


H be a transfer Krull monoid over a finite abelian group G . Then each nonunit a∈H can be written as a product of irreducible elements, say k is called the length of the factorization. The set L(a) of all possible factorization lengths is the set of lengths of a. It is classical that the system ℒ(H) = {L(a)∣a∈H} of all sets of lengths depends only on the group G, and a standing conjecture states that conversely the system ℒ(H) is characteristic for the group G. Let H′ be a further transfer Krull monoid over a finite abelian group G′ and suppose that ℒ(H) = ℒ(H′). We prove that, if r≤n−3 or (r≥n−1≥2 and n is a prime power), then G and G′ are isomorphic.Let  Then each non-unit a∈H can be written as a product of atoms, and if H, then k is called the length of the factorization. The set L(a) of all possible factorization lengths is the set of lengths of a, and ℒ(H) = {L(a)∣a∈H} is called the system of sets of lengths of H (for convenience we set L(a) = {0} if a is an invertible element of H). Under a variety of noetherian conditions on H  all sets of lengths are finite. Sets of lengths  are a well-studied means for describing the arithmetic structure of monoids.Let H be a transfer Krull monoid over a finite abelian group G. Then, by definition, there is a weak transfer homomorphism 𝜃:H→ℬ(G), where ℬ(G) denotes the monoid of zero-sum sequences over G, and hence ℒ(H) = ℒ(ℬ(G)). We use the abbreviation ℒ(G) = ℒ(ℬ(G)). By a result due to Carlitz in 1960, we know that H is half-factorial ) if and only if |G|≤2. Suppose that |G|≥3. Then there is some a∈H with |L(a)|>1. If k,ℓ∈L(a) with k<ℓ and m∈ℕ, then L(am)⊃{km+ν(ℓ−k)∣ν∈} which shows that sets of lengths can become arbitrarily large. Note that the system of sets of lengths of H depends only on the class group G. The associated inverse question asks whether or not sets of lengths are characteristic for the class group. In fact, we have the following conjecture .Conjecture 1.1.G be a finite abelian group with D(G)≥4. If G′ is an abelian group with ℒ(G) = ℒ(G′), then G and G′ are isomorphic.Let D(G) = 3, then we have ℒ(G) is studied with methods from Additive Combinatorics. In particular, zero-sum theoretical invariants (such as the Davenport constant or the cross number) and the associated inverse problems play a crucial role. Most of these invariants are well-understood only in a very limited number of cases  is known and the associated inverse problem is solved; however, if n is not a prime power and r≥3, then the value of the Davenport constant Note if refer to , 9).r,n∈ℕ and 2r<n−2 (this is done by Geroldinger and Zhong  = {x∈ℤ∣a≤x≤b} the discrete, finite interval between a and b. If L⊂ℕ is a subset, then Δ(L) denotes the set of (successive) distances of L  if and only if d = b−a with a,b∈L distinct and ∩L = {a,b}) and ρ(L) = supL∕minL denotes its elasticity  = 1).Our notation and terminology are consistent with , 9. For r∈ℕ and let r-tuple of elements of G. Then ei≠0 for all i∈ and if for all i∈. Furthermore, G if it is independent and n∈ℕ, we denote by Cn an additive cyclic group of order n. Since r = r(G) is the rank of G and nr = exp(G) is the exponent of G.Let 2.1.H be a monoid with unit element 1 = 1H∈H. An element a∈H is said to be invertible (or an unit) if there exists an element a′∈H such that H will be denoted by H×, and we say that H is reduced if H× = {1}. The monoid H is said to be unit-cancellative if for any two elements a,u∈H, each of the equations au = a or ua = a implies that u∈H×. Clearly, every cancellative monoid is unit-cancellative.By a monoid, we mean an associative semigroup with unit element, and if not stated otherwise we use multiplicative notation. Let H is unit-cancellative. An element u∈H is said to be irreducible (or an atom) if u∉H× and for any two elements a,b∈H, u = ab implies that a∈H× or b∈H×. Let 𝒜(H) denote the set of atoms, and we say that H is atomic if every non-unit is a finite product of atoms. If H satisfies the ascending chain condition on principal left ideals and on principal right ideals, then H is atomic .If We say a monoid homomorphism 𝜃:H→B be a weak transfer homomorphism between atomic unit-cancellative monoids. It follows that for all a∈H, we have ℒ(H) = ℒ(B).Let H is a transfer Krull monoid if one of the following equivalent conditions holds:𝜃:H→ℋ* for a commutative Krull monoid ℋ* .There exists a weak transfer homomorphism G0 of an abelian group.There exists a weak transfer homomorphism We say H is a transfer Krull monoid over G0. If G0 is finite, then H is said to be a transfer Krull monoid of finite type.If the second condition holds, then we say R is a transfer Krull domain (of finite type) if its monoid of cancelative elements R∙ is a transfer Krull monoid (of finite type).We say a domain Let In particular, commutative Krull monoids are transfer Krull monoids. Rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains are commutative Krull monoids with finite class group such that every class contains a prime divisor , 2, and v-noetherian nor completely integrally closed. To give a noncommutative example, let 𝒪 be a holomorphy ring in a global field K, A a central simple algebra over K, and H a classical maximal 𝒪-order of A such that every stably free left R-ideal is free. Then H is a transfer Krull monoid over a ray class group of 𝒪 , and If Lemma 2.4.A∈ℬ(G) with