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Water Research 36(2002)469–481Nitrifying and heterotrophic population dynamics in biofilm reactors:effects of hydraulic retention time and the presence oforganic carbonRegina Nogueira a,b ,Lu !ıs F.Melo a ,Ulrike Purkhold b ,Stefan Wuertz c ,Michael Wagner b,*aDepartment of Chemical En g ineerin g ,Faculty of En g ineerin g of Porto,LEPAE,Porto,PortugalbLehrstuhl fur Mikrobiolo g ie,Technische Uni v ersit .at M .u nchen,Am Hochan g er 4,D-85350Freisin g ,Germany cTechnische Uni v ersit .a t M .u nchen,Am Coulombwall,D-85748Garchin g ,GermanyReceived 29September 2000;received in revised form 09January 2001;accepted 1February 2001AbstractTwo biofilmreactors operated with hydraulic retention tim es of 0.8and 5.0h were used to study the links betweenpopulation dynamics and reactor operation performance during a shift in process operation from pure nitrification to combined nitrification and organic carbon removal.The ammonium and the organic carbon loads were identical for both reactors.The composition and dynamics of the microbial consortia were quantified by fluorescence in situ hybridization (FISH)with rRNA-targeted oligonucleotide probes combined with confocal laser scanning microscopy,and digital image analysis.In contrast to past research,after addition of acetate as organic carbon nitrification performance decreased more drastically in the reactor with longer hydraulic retention time.FISH analysis showed that this effect was caused by the unexpected formation of a heterotrophic microorganism layer on top of the nitrifying biofilm that limited nitrifiers oxygen supply.Our results demonstrate that extension of the hydraulic retention time might be insufficient to improve combined nitrification and organic carbon removal in biofilm reactors.r 2002Elsevier Science Ltd.All rights reserved.Keywords:Nitrification;Biofilm;Organic carbon;Hydraulic retention time;Fluorescence in situ hybridization1.IntroductionThe competition between heterotrophic and nitrifying bacteria for substrates (oxygen and ammonia)and space in biofilms is of major practical importance and thus has been the subject of several previous studies (for example [1–3]).According to these investigations,competition in biofilms results in a stratified biofilm structure,the fast growing heterotrophic bacteria being placed in the outer layers,where both substrate concentration and detach-ment rate are high,while the slow growing nitrifying bacteria stay deeper inside the biofilm.Thus a hetero-trophic layer can formabove the nitrifiers in the biofilm ,which constitutes a disadvantage to themwhen the bulk liquid oxygen concentration is low.In this case oxygen limitation resulting from consumption and resistance to mass transfer within the heterotrophic layer affects the nitrification performance negatively.As long as the bulk oxygen concentration is high enough to preclude its depletion in the biofilm,however,the heterotrophic layer can also have a positive effect on the nitrifiers by protecting themfromdetachm ent [4].One possible approach to minimize the competition of heterotrophs and nitrifiers for oxygen is their spatial separation into a nitrifying biofilmpopulation and a*Corresponding author.Tel.:+49-8161-71-5444;fax:+49-8161-71-5475.E-mail address:wagner@mikro.biologie.tu-muenchen.de (M.Wagner).0043-1354/02/$-see front matter r 2002Elsevier Science Ltd.All rights reserved.PII:S 0043-1354(01)00229-9heterotrophic population in suspension.This separation can be achieved by extending the reactor’s hydraulic retention time.At comparatively long hydraulic reten-tion times,the fast growing heterotrophic microorgan-isms(with reciprocal maximum specific growth rates smaller than the selected hydraulic retention time)grow mainly in suspension while the slow growing nitrifiers formbiofilm s[2].However,in this study the reactor was operated at a high oxygen concentration of6mg lÀ1,a condition which did also allow full nitrification in a conventional biofilmreactor(nitrifiers and heterotrophs in the biofilm)[2].Reactors with a spatial separation of heterotrophic and nitrifying activity for the treatment of effluents with a high chemical oxygen demand(COD)/NH4+-N ratio can be very attractive for future practical applications if (i)it can be demonstrated that these reactors allow complete nitrification at a relatively low dissolved oxygen concentration and if(ii)the energy savings in aeration relatively to the traditional process compensate the investment costs in the construction of a bigger reactor required due to its longer retention time.So farthe effects of influent composition and operational conditions were studied with respect either to total reactor performance(macroscale studies)or to biofilm composition and structure(microscale studies),but hardly to both simultaneously(for example[3]).The research subjects of this study were the effects of different hydraulic retention times and changes in the organic carbon dosing on the population dynamics of nitrifying biofilmreactors operated at around2m g lÀ1 dissolved oxygen.The specific objectives were:(i)to identify and