大肠杆菌E.coli CVCC249在分批和连续培养条件下的生长动力学和生理特性比较研究
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摘要
在限制性培养条件下生长形成的微生物群体组成具有异质性的特点，其生长是一非线性并涉及多变量的复杂性的动态过程。因此，如果通过构建微分速率模型来概括这一复杂的动力学过程，势必不得不做出若干简化的假定，例如，在Malthus指数增长模型(dN/dt=rN或N(t)=N_0d~(-rt))基础上，Logistic方程(dN/dt=rN-βN~2)中加入阻滞因子β，Gompertz方程(dN/dt=μ_0Ne~(-st))中加入衰减参数s，以及Richards方程(Vt=A(1-Be~(-kt))~(1/(1-m)))中再加入调整曲线形状的参数m等，但它们也都只能表征一定情况下的生长过程，而难以概括全部可能发生的情况。尽管可在多项式生长速率方程中加入高次项来适合生长过程的变化：dN/dt=F(N)=C_0+C_1N+C_2N~2+…+C_nN~i，但引入高次项CnN~i(i≥3)后，模型参数的生物学意义经常很难符合。建立多参数的非线性微分方程组也一直是此领域的研究方向，但模型的求解困难，以及其参数的生物学意义不明，仍然是一难题。用分段函数方程来描述生长过程，并以其作为生长阶段划分的依据仍然有待讨论。研究细菌增殖规律时，也通常假定细菌群体的组成是均质性的，认为细菌以“一分为二”的模式进行增殖，即产生同质的2个子细胞后，母细胞不复存在，这又与细菌群体组成的异质性相矛盾，无法体现细菌群体增殖的内在规律。
本论文针对上述存在的问题，通过对E.coli CVCC249群体瞬时生长速率的研究，分析了不同生长条件下菌群生长的动力学特性，表明用菌群瞬时生长速率，可揭示出菌群异质性形成的机理。通过对生长动力学过程的分析，指出了经典方程在限制性培养条件下表征生长动力学过程的局限性，建立了新的区分生长阶段、估算生长参数，以及准确表征菌群异质性对生长动态过程的影响的新方法。进一步用Fibonacci数列分析，以及连续培养法与流式细胞技术相结合的研究策略，提出了细菌群体增殖的新模式。测定了均质性和非均质性细菌群体对抗菌药物的敏感性。并在此基础上，将数值微分法成功应用于菌群生长过程中酶催化的研究。与此领域中广为认可的研究结果相比较，本论文在以下5个方面取得了具有一定原始创新性的结果：
1．提出了分批培养条件下细菌群体生长阶段的区分及确定生长参数的新方法
为减少光吸收测定数据的误差，应用3次样条插值法(Spline Interpolation Method)将观测数据的离散点转化为连续光滑曲线，以真实反映动态过程，再用数值微分法(Numerical Differential Method)直接求得瞬时生长速率(instantaneous growth rate，v_(inst))。然后用Gaussian分布函数拟合过程曲线，求解出具有明确生物学含义的参数，简称该方法为SNG法(Spline-Numerical-Gaussian)。由此提出了区分群体生长阶段的新方法：即依据瞬时生长速度和瞬时生长加速度规律性的变化，把细菌群体生长过程区分为加速生长期、恒速生长期、减速生长期和衰亡期四个阶段(图1)，并得到新的生长参数。测定了群体生长过程中DNA、蛋白质含量，以及脱氢酶比活力的变化与瞬时生长速率的相关性。
在上述研究过程发现，在经典的表征群体生长的Logistic方程中，通常都假定群体中每个个体都是同质的，因此，解方程时的初值仅限定为N(t_0)=N_0，这样，对N_0相同而生理状态不同的接种物的生长动力学就不能区分，这一影响生长方程正确求解的思路在广泛应用的生物生长方程中都存在。事实上，在菌群生长的任一时刻，只有一定比例的细胞在进行分裂(设为θ)，这样，dN/dt不仅由N，而且实际上应由N和θ共同决定。因此，解方程的初值条件还应加入θ(t_0)=θ_0，而这一重要和必需条件又被在构建生长方程时所忽略，但这一附加条件使Logistic方程的求解甚为复杂。基于细菌生长过程的瞬时生长速率v_(inst)的时间过程曲线都呈Gauss分布形状这一特点，在用瞬时生长速率成功地表征限制性条件下区分群体生长阶段的基础上，进一步用Gaussian分布近似函数表达式(Y_i=αe~([-0.5((x_i-x_0)/b)~2]))以求得异质性群体的生长参数，这些参数可真实地表征不同生理状态细菌群体的生长动态，为微生物学基础研究和在生物工程等方面的应用提供了一个新方法。将新的表征群体生长的建模方法(SNG法)与依据Logistic方程划分生长阶段的传统方法作了比较，表明此方法可克服Logistic方程中存在的问题，而能准确表征E．coli CVCC249在分批培养下的生长动力学过程。
2．根据在分批和连续培养条件下菌群对药敏试验的不同影响，提出了一个准确进行药敏试验的新方法