Δ(supp(A))≠∅, the set Δ(supp(A)), length at least 1, and bound M2.There exist constants Proof.See  and k(Vj)≤1 for all j∈.Let Let Proof.G),exp(G)k(A)}⊂L(AG)exp, we haveand hence1. Since {exp(Δ(G0) | minΔ(supp(A)) and minΔ(G0)≥⌊exp(G)∕2⌋, it follows thatSince minΔ(G0) = minΔ(supp(A)) = exp(G)−exp(G)k(A) and hence Therefore ming∈supp(A) such that B1∈ℬ(supp(A)). If there exists an atom A1 with k(A1)<1 such that A1 divides B1, then ord(h) = exp(G) and vh(A) = 1 for all h∈supp(A).Let supp(A) is not an LCN-set. Then there exists A1∈𝒜(supp(A)) with k(A1)<1. It follows by 1 that h∈supp(A1) which implies that A1 | A, a contradiction. Thus supp(A) is an LCN-set and hence minΔ(supp(A))≤|supp(A)|−2.2. Assume to the contrary that G0| = r(G)+1 and r(⟨G0⟩) = r(G). If h∈⟨G0∖{h}⟩, then r(⟨G0∖{h}⟩) = r(G). Otherwise let d = min{k∈ℕ∣kh∈⟨G0∖h⟩} and hence (G0∖{h})∪{dh} is also a minimal non-half-factorial LCN-set with r(⟨G0∖{h}⟩) = r(G).3. By Geroldinger and Zhong , say i = 1, such that k(U1)<1. Then exp(G)k(U1)<exp(G). Since 4. Assume to the contrary that there exists L(BG)exp≥exp(G)ℓ that It follows by maxj∈, say j = 1, such that k(V1)>1. Then exp(G)k(U1)>exp(G). Since Assume to the contrary that there exists L(BG)exp≤exp(G)k that It follows by min3.First, we introduce new invariants which play a crucial role in the proof of Theorem 1.2 (see Proposition 3.5).Definition 3.1.d∈Δ1(G) and k∈ℕ. We defineLet ρ the limit of Then ρ = ρ(G) if and only if G is not a cyclic group of order 4,6 or 10.It follows by Geroldinger and Zhong , say i = 1, such that t1≥M. Set I = {i∈∣ti≥M} and M thatSince minL(Bk) is an AAP with difference d and Bk, we obtainSince d | minΔ(G0) and hence J⊂ with |J|≥maxL(Bk)−M|𝒜(G)|D(G) such that Therefore Thereforeρ≤ρ*.By definition, we infer G0⊂G with d | minΔ(G0) and B∈ℬ(G0) such that ρ(L(B)) = ρ*. Since supp(B)⊂G0, we infer minΔ(G0) divides minΔ(supp(B)) and hence d divides minΔ(supp(B)).2. Let M such that minΔ(supp(B))∈Δ(L(BM)). It follows by Lemma 3.2 and Lemma 2.2.6 thatBy Lemma 2.4, there exists a constant d divides minΔ(supp(B)).Thus the assertion follows by k∈ℕ such that kd∈Δ1(G), we have 3. For every Proposition 3.5.ℒ(G) = ℒ(G′) for some finite abelian group G′ and let d∈Δ1(G). Then ρ = ρ, and Suppose Proof.ℒ(G) = ℒ(G′), it follows by definition that ρ = ρ for every k∈ℕ such that kd∈Δ1(G). By Lemma 3.4.3, we obtain Since Lemma 3.6.d∈Δ1(G). Then ρ*≥K. In particular, if d∈, then Let Proof.G0⊂G with d dividing minΔ(G0) and K(G0) = K. Then there exists A∈𝒜(G0) such that k(A) = K≥1. Sincewe haveSuppose d∈, we let n1. Set Δ(G0) = d by Lemma 2.3.1.In particular, if Since Lemma 3.7.G0⊂G be a subset with d | minΔ(G0), where d∈Δ1(G) satisfies A∈𝒜(G0) with k(A)>1, then s∈ℕ.Suppose nr is a prime power, then if n1 is not a prime power, then K>r.if In particular,Proof.t = r−d and we start with the following claim.Let Claim A.B∈ℬ(G0) with g∈supp(B) and supp(B) is an LCN-set. Then there exists B0 is a product of atoms having cross number 1 and g∈supp(B0) such that BB0 is a product of atoms having cross number 1.Let Proof of Claim A.B∈ℬ(G0) with g∈supp(B) and supp(B) is an LCN-set, such that the assertion does not hold. Suppose |supp(B)| is minimal in all the counterexamples.Assume to the contrary that there exists a G1 = supp(B). If for all g∈G1, ord(g)∣vg(B), then B is a product of atoms having cross number 1, a contradiction. Therefore there exits g∈supp(B), we infer Set i∈ and and Hi is an qi-group of rank r(Hi)≤r−t = d which implies that there exists E|≤r(Hi)≤d such thatLet Therefore there exists Since Δ(supp(Bi)) = 0 and hence supp(Bi) is half-factorial. Therefore Bi is a product of atoms having cross number 1.Note that Since y∈ℕ and i∈ and every g∈supp(C). Since |supp(C)|<|supp(B)|, it follows by the minimality of |supp(B)| that there exists C0 is a product of atoms having cross number 1 and g∈supp(C0), such that CC0 is a product of atoms having cross number 1. Let BB0 is a product of atoms having cross number 1, a contradiction to our assumption.Therefore A∈𝒜(G0) be with k(A)>1. Then Lemma 2.6.2 implies that supp(A) is an LCN-set. Set Claim A implies that there exist atoms ℓ∈ℕ0, such that d | minΔ(G0), we infer s∈ℕ.Let Now we begin to prove the “in-particular’’ parts.j∈ and every m∈ℤ, we denote by ∥m∥j the least positive residue of m modulo 1. For every K, we have s∈ℕ. Let H be a subgroup of G withand let H with ord(e) = n1 and i∈, j∈. Set W∈𝒜(G2), we haveandBy definition of We suppose that Since for each j∈, whenceThereforer−1 | minΔ(G2) which implies that K≥K(G2). Let Since s∈ℕ, it follows that Since nr is a prime power, then K(G)<r by Lemma 2.1. It follows by 1. that s∈ℕ. Therefore 2. If n1 is not a prime power, then u≥2 and K>1+r−1 = r. □3. If Proposition 3.8.s∈ℕ be maximal such that G has a subgroup isomorphic to d∈Δ1(G) with d≥r, then If d≥nr−1, then ρ* = K.If r = nr−1≥3. If p and k∈ℕ, then K = ρ*.Suppose Let Proof.d≥r, it follows by Lemma 2.5(items 3 and 4) that d>m(G) and 1. Since nr and let Let G0⊂G with d | minΔ(G0) be such that B∈ℬ(G0) with ρ(L(B)) = ρ*. Setwhere k = minL(B), ℓ = maxL(B), and Let i∈ such that k(Ui)>1, then Lemma 2.6.2 implies that supp(Ui) is an LCN-set and hence minΔ(supp(Ui))≤m(G). Sincewe get a contradiction to d>m(G). Therefore k(Ui)≤1 for all i∈.If there exists i∈ such that k(Vi)<1, then Lemma 2.6.1 implies that i∈. It follows thatIf there exists Then G0⊂G be such that d | minΔ(G0) and A∈𝒜(G0) such that k(A)<1, then Lemma 2.6.1 implies that A|≥2. Thus G0 is an LCN-set. Let B∈ℬ(G0) such that which implies that ρ* = ρ(L(B))≤K.2. Let G0⊂G be such that d | minΔ(G0) and K(G0) = K. Then there exists an atom A∈𝒜(G0) such that k(A) = K≥1. Since and hence ρ* = K.Let r = nr−1≥3 and we proceed to prove the following claim.3. Let Claim B.