quantify nitrifying and heterotrophic bacter-ia in the biofilmand in suspension using a set of rRNA-targeted oligonucleotide probes forfluorescence in situ hybridization(FISH),(ii)to correlate changes in micro-bial community composition in the biofilm and suspen-sion with reactor performance during a shift from pure nitrification to combined nitrification and organic carbon removal,and(iii)to asses the recovery of the nitrification process after a shift back to pure nitrification.2.Materials and methods2.1.Biofilm reactorsTwo laboratory-scale circulating bed reactors(CBR) of1.2l each[5]were employed for this study.This type of airlift reactor has a rectangular geometry of310mm height and60mmÂ60mm cross-section and is sepa-rated in an up-flow aerated section and a down-flow non-aerated one by a vertical wall(Fig.1).High density polyethylene granulate with a particle size of1mm and a density of731kg mÀ3was used as support material for biofilmgrowth.The superficial air velocity in both reactors(defined as the airflow rate divided by the aerated reactor cross-section)was set at0.003msÀ1.The temperature was maintained at301C,and the pH was kept at7.5by adding sodiumhydroxide(1M).The experimental protocol included4phases(Fig.2).The corresponding experimental conditions are summarized in Table1.Phase I.A CBR(hereafter reactor R0)wasfilled with nitrifying biofilmparticles(23volum e percent)obtained froma nitrifying circulating bed reactor,which was maintained at identical operating conditions.The biofilmparticles had been stored at41C for90days prior to inoculation of reactor R0.An ammonium solution was supplied continuously(N operation mode), with a retention time of0.70h.During phase I a stable nitrifying biofilmwas established.Phase II.Half of the biofilmparticles fromreactor R0 were transferred to a second identical reactor.Both CBRs(hereafter reactors R1and R2)were operated simultaneously with retention times of0.8h for reactor R1and 5.0h for reactor R2.The ammonium load supplied to each reactor was approximately half of the value used in phase I in order to maintain a constant NH4+-N load to biomass ratio(0.46kg kgÀ1dÀ1),and still no organic carbon was added(N operation mode). Phase III.Acetate as an organic carbon source was supplied to reactors R1and R2at the same COD/NH4+-N mass ratio(C+N operation mode)in order to investigate the effect of organic carbon on the nitrifica-tion performance of reactors operating at different retention times.Phase IV.Discontinuation of the supply of acetate to reactors R1and R2in order to study the recovery of the nitrification process(N operationmode).Fig.1.Schematic diagram of continuous circulating bed reactor set-up.R.No g ueira et al./Water Research36(2002)469–481 470Phases I,II and IV are characteristic of a pure nitrification process,while phase III corresponds to a combined nitrification and organic carbon removal process.2.2.MediaDuring the experimental phases I,II and IV (N operation mode)3.4ml min À1of an ammonium medium was delivered to each reactor.The medium contained (NH 4)2SO 4(0.59g l À1),NaHCO 3(0.19g l À1),KH 2PO 4(0.06g l À1),CaCl 22H 2O (0.014g l À1)and trace elements.In reactors R 0and R 1,also 15.9ml min À1of deionized water was added to the reactors to obtain the desired hydraulic retention time.During phase III (C+N operation mode) 1.7ml min À1of a sterilized acetate solution (CH 3COONa 3H 2O,0.71g l À1)was added separately to reactors R 1and R 2.In order to maintain the retention time in both reactors constant,the ammonium medium’s volumetric flow rate was reduced to half,and its concentration was doubled.2.3.O v erall kineticsThe macroscale reactor performance was evaluated fromeffluent sam ples filtered with 0.22m m membrane filters.COD,ammonia plus ammonium,nitrite and nitrate were determined photometrically (LCK,nge).Biofilmand suspended biom ass total solids were measured according to APHA [6]using 0.22m m membrane filters.Prior to this analysis,the biofilm was detached fromthe support m aterial by an ultrasonic homogenizer (Bandelin electronics D-1000,Berlin)treatment (120s at 50W).The dissolved oxygen concentration was measured with an oxygen electrode (WTW,model Oxi 340-A).2.4.Fluorescence in situ hybridization,microscopy and quantification of probe-tar g eted bacteriaMicrobial population dynamics in biofilm particles and suspended biomass was evaluated using FISH with rRNA-targeted oligonucleotide probes.Samples 1–6Table 1Operating conditions and performance of reactors R 0,R 1,and R 2.For all parameters average values are given for the different phases of operation Phase/mode of operationInfluentNH 4+-N loadNH 4+-Nremoval (%)COD loadCOD removal (%)COD/N applied (g g À1)NH 4+-N COD (kg m À3)AppliedRemoved(kg m À3d À1)AppliedRemoved (kg m À3d À1)Reactor R 0hydraulic retention time of 0.7h I N 0.037F a 1.26 1.2095FFFF Reactor R 1hydraulic retention time of 0.8h II N 0.0200.010.610.60980.330.04120.53III C+N 0.0220.030.650.45690.850.6172 1.30IV N 0.0190.010.570.51900.170.05290.30Reactor R 2hydraulic retention time of 5.0h II N 0.1000.020.480.481000.100.04400.21III C+N 0.1090.140.530.53-0100-00.680.6190 1.28IV