根据菌群瞬时生长速率的变化，将E．coli CVCC249的分批培养过程区分为加速生长期、恒速生长期、减速生长期和衰亡期四个阶段，各阶段中菌群的核酸、蛋白质含量和脱氢酶活力都有明显差别；分别取样，测定各阶段菌群的浓度-杀菌曲线(Concentration-killing Curve，CKC)，检测结果表明这种生理异质性会导致各阶段菌群对药物敏感性的不同，同时，菌群内个体间药物敏感性也存在差异，呈现正态分布。以加速生长期菌群的CKC曲线为参照，对于庆大霉素，恒速生长期和衰亡期菌群药物敏感性增强；对于恩诺沙星，恒速生长期反而减弱，其它阶段差别不大。这是常规药敏试验方法难以显示的，表明对不同生理状态菌群的药敏试验测定结果可反映出药物的不同杀菌机理，但这同时也表明常规的药敏检测不能给出一个稳定的药物敏感性结果(图2)。而在连续培养下得到的“同步生长”状态的菌群，则可得到稳定的药敏试验结果(图3)，这有助于常规药敏试验方法的改进，并可为抗菌药物的合理使用和抗菌药物的筛选提供新方法。
3．证实了连续培养下细菌菌群以1→1+1'…模式进行增殖
用Fibonacci数列分析方法对在分批和连续培养下E．coli CVCC 249菌株的群体增殖过程进行了分析。Fibonacci数列模式是一经典递增函数，已广泛应用于表征生物生长发育过程。其数列分析属于数列的差分分析方法，它可把一个复杂的动力学、动态学过程简化求解。本文首次由数列的递推系数(recurrent coefficient for sequence)、递增系数(incremental recurrent coefficient for sequence)和净增率(net increment％)(或增长百分比)表征群体的增殖规律。分析结果表明，E．coli CVCC 249在连续培养下，群体中的细胞以1_(-M)(母细胞)→1_(-M)(母细胞)+1_(-D)(子细胞)→…的分裂模式进行增殖，即子细胞形成后，母细胞仍可继续进行同样的分裂增殖，简称为1→1+1'…(1…母细胞，1'…子细胞)模式。在分批培养条件下，菌群的净增率逐渐下降，最终成为不再进行增殖的状态，这是在营养供给受限和产物抑制双重作用下，代时逐渐延长的结果。在分批培养下，菌群的瞬时生长率，DNA复制和蛋白质合成的瞬时增长率，以及表征代谢还原能力的脱氢酶的比活力，都呈现正态分布，其数列分析结果也与活细胞数的分析结果一致。在相同的分批培养条件下，低密度接种可得到高的细胞净增率，而高密度接种则可迅速达到最高生长量。稳态接种材料的生长动力学类似于低密度接种。在连续培养条件下，稀释率的增加可加速淘汰最快和最慢增殖的个体，更快达到同步生长状态，从而使群体增殖率分布更趋于正态化，这与以流式细胞技术检测的不同稀释率下菌群内细胞尺度分布的结果相一致。数列分析结果表明，稳态下群体净增率为零，存在于培养罐中菌群的递增系数恒定在1．0(图4)，表明在开放系统条件下E．coli CVCC 249群体以1→1+1'…模式增殖。用Fibonacci数列分析方法对大肠杆菌以1→1+1'…(即类似于DNA的半复制模式)这一增殖模式的确定，涉及细菌基本增殖规律性的认识。对细菌突变率的测定、药敏测定中不确定性等难题的解决，以及生物工程中条件优化设计等，都会提供新的策略。
4．用瞬时速率可准确表征β-半乳糖苷酶的催化效率并证实了活性中间体的存在
通常依Michaelis-Menten方程进行测试时，酶的最大反应速率v_(max)和最初反应速率v_0决定着酶催化反应速率的常数k_(cat)。但由于v_0和v_(max)测试和拟合中的不确定性，从而导致k_(cat)结果的不确定性。因此本文采用一种改进的方法即数值微分法来准确的估算β-半乳糖苷酶的催化反应速率常数。依照IUPAC(International Union of Pure andApplied Chemistry)用ξ来定义反应速率的建议和国标的类似规定，dξ／dt定义为瞬时反应速率(v_(inst))。用实验数据直接得到瞬时反应速率，并以其作为目标函数，来确定β-半乳糖苷酶水解邻硝基苯酚-β-D-半乳糖苷(ONPG)的最适反应条件(如最适温度、时间、pH值和[S]／[E]等)。β-半乳糖苷酶的活性可以由反应过程中不断生成的邻硝基苯酚(ONP)的UV-Vis吸收光谱来表征。因此，在最适反应条件下，得到的瞬时反应速率的最大值(v_(inst-max))与达到瞬时反应最大值时的酶浓度([E]_(inst-max))之比，可准确的估算β-半乳糖苷酶的催化反应速率常数。未经纯化的样品中的β-半乳糖苷酶也可以通过这种方法估测。由这种改进方法与广为应用的Michaelis-Menten方程(v=v_(max)[s]/k_m+[s])得到的v_(max)和k_m求k_(cat)进行了比较，进一步剖析了Michaelis-Menten测试方法中不能得到稳定的k_(cat)等问题的原因。
通常用固定温度下求不同时间过程对反应速率影响的方法，或固定时间而改变温度的方法，来表征温度和时间对反应速率的影响。此方法属于求偏导数方法，因而无法全面表征温度和时间对速率的复合作用。而这2个变量的复合函数c-v_(inst)=[(?)v/(?)t]_T+[(?)v/(?)T]_t则可表征2者的共同影响。β-半乳糖苷酶降解ONPG过程中，ONPG的减少及ONP的增加可以用UV动态光谱图来测定，此过程的净转化速率(d[ONPG]／dt-d[ONP]／dt)即可表示活性中间产物(ES-复合物)的存在(图5)。温度和时间对ES-复合物形成的v_(inc)的复合影响，可用等高线作图法进行展示(图6)。通过这种途径，可以得出ONPG转化成ONP酶解过程中一些新的信息，即活性中间物并不是动力学曲线中的一个点，而是在三维空间上从反应物到产物酶解过程中的一个运动聚集体，并且受温度和时间的综合影响。活性中间物的动力学行为也可以在三维图上直观地表征出来。研究结果可为酶促反应动力学研究的深入开展提供一种新思路。