ρ*>K and let G0⊂G be such that (r−1) | minΔ(G0) and ρ(G0)>K.g∈G0 with ord(g) = nr such that −g∈G0.There exists G2| = r. If there exists a∈ such that ag∈⟨G2⟩, then G2 is a basis of G.Let G, and Suppose Proof of Claim B.By Lemma 2.5.2, we infer that a. If G0 is an LCN-set, then for every B∈ℬ(G0), we havewhich implies that ρ(L(B))≤K. Therefore ρ(G0)≤K, a contradiction. Thus there exists A∈𝒜(G0) such that k(A)<1. Lemma 2.6.1 implies that A = g(−g) for some g∈G0 with ord(g) = nr. Hence {g,−g}⊂G0.b. Let E⊂G2 be minimal such that there exists a∈ such that ag∈⟨E⟩ and let dgg∈⟨E⟩. Then there exists an atom V∈𝒜(E∪{g}) with supp(V)|≤r+1. Let T∈ℱ(E). Thenr−1 | dg+ℓ−2. Therefore dg = 1 and Note that for each g)Ti, g∈⟨E⟩ which implies that dg = 1 by the minimality of dg. The minimality of E implies that supp(Ti) = E for each i∈. Then for every h∈E, ord(h) = nr and Suppose k((−g)T1)<1, then Lemma 2.6.1 implies that T1 = g, a contradiction. ThereforeIf E|≤|G2|≤r and r = nr−1, we have |E| = |G2| = r. Let G, Since |dg = 1 and V = gT and hence |T|≥2. We infer g}∪E is an LCN-set and minΔ({−g}∪E)≤|E|−1. Since (r−1) | minΔ({−g}∪E) and |E|≤|G2| = r, we haveSuppose Let E1| = r+1 and hence It follows by Geroldinger and Zhong .Let k(V)>1, then Lemma 2.6.2 and  and every h∈G2, mg∉⟨G2∖{h}⟩. Note that nr≥4. Let andare both atoms. Since r−1 | |I|−1 and r−1 | |I|. Hence r = 2, a contradiction to r≥3.By the minimality of k(V) = 1 which implies that i∈. It follows by k(V1)>1 and  such that ag∈⟨G1∖{h}⟩. Since |G1∖{h}| = r, it follows by a and b that G such that Let t∈, i∈ and b that Assume to the contrary that there exists We only need to prove Since B∈ℬ(G0) such that ρ(L(B)) = ρ(G0) and assume thatwhere k,ℓ∈ℕ and Let k(Ui)≥1 for all i∈ and k(Vj)≤1 for all j∈. Since B by i∈ and each j∈.Note that j0∈ such that i0∈ such that j∈. If there exists i1∈ such that Since there must exist Thereforeρ(G0)>K. If there exists j1∈ such that a contradiction.It follows by Lemma 3.6 that andTo sum up, we obtainJ = {j∈∣Vj = g(−g)}. ThenandLet Therefore We distinguish two cases to finish the proof.p and k∈ℕ with pk≥4. Then G and we obtainSuppose that Claim B thatIt follows by p and any k∈ℕ. Assume to the contrary that K<ρ*. Then Claim B implies that ρ<r. Since nr is not prime power, it follows by Lemma 3.7.3 that K>r, a contradiction. □Suppose that 4.G′ is a further finite abelian group with From now on, we assume Lemma 4.1.ℒ(G) = ℒ(G′) and d∈Δ1(G). ThenG is isomorphic to a subgroup of G′, then G≅G′.If Suppose Proof.D(G) = D(G′). It follows from . Then d≥max{r,⌊nr∕2⌋}, Proposition 3.8.1 implies that Suppose that Δ*(G) is an interval, we obtain that k be maximal such that there exists a subgroup H of G with k′ be maximal such that there exists a subgroup H′ of G′ with r≥r+k≥nr−2 and 5. By 3, we have r≠r(G′) and by symmetry, we can assume r<r(G′). Thus nr is even, nr = 2r+2, and r(G′) = r+1. Since Δ*(G) and k′≥r. Then G is a subgroup of G′ which implies that G≅G′ by Lemma 4.1.1, a contradiction.Assume to the contrary that Proof of Theorem 1.2.By definition of transfer Krull monoids, it follows that r≤n−3. If Δ*(G) is not an interval, then [G≅G′.1. Let Δ*(G) is an interval. Then Theorem 4.2.5 implies that n = exp(G′) and r = r(G′). Therefore G′ is isomorphic to a subgroup of G which implies that G≅G′ by Lemma 4.1.1.Suppose n = 2, then G is an elementary two-group and the assertion follows by Geroldinger and Halter-Koch [n≥3 is a prime power. Then Theorem 4.2.2 implies that r = r(G′) and G is isomorphic to a subgroup of G′ which implies that G≅G′ by Lemma 4.1.1.2. If 5.D(G)≥4 and Throughout this section, let Conjecture 5.1.r≥n−1. Then the following hold:s∈ℕ with gcd = 1.Suppose C2 implies C1. By Lemma 3.7.1, we have s∈ℕ and if n is a prime power, then C2 holds.It is easy to see that Proposition 5.2.G′ be a finite abelian group with ℒ(G′) = ℒ(G). If r≥n−1 and C1 holds, then G≅G′.Let Proof.n is a prime power, then the assertion follows by Theorem 1.2.If n is not a prime power. Then r≥n−1≥5. Let ℒ(G) = ℒ(G′) and let s∈ℕ. If s′∈ℕ. Thus s,n) = 1. Therefore G is isomorphic to a subgroup of G′ which implies that G≅G′ by Lemma 4.1.1.Suppose s,n) = 1, we have n = r+2, a contradiction.Suppose G′ and there is known no group G′ with Recall that Proposition 5.3.G′ be a finite abelian group with ℒ(G′) = ℒ(G). If r≥max{(u−1)n+1,n}≥3 and K(G) = K*(G), then G≅G′.Let Proof.u = 1, then the assertion follows by Theorem 1.2.2. Suppose u≥2. Note r≥(ω(n)−1)n+1, we have s′∈ℕ. Therefore s,n) = 1. The assertion follows by Proposition 5.2.If


Scientific Reports 10.1038/s41598-019-39429-2, published online 11 February 2019Correction to: The Article contains errors in Table 1, where the reported second column heading under Urea CD was incorrectly titled. Additionally, there are errors in four of the values listed under ‘Urea Fluorescence’ and ‘Urea CD’.The correct Table Additionally, there is an error in Table 3, in which the value “137 ± 2x” under FXN L198C should read “137 ± 2”.