N 0.1170.020.560.53940.080.05630.14aF =notdetermined.Fig.2.Schematic diagram describing the experimental phases.The arrows indicate biofilm and suspended biomass sampling times.R.No g ueira et al./Water Research 36(2002)469–481471(Fig.2)were taken from the reactors and immediately fixed with paraformaldehyde.During phases I and II(N operation)only biofilmwas sam pled since the suspended biomass concentration was very low,while during phase III(acetate addition)both biofilmand suspended biomass samples were taken.In situ characterization of microbial populations followed a top to bottomapproach(F ig.3).F irst the samples were hybridized with a probe set(EUB338, EUB338-II,EUB338-III)designed to target almost all bacteria[7],then with previously published group specific probes(Fig.3;[8,9]).The ammonia-oxidizing and nitrite-oxidizing bacteria were identified using previously published probes(Fig.3;[10–14]). Oligonucleotide probes were purchased as derivatives labeled with thefluorescent dyes Cy3,Cy5,and5(6)-carboxyfluorescein-N-hydroxysuccinimide-ester (FLUOS),respectively(Interactiva,Ulm,Germany). FISH was performed using the hybridization and washing buffers as described by Manz et al.[8].A Zeiss LSM510laser scanning confocal microscope(Zeiss, Jena,Germany)was used for image acquisition.For quantification of probe-targeted bacteria,simultaneous hybridizations were performed with Cy3labeled specific probes and the FLUOS labeled bacterial probe set.The relative biovolume defined as the ratio between the area of probe-targeted bacteria to the area of all bacteria detectable by FISH was determined for each probe in20 randomly recorded confocal images(thickness1m m) using the procedure described by Schmid et al.[15]. Biofilmthickness was determ ined for fresh,unfixed biofilmsam ples which were stained with a0.25g lÀ1fluorescein isothiocianate solution for3h at room temperature,using CLSM optical sectioning in the sagittal(xz)direction.parati v e sequence analyses of the amoA g ene High resolution analyses of ammonia-oxidizer diver-sity in reactor R0was performed using the gene encoding the catalytic subunit of the ammonia-mono-oxygenase enzyme(amoA)as a marker.Amplification, cloning,sequencing and phylogenetic analyses of the biofilm-derived amoA fragments was performed as described by Purkhold et al.[16].3.Results3.1.Reactor performance3.1.1.Phase IOne of the main drawbacks of biological nitrification processes is the requirement of a long start-up period. By using biofilmparticles as inoculum,the start-up period of reactor R0in phase I could be reduced to4 days after which the ammonium removal efficiency had reached95%.Subsequently,R0’s biofilmparticles were split between reactors R1and R2to ensure that both reactors had the same original microbial populationcomposition.3.1.2.Phases II to IVFig.4(A–D)depicts the performance of reactors R1 and R2operated with retention times of0.8and5.0h, respectively,during pure nitrification(phases II and IV) and combined nitrification and organic carbon removal (phase III).Tables1and2summarize the experimental results obtained during reactors operation and the characterization of biofilmand suspended biom ass. Though acetate as organic carbon source was only added during phase III,there was a certain background COD in the influent during phases I,II and IV(Table1;F igs.4B and D)deriving fromoxidizable m atter in the deionized water source.However,during these phases maximal63%of the incoming COD was removed demonstrating that a significant fraction of these compounds were not degraded in the reactors.During phase II,both reactors had a stable perfor-mance,no nitrite accumulation was observed and the NH4+-N effluent concentration was below 1.0mg lÀ1 corresponding to an ammonium removal efficiency higher than95%.The biofilmcharacterization at the end of phase II showed similar biofilm thickness(41and 42m m),and biofilm mass concentration(2.48and 2.43kg mÀ3)in reactors R1and R2.Shortly after the addition of acetate to reactor R1 (start of phase III),the ammonium removal rate decreased from0.65to0.45kg mÀ3dÀ1(69%)and hereafter,was constant(Fig.4A).Due to a mechanical problem in the ammonium dosing pump,reactor R1 received an ammonium overload10days after the start of phase III(gray area in Fig.4A).The ammonium removal rate,however,remained constant.In reactor R2 Fig.4.Time changes of NH4+-N,COD and oxygen during operation of reactors R1and R2with pure nitrification(phases II and IV) and combined nitrification and organic carbon removal(phase III).Closed symbols correspond to influent concentrations and open symbols to effluent concentrations.