5．提出了β-半乳糖苷酶催化邻硝基苯酚-β-半乳糖苷(ONPG)水解机理的新模式
以固定浓度的β-半乳糖苷酶，对不同浓度的ONPG在0℃下进行结合。以紫外线光谱、荧光光谱的变化分别确定酶的饱和结合位点数SBP。结果表明，β-半乳糖苷酶的SBP数为8，即酶分子的一个亚基可结合2个ONPG分子。通过用新的de Donder方程v_f/v_r=e~(A_i/RT)和公式dG_i=-RT×ln(d[ONPG]/dt/d[ONP]/dt)计算Gibbs自由能变化，表明随着温度的升高和时间的延长，自由能由相对高的区域逐渐低移，至最低点时催化反应不再进行，体现了热力学过程和动力学过程的一致性(图7)。综合上述结果，提出了糖苷酶类催化底物水解的新机理，即(?)，表明在水解反应不能进行的条件下，结合可导致ONPG的分子构型转变为ONP分子构型，而β-半乳糖苷酶的构象变化则可循环发生。“ONPG-β-半乳糖苷酶复合物”与β-半乳糖苷酶和ONPG之间不存在可逆性的平衡过程，在酶分子不断循环进行的结合／解离过程中，由β-半乳糖苷酶构象变化提供的能量，使ONPG分子不断转化为ONP分子构型，结合复合物解离出ONP与半乳糖后，再生的β-半乳糖苷酶又可与ONPG结合，而形成的产物半乳糖作为抑制性因子在水解过程中起作用。这是与通常假定的酶催化过程为一系列可逆平衡反应组成，不可逆反应仅发生于ES复合物分离出产物P的阶段这一传统认识所不同的新观点。
Under batch culture conditions, a bacterial population may be heterogeneous in many important characteristics, such as age-specific reproduction and death rates, cell shape and size and biochemical activities; therefore, the growth of bacterial population presents a nonlinear process with very complicated dynamics. Thus, due to such complexity, the construction of the growth model has to be based on some plausible or implicit assumptions, such as introducing the "β" factor into the Malthus equation (dN/dt=rN ) asin Logistic equation (dN/dt=rN -βN~2), parameter "s" as in Gompertz equation(dN/dt=μ_0Ne~(-st)) and parameter "m" (shape factor) as in Richards equation(Vt = A(1- Be~(-kt))~(1/(1-m))). Generally, these models are constructed based on some simplisedassumptions that may not reflect the well known properties of a biological system and cannot be adapted for all circumstances.
There is a tendency in mathematics to rush into the analysis of a growth model without truly understanding the underlying biology. For example, the term CnN~i (i≥3) hasbeen introduced to the polynomial growth equation(dN/dt=F(N)=C_0+C_1N+C_2N~2+…+C_nN~i),however, the biological significance of theparameters in the model is unclear. Similarly, non-linearity equations with multi-parameters were proposed in biological studies, but did not serve the purposes. It was difficult or impossible to find a sufficient scheme to establish an adequate definition of biological parameters. An exponential polygonal model for population growth has been recently proposed, but it does not provide a sufficient basis for distinction of the growth curve. Microbiologically, it is usually assumed that bacteria divide by a process referred to as binary fission in which one cell divides to produce two cells and a bacterial population is homogeneous. Naturally, there is no complete homogeneous system for a bacterial population and it is only a hypothesized state.