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C−H∙∙∙π and N−H∙∙∙π interactions can have an important contribution for protein stability. However, direct measurements of these interactions in proteins are rarely reported. In this work, we combined the mutant cycle experiments and molecular dynamics (MD) simulations to characterize C−H∙∙∙π and N−H∙∙∙π interactions and their cooperativity in two model proteins. It is shown that the average C−H∙∙∙π interaction per residue pair is ~ −0.5 kcal/mol while the N−H∙∙∙π interaction is slightly stronger. The triple mutant box measurement indicates that N−H∙∙∙π∙∙∙C−H∙∙∙π and C−H∙∙∙π∙∙∙C−H∙∙∙π can have a positive or negative cooperativity. MD simulations suggest that the cooperativity, depending on the local environment of the interactions, mainly arises from the geometric rearrangement when the nearby interaction is perturbed. Theoretical studies show that N−H∙∙∙π, O−H∙∙∙π and C−H∙∙∙π can have very different optimum geometries, with the interaction strength order O−H∙∙∙π > N−H∙∙∙π > C−H∙∙∙π11. The S−H∙∙∙π interaction can be weaker9 or stronger12 than O−H∙∙∙π, but is generally stronger than N−H∙∙∙π and C−H∙∙∙π12. The computational interaction energy of the indole-benzene dimer where the N−H∙∙∙π interaction exists can reach −5.2 kcal/mol13. The computational interaction energies between benzene and CH4, NH3, H2S, and H2O are −1.4, −2.5, −2.9 and −3.0 kcal/mol, respectively9. The computational binding energies between indole and CH4, NH3, H2S, and H2O are −2.0, −2.6, −4.9 and −3.6 kcal/mol, respectively12. The importance of the S−H∙∙∙π and C−H∙∙∙π interactions in proteins has also been highlighted by their occurrence in the PDB database search15. The C−H∙∙∙π interaction has been observed directly in proteins by nuclear magnetic resonance (NMR) spectroscopy methods where the across C−H∙∙∙π J-coupling is detected16. Quantification of C−H∙∙∙π in calix[4]pyrrole receptors yields a magnitude of −1 kcal/mol17. The C−H∙∙∙π interaction in benzene−methane, ethane, propane, and butane, increases monotonically from −1.1 to −2.7 kcal/mol20. The measurement of C−H∙∙∙π interactions in a cyclohexylalanine−phenylalanine pair in the core of a synthetic peptide indicates that each C−H∙∙∙π contact can contribute about −0.7 kcal/mol to peptide stability21. In real proteins, C−H∙∙∙π mainly occurs between an aliphatic side chain and an aromatic ring, or between two aromatic rings14. Although C−H∙∙∙π interactions are well documented in proteins1, direct measurements of C−H∙∙∙π and N−H∙∙∙π strength in proteins are scarce.X−H∙∙∙π interactions in biomolecules, where X can be C, N, O, or S are weak and attractive interactions between the X−H component and aromatic groups. The high incidence in biomolecules makes X−H∙∙∙π interactions an important contributor to the structure and function, and has led to an increasing number of theoretical and experimental studies devoted to characterization of such interactions22. By forming networks of weak interactions that compete against the entropy of flexible polypeptides, proteins fold into their biologically functional three-dimensional structures23. As a part of the interaction network, how X−H∙∙∙π interactions coexist and cooperate in proteins is an important question. Only a few studies have addressed the X−H∙∙∙π cooperativity, mainly in small molecules. The cooperativity of C−H∙∙∙π interactions in small molecules is studied using molecular torsional balances24. The average C−H∙∙∙π interaction strength increases as more C−H∙∙∙π pairs are formed, suggesting a positive cooperativity. This is opposite to the findings of an earlier computational study where the negative cooperativity is concluded for the same complexes25. The C−H∙∙∙π and N−H∙∙∙π cooperativity in proteins remains largely unexplored.Another important issue about X−H∙∙∙π interactions is their cooperativity. Cooperativity is a central concept for understanding molecular recognition and supramolecular self-assembly26. SNase is an enzyme that hydrolyzes nucleotides in DNA or RNA. A stable mutant of SNase, Δ + PHS, is selected as the test system27. It is found experimentally that the C−H∙∙∙π interaction on average is about −0.5 kcal/mol and the N−H∙∙∙π interaction on average is about −0.6 kcal/mol. N−H∙∙∙π…C−H∙∙∙π and C−H∙∙∙π…C−H∙∙∙π can have different cooperativities. Molecular dynamics (MD) simulations can reproduce N−H∙∙∙π and C−H∙∙∙π interactions and their cooperativities with reasonable accuracy. Geometric parameters that are important for C−H∙∙∙π and N−H∙∙∙π interactions are discussed. Their contribution to cooperativity is illustrated. With the combination of experimental and computational results, a better view of C−H∙∙∙π, N−H∙∙∙π and their cooperativity is obtained.In this work, we attempt to measure the C−H∙∙∙π and N−H∙∙∙π interactions in protein GB3 and staphylococcal nuclease (SNase). GB3 is the third immunoglobulin binding domain of protein G, a model protein that has been extensively studiedBased on the X-ray crystal structures, a series of X−H∙∙∙π interactions can be identified in GB3 and Δ + PHS . GB3 has five residue pairs that may form C−H∙∙∙π interactions, L5−F30, T18−F30, L5−Y33, I7−Y33, and T16−Y33, and one residue pair N37−Y33 that can form the N−H∙∙∙π interaction Fig. . Δ + PHSG of all proteins were derived from the denaturation curves. The values of [D]50%, m values for the wild type and mutant proteins are given in Supplementary Table 28. The values of C−H∙∙∙π interactions are shown in Table The folding free energies Δx and y in the X−H∙∙∙π pair to alanine residues to completely remove the interactions between the two. However, eliminating an aromatic residue in a protein core can be detrimental to protein stability. Instead, we only mutated the aromatic side chain (y) to a leucine (y′) which is still hydrophobic but disrupts the X−H∙∙∙π interaction . For the X−H component (x), conservative mutations are introduced (x′) to remove the X−H∙∙∙π interaction and maintain the protein folding at the same time. These mutations may create residual pairwise side chain interactions in x′y′, xy′, and x′y. Furthermore, for a residue like leucine  which has two CH3 and one CH, it can form multiple C−H∙∙∙π interactions which complicate the interpretation of the experimental results. These problems can be solved with the assist of MD simulations.According to DMC, it is preferable to mutate the two side chains 29, Charmm2730, and Gromos53a631. The experimental C−H∙∙∙π and N−H∙∙∙π interaction energies were used as a benchmark to evaluate the accuracy of different force fields. The root mean square deviation (RMSD) between the experimental and predicted X−H∙∙∙π interactions was calculated:N is 31, the total number of measured residue pairs that form X−H∙∙∙π interactions, ΔΔGexp is the experimental X−H∙∙∙π interaction energy, and ΔΔEMD is the calculated interaction energy. Charmm27 appears to perform better than the other two force fields. Its RMSD value is 0.27 kcal/mol (after removing two apparent outliers), while the RMSDs of Amber99sb and Gromos53a6 are 0.41 and 0.47 kcal/mol, respectively , and ω is the ∠C−H−X angle  versus ΔΔECH3∙∙∙π shows that the geometries with shorter dCX and larger ω have more negative interaction energies , although the absolute value of ∆ΔΔE is generally larger than that of ∆ΔΔGcoop  and thus not pursued. As discussed above, the residual interactions caused by the experimental non-alanine mutations complicate the interpretation of ∆ΔΔGcoop. To solve this problem, we rebuilt TMBs by mutating the three side chains, for example L25, F34, and V74 in L25−F34−V74, to alanines systematically in MD simulations. The cooperativity energy ∆ΔΔE′ was calculated for the residue groups listed above with the same procedure  can vary from −40% (cooperative) to +60% (anticooperative)  distance when the aliphatic side chain of the second C−H∙∙∙π (or N−H∙∙∙π) is mutated to alanine. In other words, removing the second C−H∙∙∙π (or N−H∙∙∙π) interaction weakens the first C−H∙∙∙π (or N−H∙∙∙π) interaction, suggesting a positive cooperativity. For 9 out of 10 groups, the distance change Δd predicts the cooperativity consistent with the interaction energy result  Fig. . So it iult Fig. , indicat33. However, measuring C−H∙∙∙π interactions in the protein interior using DMC can be challenging because removing an aromatic side chain can destabilize and even unfold the protein. In this work, we only mutate the aromatic residue to leucine which maintains the protein folding and removes the C−H∙∙∙π interaction. Two very stable proteins GB3 and Δ + PHS were selected for the purpose. One caveat of the F or Y to L mutation is that residual interactions with leucine complicate the data interpretation. Molecular dynamics simulations were used to decompose the various contributions and help us focus on the C−H∙∙∙π interactions. The good agreement between experimental and computational interaction energies validates the procedure which provides important insights about the C−H∙∙∙π and N−H∙∙∙π interactions.DMC experiments are commonly used to measure residue−residue interactions, such as salt bridges and hydrogen bonds21. It is likely that different interactions compete with each other in proteins so that the C−H∙∙∙π interaction of a specific residue pair is not in an optimum geometry. This is evident from the interaction energy landscape of methyl−aromatic ring pair  calculations, the cooperativity energy of MMB is 0.74 kcal/mol, indicating that C−H∙∙∙π…C−H∙∙∙π is anticooperative in this model, while the cooperativity energy of MBM is 0.03 kcal/mol, suggesting that there is no cooperativity in this model. Similar to the result in the MD simulations, the geometric reorganization occurs in the MMB model where the two methanes compete for the binding site. No such competition exists in the MBM model where the cooperativity energy is close to zero. The QM calculations highlight the importance of geometric reorganization to cooperativity.Two simpler cooperativity models were built using two methane and one benzene molecules, with methanes on the same side (MMB) or opposite side (MBM) of the benzene. The cooperativity energies of MMB and MBM models were calculated at the MP2/aug-cc-pvtz leveldC(N)X and ω. The experimental TMB analysis suggests that the cooperativity of X−H∙∙∙π interactions can be either positive or negative, depending on the local environment. The cooperativity trend is successfully captured by MD simulations where the cooperativity energy can reach ~ −40% to 60% of C−H∙∙∙π or N−H∙∙∙π interactions, highlighting its importance in proteins. The geometric rearrangement is the main cause for the cooperative interactions. It is worth noting that the C−H∙∙∙π and N−H∙∙∙π interactions and the cooperativity were only measured for two proteins GB3 and Δ + PHS. More measurements will be needed to see whether the conclusions also hold for other proteins. But we expect that the mechanism behind the interactions is universal for all protein molecules.In this study, we measured the strength of C−H∙∙∙π and N−H∙∙∙π interactions in GB3 and SNase. The C−H∙∙∙π interaction is about 0.3 to −0.9 kcal/mol whereas the N−H∙∙∙π interaction is about −0.2 to −0.9 kcal/mol. The energy decomposition from MD simulations helps determine the C−H∙∙∙π and N−H∙∙∙π interactions for individual methyl−aromatic and amino−aromatic pairs and identify important geometric parameters E. coli strain BL21 (DE3) cells for protein expression. The purification procedure for GB3 and its variants has been described previously35. Δ + PHS and its variants were purified using the same procedure as described by Shortle and Meeker36.The wild type and mutants of GB3 and Δ + PHS were prepared with the PCR-based