*Corresponds to oxygen concentrations.R.No g ueira et al./Water Research36(2002)469–481473the ammonium removal continuously decreased to zero within 50days (Fig.4C),while the dissolved oxygen concentration simultaneously increased (Fig.4B).A constant COD removal rate of 0.61kg m À3d À1was reached within 5days in both reactors (Figs.4B and D)and no nitrite accumulation was observed during the entire phase III.The increase of biofilmthickness and mass after carbon addition was more pronounced in reactor R 2than in R 1(Table 2).The amount of biomass found in the reactors during phase III was by a factor of 10to 40higher than the value expected for theoretical equilibriumbetween bacteria growth rate and dilution rate (Table 2).This inconsistency was obviously caused by an accumulation of mostly heterotrophic biomass in the reactors’bottom and effluent tubes increasing the actual sludge retention time considerably.In reactor R 2this effect was supported by the lower hydraulic retention time,leading to a higher biomass accumulation than in R 1.The accumulated biomass was removed once per week and included in the samples for suspended solids quantifica-tions and FISH analyses of suspended cells.After the discontinuation of acetate addition (phase IV)the ammonium removal in both reactors recovered and after a period of 14days approx.90%of the influent ammonium load was nitrified.In reactor R 2the dissolved oxygen concentration decreased back to the value at the beginning of phase III.3.2.Di v ersity of nitrifyin g bacteria in the reactors The ammonia-oxidizing cells in all biofilm samples could be labeled simultaneously with probes BET42a,Nso1225and Nso190.No ammonia-oxidizers belonging to the Nitrosospira -cluster were detected.A fraction ofthe ammonia-oxidizing population was detectable with probe NEU,while Nitrosococcus mobilis was absent.The NEU-positive subpopulation of ammonia-oxidizers is most likely affiliated with the Nitrosomonas europaea/eutropha group [17].Comparative sequence analyses of amoA clones derived fromthe biofilmof reactor R 0independently confirmed the presence of two different groups of ammonia-oxidizers (Fig.5).One amoA sequence cluster is closely related to Nitrosomonas europaea ,most likely representing the NEU-positive ammonia-oxidizers,while the other amoA cluster is not closely related with any described ammonia-oxidizer reference strain.In both reactors the nitrite-oxidizing cells in the biofilmwere affiliated with the genus Nitrospira during all phases of operation.Members of the genus Nitrobacter were only detected in the biofilm fromreactor R 2(5.0h retention time)during operation with acetate addition (Table 3).3.3.Di v ersity of heterotrophic bacteria in the reactorsThe heterotrophic bacteria present in the biofilmof reactor R 2were Proteobacteria of the alpha-and beta-subclasses while in reactor R 1only beta-subclass Proteobacteria could be detected (Table 3).In both reactors the microbial populations in the suspended biomass during phase III were Proteobacteria of the alpha-,beta-and gamma-subclasses and bacteria belonging to Cytopha g a-Fla v obacterium -cluster (Table 4).The dominating microbial populations both in biofilm and suspended biomass during combined organic carbon and ammonia oxidation belong to the beta-subclass of Proteobacteria .Concerning the two reactorTable 2Characterization of biofilmand suspended biom ass in reactors R 0,R 1,and R 2during the different phases of operation.Values listed in the table are the average 795%confidence interval Mode of operation/sampleBiofilmm ass perreactor volume (kg m À3)Suspended solidsconcentration (kg m À3)Biofilmthickness (m m)Reactor R 0hydraulic retention time of 0.7h N 3 2.7070.80F a3375Reactor R 1hydraulic retention time of 0.8h N 4 2.4870.37F4172C+N 5 2.4970.37FF C+N 6 2.5470.410.4670.014473Reactor R 2Hydraulic retention time of 5.0h N 4 2.4370.40F 4273C+N 5 2.7270.40FF C+N 6 3.6370.380.6170.045675aF =not determined.R.No g ueira et al./Water Research 36(2002)469–481474operation modes (N and C+N operation)the major difference is that during pure nitrification all beta-subclass Proteobacteria were ammonia-oxidizers,while during acetate dosing an additional presumably hetero-trophic beta-subclass population developed.With re-spect to the different hydraulic retention times the major difference is a presumably heterotrophic alpha-subclass Proteobacteria population that occurred during pure nitrification only in reactor R 2biofilmand disappeared after introduction of acetate.3.4.Population dynamics in biofilm and suspended biomassValues for each probe’s targeted-bacteria are depicted in Table 3for biofilmand in Table 4for suspended biomass.The relative biovolumes of the different microbial populations (determined by FISH)were normalized by taking into account the differences in biomass (measured as dry weight)in the different reactors and samples.The FISH biovolume percentages for the sample with the highest biomass per reactor volume (sample 6)were kept unchanged and the FISHpercentages of the other samples were reduced according to the biomass differences.The fraction of the heterotrophic beta-Proteobacteria population (HET)present in the biofilmof reactors R 1and R 2can be determined as follows:ðArea BET42a ÞHET ¼Area BET42a ÀArea Nso1225:ð1ÞSimultaneous hybridization of biofilm samples with probes ALF1b and Ntspa662designed for specific detection of the alpha-subclass of Proteobacteria and the genus Nitrospira ,respectively,demonstrated that members of the genus Nitrospira are non-specifically targeted by probe ALF1b.Consistent with this finding,a recent data base inspection demonstrated