Experimental design and investigation of this thesis focuses on the above mentioned problems. Using the instantaneous growth rate as an index, we analyzed the kinetics characteriactic of a bacterial population at different conditions, discussed the formation mechanism of the population kinetics, and elucidated the relationship between the population heterogeneity and the instantanneous growth rate. Based on the analysis of growth kinetics, a new method was developed to study the growth kinetics of bacterial populations and to distinguish the growth phases in batch culture. We extensively discussed many difficulties confronted by Logistic growth model when describing microbial growth. Five new discoveries were established during my Ph.D. studies, as follows:
1. A novel approach was developed for estimating growth phases and parameters of bacterial population in batch culture
Using mathematical analysis, a new method has been developed to study the growth kinetics of bacterial populations in batch culture. First, sample data were smoothed with the spline interpolation method. Second, the instantaneous rates were derived by numerical differential techniques and finally, the derived data were fitted with the Gaussian function to obtain growth parameters. We named this the Spline-Numerical-Gaussian or SNG method. This method yielded more accurate estimates of the growth rates of bacterial populations and new parameters. It was possible to divide the growth curve into four different but continuous phases based on changes in the instantaneous rates. They are the accelerating growth phase, the constant growth phase, the decelerating growth phase and the declining phase (Fig.1.). Total DNA content was measured by flow cytometry and it varied with the growth phase. The SNG system provides a very powerful tool for describing the kinetics of bacterial population growth. The SNG method avoids the unrealistic assumptions generally used in traditional growth equations.