site-directed mutagenesis on vector pET-11b. These plasmids were transformed into the S is the measured Fluo340nm or Fluo348nm, αN and αU are the intercepts and βN and βU are the slopes of the Fluo340nm or Fluo348nm baselines at low (N) and high (U) denaturant concentrations, R is the Boltzmann constant, T is the temperature, [D] is the denaturant concentration, [D]50% is the denaturant concentration at which the protein is 50% denatured.All the denaturation measurements were performed using a HITACHI f-4600 fluorescence Spectrophotometer. Mixtures consisted of up to 6.0 M GdnHCl and 50 µM proteins  were incubated for 30 min at 30 °C. The signal intensity at 340 nm for GB3 and 348 nm for SNase was extracted and fitted using the following equation,38. Double mutant cycles were performed to quantify C−H∙∙∙π interactions and N−H∙∙∙π interactions in this work. To build the DMC, dozens of single and double mutants were prepared. Single mutants included L5V, I7V, T16A, T18A, N37A, F30L, Y33L, Y33F in GB3 and L25V, V74A, I92V, F34L in Δ + PHS. Double mutants contained two substitutions, L5V−F30L, L5V−Y33L, I7V−Y33L, T16A−Y33F, T16A−Y33L, T18A−F30L, N37A−Y33L and N37A−Y33F in GB3, and L25V−F34L, V74A−F34L, I92V−F34L in Δ + PHS. The folding free energy for each mutant was determined from the denaturation curve monitored by fluorescence. The C−H∙∙∙π or N−H∙∙∙π interaction energy with the aromatic ring was then calculated using:Gxy, ΔGx′y, ΔGxy′, and ΔGx′y′ are the folding free energy for the wild type protein xy, single mutants x′y and y′x, and the double mutant x′y′, respectively. The symbols x and y denote the aliphatic and aromatic side chains in the C−H∙∙∙π or N−H∙∙∙π pair. This expression can be defined for both GB3 and Δ + PHS proteins.Double mutant cycle (DMC), proposed by Fersht and co-workers, can eliminate the contribution of the secondary interactions and obtain accurate binding energy for the interaction between two residues39. Double mutants of GB3  and Δ + PHS  were used to set TMBs. All of these double mutant proteins could be expressed except L5V-I7V of GB3. Triple mutants were prepared, including L5V-T16A-Y33L, L5V-T18A-F30L, I7V-T16A-Y33L, L5V-N37A-Y33L, I7V-N37A-Y33L, T16A-N37A-Y33L, and T16A-N37A-F33L for GB3, and L25V-V74A-F34L, L25V-F34L-I92V, and V74A-I92V-F34L for Δ + PHS. These mutants were used to quantify the cooperativity in C−H∙∙∙π∙∙∙C−H∙∙∙π interactions and C−H∙∙∙π∙∙∙N−H∙∙∙π interactions. The folding free energy for each mutant was measured using the same method mentioned above. The cooperativity energy was then calculated using:y represents the aromatic residue, x and z represent nonaromatic residues, ∆Gxyz, ∆Gx′yz, ∆Gxy′z, ∆Gxyz′, ∆Gx′y′z, ∆Gx′yz′, ∆Gxy′z′, and ∆Gx′y′z′ are the folding free energy of the wild type protein xyz, single mutants x′yz, xy′z and xyz′, double mutants x′y′z, x′yz′, xy′z′ and triple mutants x′y′z′, respectively.Two double mutant cycles can be combined to produce a TMB, which can be used for quantification of cooperative effects. Extensive studies have been performed by Hunter and co-workers using triple mutant box experiments to evaluate cooperativity in non-covalent interactions40 with Amber99sb29, Charmm2730, or Gromos53a631 force fields. The structures of all variants of GB3 and Δ + PHS were produced by FoldX41 with the protein backbone fixed. Each protein was solvated by adding 10.0 Å TIP3P water42 (or SPC water when the Gromos53a6 force field was used) in a rectangular box, and counter ions were used to neutralize the system. 500,000 steps of energy minimization followed by 1 ns MD simulation at constant pressure (1 atm) and temperature (303 K) were performed to equilibrate the system before the production running. Three 10 ns MD production runs with different random starting velocities were performed with snapshots saved every 50 ps which were then used in the data analysis and error estimation. All backbone heavy atoms are restrained in the equilibrium and production runs. Temperature was regulated by a modified Berendsen thermostat43 and pressure was controlled by the extended ensemble Parrinello-Rahman approach45. The long-range electrostatic interactions were evaluated by the Particle mesh Ewald method47. The nonbonded pair list cutoff was 10 Å and the list was updated every 10 fs. The LINCS algorithm48 was used to constrain all bonds linked to hydrogen in the protein, whereas the SETTLE algorithm49 was used to constrain bonds and angles of water molecules, allowing a time step of 2 fs. In the energy decomposition analysis, only the interaction energy between the paired residues of C−H∙∙∙π or N−H∙∙∙π was calculated. The computational interaction energy ΔΔE was calculated by,Exy, ΔEx′y, ΔExy′, and ΔEx′y′ are the x−y interaction energy in the wild type protein, x′−y in the single mutant x′y, x−y′ in the single mutant y′x, and x′−y′ in the double mutant x′y′, respectively. The symbols x and y are the same as those in Eq. ε of 4.0 was used for electrostatic interaction energy calculations. The computational cooperativity energy ΔΔΔE was calculated by,y represents the aromatic residue, x and z represent nonaromatic residues, ∆Exyz, ∆Ex′yz, ∆Exy′z, ∆Exyz′, ∆Ex′y′z, ∆Ex′yz′, ∆Exy′z′, and ∆Ex′y′z′ are the interaction energy of x−y−z, x′−y−z, x−y′−z, x−y−z′, x′−y′−z, x′−y−z′, x−y′−z′, and x′−y′−z′ in the wild type protein xyz, single mutants x′yz, xy′z and xyz′, double mutants x′y′z, x′yz′, xy′z′ and triple mutants x′y′z′, respectively.MD simulations were performed using the GROMACS 4.5 package50 level. The energy calculations were performed at the MP2/aug-cc-pvtz34 level. All the calculations were done using the Gaussian 09 software51.Two methane and one benzene molecules were built to model the cooperativity of C−H∙∙∙π∙∙∙C−H∙∙∙π. The geometries of the two models, MMB and MBM, were optimized at the MP2/6-31 + GSupplementary information.