that Nitros-pira -like 16S rRNA-sequences retrieved fromwaste-water treatment plants possess the full match target site of probe ALF1b.Consequently,Nitrospiras have to be included in the list of non-alpha-subclass Proteobacteria targeted by probe ALF1b [8].The relative biovolume labeled with probe ALF1b was higher than the one labeled with probe Ntspa662in biofilmsam ples taken fromreactor R 2,except for the last biofilmsam ple during phase III (Table 3).This demonstrates that an ALF1b-positive population not related to Nitrospira appeared in the biofilmwhen the hydraulic retention time changed from 0.8to 5.0h,corresponding to the transition fromreactor R 0to reactor R 2(phase II F N operation mode),and disappeared after the operation of reactor R 2with acetate for 50days (phase III F C+N operation mode).The fraction of the alpha-Proteobacteria population (HET)developed in the biofilmfromreactor R 2during pure nitrification (phase II)can be estimated as follows:ðArea ALF1b ÞHET ¼Area ALF1b ÀArea Ntspa662:ð2ÞThe fraction of biofilmand suspended biom ass bacteria identified with specific gene probes (F)in relation to all bacteria (EUB338probe set)was calculated as listed below:Biofilmsam plesArea ALF1b E Area Ntspa662)F ¼Area BET42a þArea Ntspa662;ð3ÞArea ALF1b >Area Ntspa662)F ¼Area BET42a þArea ALF1b :ð4ÞSuspended biomass samplesF ¼Area ALF1b þArea BET42a þArea GAM42a þArea CF319a :ð5ÞThe population dynamics of nitrifiers and hetero-trophs in the biofilmand suspended biom ass from reactors R 0,R 1and R 2during the different phases of operation is depicted in Fig.6.For most biofilm and suspension samples,more than 80%of the bacteria detectable by the EUB338probe set couldsimulta-Fig.5.Phylogenetic FITCH tree reflecting the relationships ofthe ammonia-oxidizers in reactor R 0based on amoA sequences.The scale bar indicates the number of expected amino acid substitutions per site per unit of branch length.Numbers in brackets indicate the number of clones with almost identical sequences (>99%amino acid sequence similarity)which were retrieved fromthe reactor.R.No g ueira et al./Water Research 36(2002)469–481475neously be classified by at least one of the more specific probes applied (Tables 3and 4;Eqs.3–5).However,this does not hold true for the biofilmand suspended biomass samples taken from reactor R 1during the accidental overloading period with ammonium (sample 5).In this sample 41%of the biofilm bacteria and 24%Table 3Microbial community composition of reactors R 0,R 1and R 2biofilmduring the different phases of operation a Sample/reactorNFOligonucleotide probes FALF1bBET42a Nso1225NEU Ntspa662NIT3Phase I F N removal 1R 00.74393925380105F (5374)(5374)(3473)(5274)(0)2R 0F F F (5675)(3075)0F 3R 00.742041411920082(2773)(5574)(5574)(2673)(2773)(0)Phase II F N removal 4R 10.682130301623078(3173)(4576)(4576)(2476)(3375)(0)R 20.673521211427083(5273)(3175)(3175)(2173)(4172)(0)Phase III F C+N removal 5R 10.681226191214059(1873)(3975)(2773)(1773)(2172)(0)R 20.753029211923080(4077)(3975)(2877)(2575)(3076)(0)6R 10.70243421522081(3476)(4978)(3074)(772)(3275)(0)R 21.0029600031394(2976)(6074)(0)(0)(3176)(371)aThe relative biovolumes of probe-defined bacterial populations (values in brackets)and the respective normalized values in regard to the biomass content of the samples (bold values)are given in columns 3–8.No normalized values are given for sample 2since the biomass content was not determined for this sample.The fraction of bacteria detectable with the bacterial probe set,which were identified with specific oligonucleotide probes,is given in column 9.Values listed in the table are the average percentage 795%confidence interval.F or all biofilmsam ples no signals were observed with probes specific for the Cytopha g a-Fla v obacterium -cluster (CF319a)and the gamma-subclass of Proteobacteria (GAM42a),respectively.NF=normalization factor;F =not determined.Table 4Microbial community composition of reactors R 1and R 2suspended biomass during combined nitrification and organic carbon removal (columns 3–6)a Sample/reactorNFOligonucleotide probes FALF1bBET42a GAM42a CF319a Phase III F C+N removal 5R 10.18 2.211.500.276(1271)(6472)(0)(170)R 20.17 1.6140.50.999(1071)(8171)(370)(571)6R 10.18 3.114.60.20.2100(1773)(8172)(170)(170)R 20.171.2131.9 1.5104(771)(7774)(1171)(972)aColumn 7displays the fraction of bacteria detectable with the bacterial probe set which were identified with specific oligonucleotide probes.Values listed in the table are the average percentage 795%confidence interval (values in brackets).NF=normalization factor.R.No g ueira et al./Water Research 36(2002)469–481476of the suspended bacteria which were detectable by FISH could not further be classified with the more specific probes.Whether these bacteria appeared in the reactor due to the acetate addition or the elevated ammonia concentrations could not be clarified.3.5.Population dynamics durin g phase IBiofilmsam ples were taken before and after the storage period preceding operation of reactor R 0(samples 1and 2)and after 112days of operation (sample 3),in order to evaluate the effects of the storage period on the microbial