During past 50 years, different types of the Logistic model are constructed based on a simple assumption that the microbial populations are all composed of homogeneous members and consequently, the condition of design for the initial value of these models has to be rather limited in the case of N(t_0)=N_0. Therefore, these models cannot reflect the different dynamic behavior of the populations possessing the same No from heterogeneous phases. In fact, only a certain ratio of the cells in a population is dividing at any time duringgrowth progress, termed asθ, and thus,dN/dt not only depends on N, but also onθ. SoN(t_0)=N_0 is a necessary element for the condition design of the initial value. Unfortunately, this idea has long been neglected in widely used growth models. However, combining together the two factors (N_0 andθ) into the initial value often leads to the complexity in the mathematical solution. This difficulty can be overcome by using instantaneous rates (V_(inst)) to express growth progress. Previous studies in our laboratory suggested that the V_(inst), curve of the bacterial populations all showed a Guassian function shape and thus, the different growth phases can be reasonably distinguished. In the present study, the Gaussian distribution function was transformed approximately into an analytical form(Y_i=αe~([-0.5((x_i-x_0)/b)~2]))that can be conveniently used to evaluate the growth parameters and inthis way the intrinsic growth behavior of a bacterial species growing in heterogeneous phases can be estimated. In addition, a new method has been proposed, in this case, the lag period and the doubling time for a bacterial population can also be reasonably evaluated. This approach proposed could thus be expected to reveal insight of bacterial population growth. Some aspects in modeling population growth are also discussed.
2. A new method was proposed for estimating the effect of physiological heterogeneity of E.coli population on antibiotic susceptivity test
According to the instantaneous growth rate (dN/dt) of E.coli CVCC249 growing in batch culture, the entire growth curve was distinguished into four phases: accelerating growth phase, constant growth phase, decelerating growth phase and declining phase.Each of four phases have obvious variation in physiological and biochemical properties, including total DNA content, total protein content and MTT-dehydrogenase activity, etc. that leaded to the difference in their antibiotic susceptivity. Antibiotic susceptivity of a population sampled from each phase was tested respectively by Concentration-killing Curve (CKC) approachfollowing the formula N =N_0/1+e~( r(x-BC_(50))), showing as normal distribution at individual cell levelfor an internal population, in which the median bactericidal concentration BC_(50) representedthe mean level of susceptivity, while the bactericidal span BC_(1-99)=2/r lnN_0 indicated thevariation degree of the antibiotic susceptivity. Furthermore, tested by CKC approach, the antibiotic susceptivity of E.coli CVCC249 population in each physiological phase to gentamicin or enoxacin varied: susceptivity of population in constant growth phase and declining phase all increased, compared with that in accelerating growth phase, for gentamicin but declined for enoxacin. This primary investigation revealed that physiological phase should be taken account of in the context of both antibiotic susceptivity test and research into antimicrobial mechanism and bacterial resistance (Fig.2.). However there are few reports regarding this aspect, therefore, further research using different kinds of antibiotics with synchronized continuous culture of different bacterial strains is necessary(Fig.3.).