Highly significant and positive correlations at phenotypic level were observed in grain weight per hill (0.796), filled grains per panicle (0.702), panicles per hill (0.632), and tillers per hill (0.712) with yield per hectare, while moderate positive correlations were observed in flag leaf length to width ratio (0.348), days to flowering (0.412), and days to maturity (0.544). By contrast, unfilled grains per panicle (-0.225) and plant height (-0.342) had a negative significant association with yield per hectare. Filled grains per panicle (0.491) exhibited the maximum positive direct effect on yield followed by grain weight per hill (0.449), while unfilled grain per panicle (-0.144) had a negative direct effect. The maximum indirect effect on yield per hectare was recorded by the tillers per hill through the panicles per hill. Therefore, tillers per hill, filled grains per panicle, and grain weight per hill could be used as selection criteria for improving grain yield in rice.The associations among yield-related traits and the pattern of influence on rice grain yield were investigated. This evaluation is important to determine the direct and indirect effects of various traits on yield to determine selection criteria for higher grain yield. Fifteen rice genotypes were evaluated under tropical condition at five locations in two planting seasons. The experiment was laid out in a randomized complete block design with three replications across the locations. Data were collected on vegetative and yield components traits. The pooled data based on the analysis of variance revealed that there were significant differences ( Oryza sativa L.) is an important staple food that constitutes a dominant portion of a world standard diet. In spite of its position among the highly rated cereals, the geometric growth rate of the global population has called for improving the current yield for this extremely important cereal. Thus, several methods have been attempted by scientists to combat this perennial problem. Some researchers have attempted nutritional methods such as physiological methods, breeding, and pest and disease control [Rice , and three commercial varieties were used in this study. The commercial varieties of MR219, MR220, and MR253 were developed by the Malaysian Agricultural Research and Development Institute (MARDI) and officially released in 2001. These varieties were the first set of varieties to be developed by means of a direct seeding planting system. The emphasis was on the panicle characteristics, short life cycle (105–111 days), fairly long but strong culms, and being tolerant to blast and bacterial leaf blight. VN121, VN124, and VN001 were developed by Vietnam Atomic Energy Institute (VINATOM), and the main characteristics of these varieties include good ratoon potential and semidwarf stature. Three mutant varieties were obtained from Bangladesh Institute of Nuclear Agriculture (BINA), namely, Binadhan4, Binadhan7, and Iratom. These varieties possess high-tillering capacity but are prone to lodging. Six advanced mutant lines  were promoted from preliminary studies of ion beam irradiation  with an 2, with subplot size of 4 m2 unit for each genotype in each replication. Optimum date for transplanting at each location was followed according to the farmer's schedule.The field trials were conducted in five locations in two different cropping seasons in peninsular Malaysia, namely, Kota Sarang Semut, Seberang Perai, Tanjung Karang, Sekinchan, and Serdang. The locations covered a wide range of environmental conditions differing in management practices (farmers' fields versus research field), water system (supplementary irrigation versus rainfed), and temperatures (moderate and warm climate conditions). The locations were chosen to represent major rice producing areas in Malaysia. In each environment, the experiment was laid out in a randomized complete block design with three replications. Plot size was 342 m−1. Phosphorus (in form of triple superphosphate) was applied at 15 days at the rate of 57kg per ha−1 and potassium (applied in form of Murate of potash) at 42kg ha−1.Fertilizer application was applied following the recommendation by Malaysian Agricultural Research and Development Institute (MARDI) as nitrogen was applied in form of urea in 30, 55, and 75 days after transplanting at 80, 12, and 20 kg per hectare. NPK fertilizer was also applied in triplicate starting on days 15, 55, and 75 after transplanting at the rate of 140, 107, and 50 kg ha−1) was estimated from the weight of threshed grains from all panicles in 1.5 × 1.5 m2, excluding border rows.Five hills were randomly selected for each genotype in each replication to record observations for plant height, flag leaf length to width ratio, days to 50% flowering, days to maturity, tillers per hill, panicles per hill, panicle length, filled grains per panicle, unfilled grains per panicle, 100-grain weight, and grain weight per hill. Yield in ton per hectare . Association of the various characters with yield per hectare was worked out at genotypic and phenotypic levels as described by Kashiani and Saleh . The pheEffects of Vegetative and Yield Component Variables on Yield per Hectarer112=P112+r12P212+r13P312+r14P412+r15P512+r16P612+r17P712+r18P812+r19P912+r110P1012+r111P1112 r212=r21P112+P212+r23P312+r24P412+r25P512+r26P612+r27P712+r28P812+r29P912+r210P1012+r211P1112 r312=r31P112+r32P212+P312+r34P412+r35P512+r36P612+r37P712+r38P812+r39P912+r310P1012+r311P1112 r412=r41P112+r42P212+r43P312+P412+r45P512+r46P612+r47P712+r48P812+r49P912+r410P1012+r411P1112 r512=r51P112+r52P212+r53P312+r54P412+P512+r56P612+r57P712+r58P812+r59P912+r510P1012+r511P1112 r612=r61P112+r62P212+r63P312+r64P412+r65P512+P612+r67P712+r68P812+r69P912+r610P1012+r611P1112 r712=r71P112+r72P212+r73P312+r74P412+r75P512+r76P612+P712+r78P812+r79P912+r710P1012+r711P1112 r812=r81P112+r82P212+r83P312+r84P412+r85P512+r86P612+r87P712+P812+r89P912+r810P1012+r811P1112 r912=r91P112+r92P212+r93P312+r94P412+r95P512+r96P612+r97P712+r98P812+P912+r910P1012+r911P1112 r1012=r101P112+r102P212+r103P312+r104P412+r105P512+r106P612+r107P712+r108P812+r109P912+P1012+r1011P1112 r1112=r111P112+r112P212+r113P312+r114P412+r115P512+r116P612+r117P712+r118P812+r119P912+r1110P1012+P1112.  