community composition.The samples were hybridized in situ with probes Nso1225,NEU and Ntspa662,as well as the EUB338probe set.After the storage period (sample 2)the fraction of ammonia-oxidizers was larger than during normal operation (samples 1and 3)while the relative abundance of nitrite-oxidizers decreased (data not shown),suggesting that the nitrite-oxidizers decayed faster than the ammonia-oxidizers under the selected storage conditions.The characterization of the biofilm fromreactor R 0during operation (sample 3)showed that the ammonia-oxidizing population detected with probe Nso1225was able to recover a comparablerelative abundance as before the storage period (sample 1),while the nitrite-oxidizing population decreased considerably from 38%(sample 1)to 20%(sample 3).Furthermore,the morphology of the nitrite-oxidizing clusters changed frombig clusters present before the storage period (sample 1),to small ones mixed with net-like structures afterwards (sample 3).Despite the decrease in the relative abundance of nitrite-oxidizers,no nitrite accumulation was observed during phase I.This suggests that the nitrite-oxidizers present in net-like structures in sample 3are more active (possibly due to a better accessibility to substrates)than those occurring in big clusters in sample 1.3.6.Population dynamics durin g phase IIIn the transition fromphases I to II the am ount of biofilmparticles fromreactor R 0was split up between reactors R 1and R 2,while the NH 4+-N load per reactorwas reduced from1.24to 0.48–0.61kg m À3d À1.Despite the lower amount of biofilm particles in R 1and R 2than in R 0,the biomass concentration per reactor volume was similar in both reactors during phases I and II (Table 2).This can be explained by a lower biofilmdetachm ent rate in reactors R 1and R 2caused by a decrease intheFig.6.Population dynamics of ammonia-oxidizers,nitrite-oxidizers and heterotrophs in the biofilm during process operation of reactors R 1and R 2with pure nitrification [phase I (sampling point 3)and phase II (sampling point 4)]and combined nitrification and organic carbon removal (phase III F sampling points 5and 6).In addition,the microbial community composition of the suspended biomass is displayed for both reactors for sampling point 6.R.No g ueira et al./Water Research 36(2002)469–481477。
信息熵InformationTheory信息论（Information Theory）是概率论与数理统计的⼀个分枝。
⽤于信息处理、信息熵、通信系统、数据传输、率失真理论、密码学、信噪⽐、数据压缩和相关课题。
本⽂主要罗列⼀些基于熵的概念及其意义，注意本⽂罗列的所有 log 都是以 2 为底的。
信息熵在物理界中熵是描述事物⽆序性的参数，熵越⼤则越混乱。
类似的在信息论中熵表⽰随机变量的不确定程度，给定随机变量 X ，其取值x1,x2,⋯,x m，则信息熵为:H(X)=m∑i=1p(x i)⋅log1p(x i)=−m∑i=1p(x i)⋅log p(x i)这⾥有⼀张图，形象的描述了各种各样的熵的关系：条件熵设 X ,Y 为两个随机变量，X 的取值为x1,x2,...,x m ,Y 的取值为y1,y2,...y n，则在X 已知的条件下 Y 的条件熵记做 H(Y|X) :H(Y|X)=m∑i=1p(x i)H(Y|X=x i)=−m∑i=1p(x i)n∑j=1p(y j|x i)log p(y j|x i)=−m∑i=1n∑j=1p(y j,x i)log p(y j|x i)=−∑x i,y j p(xi,y j)log p(y j|x i)联合熵设 X Y 为两个随机变量，X 的取值为x1,x2,...,x m ,Y 的取值为y1,y2,...y n，则其联合熵定义为:H(X,Y)=−m∑i=1n∑j=1p(x i,y j)log p(x i，y j)联合熵与条件熵的关系：H(Y|X)=H(X,Y)−H(X)H(X|Y)=H(X,Y)−H(Y)联合熵满⾜⼏个性质：1）H(Y|X)≥max(H(X),H(Y)) ;2）H(X,Y)≤H(X)+H(Y) ;3）H(X,Y)≥0.相对熵 KL距离相对熵，⼜称为KL距离，是Kullback-Leibler散度（Kullback-Leibler Divergence）的简称。
the theory of information and coding 英文The theory of information and coding refers to a field in computer science and mathematics that deals with the study of how information is measured, manipulated, and transmitted reliably and efficiently.This theory was first developed by Claude Shannon in 1948 as part of his groundbreaking work on communication theory. Shannon formulated a mathematical theory of communication, where he introduced the concept of entropy as a measure of information and provided a framework for systematically encoding and transmitting information over noisy channels.The theory of information and coding encompasses various concepts and techniques related to encoding, decoding, error detection, and error correction. It involves the study of different coding schemes, such as Huffman coding, Reed-Solomon coding, and Hamming coding, which are used to represent information in a more compact and robust manner.Information theory also explores fundamental limits and capacities of different communication systems, such as the maximum achievable data rate or the minimum error probability. This involves the calculation of channel capacity, information rate, and the development of practical coding schemes that approach these theoretical limits.Moreover, the theory of information and coding has applications in various fields, including telecommunications, data compression, error correction, cryptography, and network coding. It plays acrucial role in the design and development of efficient and reliable communication systems, as well as in understanding the fundamental principles underlying information processing. Overall, the theory of information and coding provides a mathematical foundation for understanding how information can be efficiently and reliably transmitted, encoded, and decoded in different communication systems. It continues to be an active and important field of research, with ongoing advancements and applications in various domains.。
关于熵的英语作文Entropy is a fundamental concept in physics, information theory, and other fields. It is a measure of the disorder or randomness in a system, and plays a key role in understanding the behavior of complex systems.In thermodynamics, entropy is often described as a measure of the degree of disorder in a system. The second law of thermodynamics states that the total entropy of a closed system always increases over time, meaning that the system becomes more disordered or random. For example, when heat is added to a system, it increases the disorder of the system, and this leads to an increase in entropy.In information theory, entropy is used to measure the amount of uncertainty or randomness in a message. The more uncertain or random a message is, the higher its entropy. For example, a message with only one possible outcome has zero entropy, while a message with many possible outcomes has higher entropy.Entropy is also important in the study of complex systems, such as biological systems or social systems. In these systems, there are many different components that interact in complex ways, and understanding the entropy of these systems can provide insights into their behavior.One example of this is the study of protein folding. Proteins are long chains of amino acids that must fold into specific three-dimensional shapes in order to function properly. The process of protein folding is highly complex and can be affected by many different factors, including temperature and pressure. Understanding the entropy of the system can help scientists predict how a protein will fold, and this can have important implications for drug design and other areas of biotechnology.In summary, entropy is a fundamental concept that plays a key role in understanding the behavior of complex systems. It is a measure of the disorder or randomness in a system, and is used in thermodynamics, information theory, and other fields. By studying entropy, scientists can gain insights into the behavior of complex systems, and this can have important practical applications in fields such as biotechnology and materials science.译文：熵是物理学、信息论和其他领域中的一个基本概念。
离散随机奇异系统的零和博弈及H∞控制周海英【摘要】针对噪声依赖于状态的It(o)型离散随机奇异系统,讨论其在有限时域下的零和博弈及基于博弈方法的H..控制问题.在最优控制(单人博弈)的基础上,利用配方法,得到了离散随机奇异系统鞍点均衡策略的存在等价于相应的耦合Riccati代数方程存在解,并给出了最优解的形式.进一步地,根据博弈方法应用于鲁棒控制问题的思路,得到离散随机奇异系统H∞控制问题的最优策略,最后根据动态投入产出问题的特性,建立相应的博弈模型,得到动态投入产出问题的均衡策略.【期刊名称】《南昌大学学报（理科版）》【年(卷),期】2017(041)006【总页数】5页(P519-523)【关键词】离散随机奇异系统;零和博弈;耦合Riccati代数方程;鞍点均衡策略【作者】周海英【作者单位】广州航海学院港口与航运管理系,广东广州 510725【正文语种】中文【中图分类】F224.32奇异系统由于其广泛的应用背景,自产生以来，得到了广泛研究 [1-4]。
随着研究的深入，随机奇异系统由于能更好的模拟现实实际，近年来，引起了众多研究者的兴趣。
在随机奇异系统的稳定性、最优控制及鲁棒控制方面都有不少成果。
Yan Z等研究了伊腾型随机广义系统的稳定性问题[5]。
Zhang W等研究了广义随机线性系统的稳定性问题[6]；Jin H等研究了随机奇异系统的虑波问题[7]。
文献[8]把神经网络法应用于随机奇异系统不定线性二次控制问题中，得到了相应的Riccati微分方程；高明等研究了离散随机Markov跳跃系统的广义Lyapunov方程解的性质[9]；张庆灵等在研究随机奇异系统的稳定性的基础上，得到了连续随机奇异系统线性二次最优控制的Riccati方程[10]。
Xing等研究了不确定广义随机线性系统的H∞鲁棒控制问题[11]。
Zhang和Zhao Y等研究了广义随机线性系统的H∞鲁棒控制问题[12-13] ；Shu Y等研究不确定连续时间奇异系统的稳定性和最优控制问题 [14]。
· 1 ·2.1 试问四进制、八进制脉冲所含信息量是二进制脉冲的多少倍？ 解：四进制脉冲可以表示4个不同的消息，例如：{0, 1, 2, 3}八进制脉冲可以表示8个不同的消息，例如：{0, 1, 2, 3, 4, 5, 6, 7} 二进制脉冲可以表示2个不同的消息，例如：{0, 1} 假设每个消息的发出都是等概率的，则：四进制脉冲的平均信息量H(X 1) = log 2n = log 24 = 2 bit/symbol 八进制脉冲的平均信息量H(X 2) = log 2n = log 28 = 3 bit/symbol 二进制脉冲的平均信息量H(X 0) = log 2n = log 22 = 1 bit/symbol 所以：四进制、八进制脉冲所含信息量分别是二进制脉冲信息量的2倍和3倍。
2.2 居住某地区的女孩子有25%是大学生，在女大学生中有75%是身高160厘米以上的，而女孩子中身高160厘米以上的占总数的一半。
假如我们得知“身高160厘米以上的某女孩是大学生”的消息，问获得多少信息量？ 解：设随机变量X 代表女孩子学历X x 1（是大学生） x 2（不是大学生） P(X) 0.25 0.75设随机变量Y 代表女孩子身高Y y 1（身高>160cm ） y 2（身高<160cm ） P(Y) 0.5 0.5已知：在女大学生中有75%是身高160厘米以上的 即：p(y 1/ x 1) = 0.75求：身高160厘米以上的某女孩是大学生的信息量 即：bit y p x y p x p y x p y x I 415.15.075.025.0log )()/()(log )/(log )/(2111121111=⎪⎭⎫⎝⎛⨯-=⎥⎦⎤⎢⎣⎡-=-=2.3 一副充分洗乱了的牌（含52张牌），试问 (1) 任一特定排列所给出的信息量是多少？(2) 若从中抽取13张牌，所给出的点数都不相同能得到多少信息量？ 解：(1) 52张牌共有52！种排列方式，假设每种排列方式出现是等概率的则所给出的信息量是：bit x p x I i i 581.225!52log )(log )(2==-=(2) 52张牌共有4种花色、13种点数，抽取13张点数不同的牌的概率如下：bit C x p x I C x p i i i 208.134log )(log )(4)(13521322135213=-=-==· 2 ·2.4 设离散无记忆信源⎭⎬⎫⎩⎨⎧=====⎥⎦⎤⎢⎣⎡8/14/1324/18/310)(4321x x x x X P X ，其发出的信息为(202120130213001203210110321010021032011223210)，求(1) 此消息的自信息量是多少？(2) 此消息中平均每符号携带的信息量是多少？ 解：(1) 此消息总共有14个0、13个1、12个2、6个3，因此此消息发出的概率是：62514814183⎪⎭⎫ ⎝⎛⨯⎪⎭⎫ ⎝⎛⨯⎪⎭⎫ ⎝⎛=p此消息的信息量是：bit p I 811.87log 2=-=(2) 此消息中平均每符号携带的信息量是：bit n I 951.145/811.87/==2.5 从大量统计资料知道，男性中红绿色盲的发病率为7%，女性发病率为0.5%，如果你问一位男士：“你是否是色盲？”他的回答可能是“是”，可能是“否”，问这两个回答中各含多少信息量，平均每个回答中含有多少信息量？如果问一位女士，则答案中含有的平均自信息量是多少？ 解： 男士：sym bolbit x p x p X H bitx p x I x p bit x p x I x p i i i N N N Y Y Y / 366.0)93.0log 93.007.0log 07.0()(log )()( 105.093.0log )(log )(%93)( 837.307.0log )(log )(%7)(22222222=+-=-==-=-===-=-==∑女士：symbol bit x p x p X H ii i / 045.0)995.0log 995.0005.0log 005.0()(log )()(2222=+-=-=∑2.6 设信源⎭⎬⎫⎩⎨⎧=⎥⎦⎤⎢⎣⎡17.016.017.018.019.02.0)(654321x x x x x x X P X ，求这个信源的熵，并解释为什么H(X) >log6不满足信源熵的极值性。
entropy名词解释
解释：熵，平均信息量;负熵
entropy 是什么意思及中文翻译：
名词(n.) 熵，平均信息量
名词(n.) 负熵
entropy 大小写变形：Entropy
entropy 词态变化：副词：entropically形容词：entropic
包含entropy的单词
Entropy [电影]崩溃
negentropy 负平均信息量，负熵（指选定的特殊符号所产生的信息的数值量度）clozentropy 克洛兹特罗普
包含entropy的短语
entropy flow 熵流
entropy flux 熵流
entropy rate 熵速率
entropy index 熵指数
entropy layer 熵层