The studied traits were further subdivided into two-stage relations: first-order components and second-order components. The first-order components include plant height, flag leaf length to width ratio, days to flowering, days to maturity, and tillers per hill. The second-order components are panicles per hill, panicle lengths, filled grains per panicle, unfilled grains per panicle, 100-grain weight, and grain weight per hill. The cause and effect relationships between the two components were worked out using additional simultaneous equations arranged in matrix notation as indicated in the equations below. Effects of First-Order Components on the Panicles per Hill, Panicle Lengths, Grains per Panicle, Unfilled Grains per Panicle, 100-Grain Weight, and Weight per HillPanicles per Hill r16  =  P16  +  r12P26  +  r13P36  +  r14P46  +  r15P56 r26  =  r21P16  +  P26  +  r23P36  +  r24P46  +  r25P56 r36  =  r31P16  +  r32P26  +  P36 + r34P46  +  r35P56 r46  =  r41P16  +  r42P26  +  r43P36  +  P46  +  r45P56 r56=r51P16  +  r52P26  +  r53P36  +  r54P46  +  P56. Panicle Lengths r17  =  P17  +  r12P27  +  r13P37  +  r14P47  +  r15P57 r27  =  r21P17  +  P27  +  r23P37  +  r24P47  +  r25P57 r37  =  r31P17  +  r32P27  +  P37  +  r34P47  +  r35P57 r47  =  r41P17  +  r42P27  +  r43P37  +  P47  +  r45P57 r57  =  r51P17  +  r52P27  +  r53P37  +  r54P47  +  P57. Grains per Panicle r18  =  P18  +  r12P28  +  r13P38  +  r14P48  +  r15P58 r28  =  r21P18  +  P28  +  r23P38  +  r24P48  +  r25P58 r38  =  r31P18  +  r32P28  +  P38  +  r34P48  +  r35P58 r48  =  r41P18  +  r42P28  +  r43P38  +  P48  +  r45P58 r58  =  r51P18  +  r52P28  +  r53P38  +  r54P48  +  P58. Unfilled Grains per Panicle r19  =  P19  +  r12P29  +  r13P39  +  r14P49  +  r15P59 r29  =  r21P19  +  P29  +  r23P39  +  r24P49  +  r25P59 r39  =  r31P19  +  r32P29  +  P39  +  r34P49  +  r35P59 r49  =  r41P19  +  r42P29  +  r43P39  +  P49  +  r45P59 r59  =  r51P19  +  r52P29  +  r53P39  +  r54P49  +  P59. Unfilled Grains per Panicle r19  =  P19  +  r12P29  +  r13P39  +  r14P49  +  r15P59 r29  =  r21P19  +  P29  +  r23P39  +  r24P49  +  r25P59 r39  =  r31P19  +  r32P29  +  P39  +  r34P49  +  r35P59 r49  =  r41P19  +  r42P29  +  r43P39  +  P49  +  r45P59 r59  =  r51P19  +  r52P29  +  r53P39  +  r54P49  +  P59. 100-Grain Weight r110  =  P110  +  r12P210  +  r13P310  +  r14P410  +  r15P510 r210  =  r21P110  +  P210  +  r23P310  +  r24P410  +  r25P510 r310  =  r31P110  +  r32P210  +  P310  +  r34P410  +  r35P510 r410  =  r41P110  +  r42P210  +  r43P310  +  P410  +  r45P510 r510  =  r51P110  +  r52P210  +  r53P310  +  r54P410  +  P510. 100-Grain Weight r110  =  P110  +  r12P210  +  r13P310  +  r14P410  +  r15P510 r210  =  r21P110  +  P210  +  r23P310  +  r24P410  +  r25P510 r310  =  r31P110  +  r32P210  +  P310  +  r34P410  +  r35P510 r410  =  r41P110  +  r42P210  +  r43P310  +  P410  +  r45P510 r510  =  r51P110  +  r52P210  +  r53P310  +  r54P410  +  P510. Grain Weight per Hill r111  = P111  +  r12P211  +  r13P311  +  r14P411  +  r15P511r211  =  r21P111  +  P211  +  r23P311  +  r24P411  +  r25P511 r311  =  r31P111  +  r32P211  +  P311  +  r34P411  +  r35P511 r411  =  r41P111  +  r42P211  +  r43P311  +  P411  +  r45P511 r511  =  r51P111  +  r52P211  +  r53P311  +  r54P411  +  P51. Effects of Second-Order Components on Yield per Hectarer612  =  P612  +  r67P712  +  r68P812  +  r69P912  +  r610P1012  +  r611P1112 r712  =  r76P612  +  P712  +  r78P812  +  r79P912  +  r710P1012  +  r711P1112 r812  =  r86P612  +  r87P712  +  P812  +  r89P912  +  r810P1012  +  r811P1112 r912  =  r96P612  +  r97P712  +  r98P812  +  P912  +  r910P1012  +  r911P1112 r1012  =  r106P612  +  r107P712  +  r108P812  +  r109P912  +  P1012  +  r1011P1112 r1112  =  r116P612  +  r117P712  +  r118P812  +  r119P912  +  r1110P1012  +P1112. The data presented in In this study, the phenotypic and genotypic correlation coefficient of vegetative, yield, and yield component are separated for a clear understanding as shown in The correlation coefficient of phenotypic characters ranges from 0.026 to 0.818 while the genotypic character correlation coefficient ranges from 0.04 to 0.919. This indicates that there is higher magnitude at the genotypic level in most cases as compared with the corresponding phenotypic level. The yield per hectare shows a strong, positive, and highly significant correlation with grain weight per hill, tillers per hill, filled grains per panicle, and panicles per hill, respectively. Selection based on these four components is effective due to their equal contribution towards grain yield increment.Grain yield is considered as the artefact of all the contributory traits , and the correlation coefficient of these contributory factors with final yield are partitioned into direct and indirect effects as presented in The present study revealed a significant interrelationship among various vegetative and yield components. These traits define the limitation to yield per hectare and the component characters through the direct and indirect effects as a result of interrelationships between them. Therefore, the use of path coefficient analysis investigates the direct and indirect relationships among the component characters through the partitioning of correlation coefficients . The patThe vegetative traits were grouped as the first-order component which includes the following: plant height, flag leaf length to width ratio, days to flowering, days to maturity, and tillers per hill. The second-order component was regarded as yield component traits which were regarded as the principal yield determining factors in rice, and these comprise panicles per hill, panicle length, filled grains per panicle, unfilled grains per panicle, 100-grain weight, and grain weight per hill. The interrelation between these two components is presented in Tables The path of the influence of first-order component on panicles per hill  revealedThe path analysis of second-order component on yield was presented in The critical analysis of path coefficient analysis and partitioning of correlation reveals that tillers per hill and grain weight per hill possessed positive direct effect and positive association with yield per hectare. Selection for the improvement of grain yield can be efficient if it is based on tillers per hill and grain weight per hill because of their contribution directly towards grain yield. Breeders in these areas should, therefore, develop early maturing genotypes focusing on tillers per hill and grain weight per hill for improving the grain yield per plant in both rainfed and irrigated area.