entropy power 熵功率
entropy union 熵联合
communal entropy 共有熵
conformational entropy 构象熵
enthalpy-entropy compensation 焓熵补偿
entropy 相关例句
The Entropy of Any System at Absolute Zero
绝对零度下系统的熵
Information Entropy Method in Actuarial Science
保险精算的信息熵方法
UNITY OF CLAUSIUS ENTROPY AND BOLTZMANN ENTROPY
克劳修斯熵与玻耳兹曼熵的统一性
NERNST THEOREM AND PLANCK ABSOLUTE ENTROPY OF BLACK HOLE。
信息论与编码学习辅导及习题详解Information Theory and Coding Learning Coaching and Exercise ExplanationInformation Theory and Coding are two closely related scientific disciplines for signal processing, data communication, and computer networks. These two studies are often used together in applications, making the learning of the concepts of Information Theory and Coding important. This essay will guide through the basic knowledge of these two fields, provide coaching on learning Information Theory and Coding, and go through exercise explanations.I. Information TheoryA. DefinitionInformation Theory is a branch of applied mathematics and electrical engineering dealing with the quantification, storage, and communication of information. It was developed in 1948 by Claude Shannon. The fundamental problem of communication is to convey a message from a sender to a receiver without any errors. Information Theory is the study of how information is transmitted over a communication medium, and how the communication medium affects the transmission process.B. Basic ConceptsSome basic concepts of Information Theory are entropy, noise, channel capacity, coding, and error control. Entropy is a measure of uncertainty, and is used to help determine the amount of information contained in a signal.Noise is any disturbances that have an impact on the transmission of information. Channel capacity is the maximum amount of data that can be transmitted through a communication channel. Coding is the process of translating the message from the source into a form that can be understood by the destination. Error control is the process of detecting, identifying, and correcting any errors that occur during transmission.II. CodingA. DefinitionCoding is a branch of mathematics and computer science dealing with the efficient representation of data. It was developed in the late 1950s and early 1960s. Coding is used in a variety of applications, including data storage, image processing, digital signal processing, and communications. Coding techniques can greatly reduce the amount of data that needs to be stored and transmitted.B. Basic ConceptsThe main concepts of Coding are coding, signaling, modulation, coding rate, coding efficiency, and entropy. Coding is the process of transforming the message from the source into a form that can be understood by the destination. Signaling is the process of conveying information through a medium over a communication link. Modulation is the process of varying some aspect of a signal in order to transmit information. The coding rate is the number of bits required to encode one message. The coding efficiency is the ratio between the actual number of bits used to encode the message, and the total number of bits used. Entropy is a measure of the amount of information contained in a signal.III. Learning CoachingA. FundamentalsThe best way to learn the fundamentals of Information Theory and Coding is to start by familiarizing oneself with the core concepts such as entropy, noise, channel capacity, coding, and error control. Taking a college course in Information Theory and Coding is also beneficial. Alternatively, reading textbooks and studying reference material is a viable option.B. PracticePracticing the concepts of Information Theory and Coding is essential to mastering them. It is important to try to understand the material rather than memorize it. Doing practice problems is the best way to learn and build an understanding of the material.IV. Exercise ExplanationA. Information TheoryFor the Information Theory part of this exercise, the main goal is to determine the maximum rate at which data can be transmitted through a given communication channel. To do this, one needs to first calculate the entropy of the signal to determine the amount of information contained in it. Then, the channel capacity needs to be calculated by taking the s ignal’s entropy, the noise of the channel, and the coding rate into consideration.B. CodingFor the Coding part of this exercise, the main goal is to encode a message into a format that can be understood by the destination. To do this, one needsto first select an appropriate coding technique. Then, the information needs to be encoded using this technique. Finally, the encoded message needs to be transmitted through a communication link.In conclusion, Information Theory and Coding are two important scientific fields for signal processing, data communication, and computer networks. This essay has guided through the basics of these two fields, provided coaching on learning Information Theory and Coding, and gone through exercise explanations. Therefore, it is essential for one to understand the fundamentals of Information Theory and Coding and practice the concepts in order to gain mastery in these fields.。