3. A new model was proposed for expressing the reproduction of a heterogeneous bacterial population
According to the Fibonacci sequence analysis, three parameters, the recurrent coefficient, the incremental recurrent coefficient, and the net increment (%), were used to analyze the growth and reproduction behaviour of E.coli under batch and continuous culture. The analysis result suggested that under continuous culture, E. coli CVCC 249 divided as in regular model of 1_(-M) (mother)→1_(-M)(mother ) + 1_(-D) (daughter)→…termed as 1→1 + 1'…model. Under batch culture, the net increment of the population growth continuously decrease and finally a zero rate will be reached due to nutrition exhaust and the production increase as inhibitor that can lead to the prolonged generation time. All curves including the instantaneous rate, DNA and protein biosynthesis, and MTT activity appear to fit in the normal distribution. Under same batch culture, the net increment (%) will decrease with the increase in the inoculum size, but the highter biomass will be obtained at larger inoculum size. As steady state cultures were used as the inoculum, the growth dynamic behaviour was similar to that lower inoculum size. Under continuous culture, individual at low growth rate wash out with the increase in the dilution rate and the population growth curve present a normal distribution which is similar to that cell size distribution. The research results showed that the incremental recurrent coefficient for sequence is 1.0 (Fig.4.), which suggests that the bacteria reproduce in regular model of 1→1 + 1'…This new model will provide a good indication for accurately determining the mutation rate and the degree susceptivity of bacteria to antibiotic et al.
4. The instantaneous reaction rate was used to express the catalytic efficiency ofβ-galactosidase
To overcome the difficulties in determining the initial rate (V_0) of enzyme-catalyzed reactions based on the classic Michaelis-Menten kinetic assumption and to avoid the uncertainty of calculating the catalytic rate constant (k_(cat)) by extrapolation of V_(max)and k_m data, the instantaneous reaction rate was used to estimate the k_(cat) ofβ-galactosidase. This was based on analysis of the changes in relationship between input data [the value of added O-nitrophenylβ-D-galactoside (ONPG)] and output data [the instantaneous rate of O-nitrophenol (ONP) formation] for entire process of the enzyme-catalyzed reaction. The amount of ONP was determined based on the online UV-visible dynamic spectrum during the entire course of the reaction and its instantaneous change rate (v_(inst)) was directly derived from the dynamic spectrum curve. v_(inst) was used as an objective function to determine the optimum assay conditions forβ-galactosidase activity with variables such as temperature, time, pH, and the ratio of [E] to [S]. Under these optimal assay conditions, the maximum value of v_(inst) (v_(inst-max)) and the corresponding concentration of enzyme ([E] inst-max) could be accurately determined. Then the k_(cat) of p-galactosidase was calculated with the formula: k_(cat) = v_(inst-max)/[E]_(inst-max) .
The amount ofβ-galactosidase in a sample could also be easily determined by this approach without a pre-purification step. The general effectiveness of this new approach and the problems of applying the Michaelis-Menten approximation, particularly for estimating the initial velocity, are also discussed in detail.
The disappearance of ONPG and the appearance of ONP were synchronously measured during the hydrolysis of ONPG byβ-galactosidase using UV-visible spectrum in situ on-line. The conversion process of ONPG to ONP was calculated using d[ONPG]/dt - d[ONP]/dt (Fig.5.). The combined effects of temperature and time on v_(inst) and v_(inc) were expressed as relative variability and visualized by the isograms method - contour plots. With this approach, new insights into the irreversible-continuous conversion of ONPG to ONP during hydrolysis can be clearly observed. That is, the intermediate was a moving-mass flow in three-dimensional space from substrate converting to product during hydrolysis and the temperature-time compensating effects. The dynamic behavior of the intermediate was effectively visualized in a two-dimensional plot (Fig.6.). The results of the present study provide evidence to support the isograms method as a useful tool in understanding the reaction mechanisms of enzyme catalysis.
5. A new model for enzyme-catalyzed reaction was proposed based on a case study on the hydrolysis of ONPG byβ-galactosidase
Based on the synchronously measuring the disappearance of ONPG and the appearance of ONP during hydrolysis of ONPG byβ-galactosidase, the existence of ES-complex as an intermediate can be estimated by the differences between d[ONPG]/dt and d[ONP]/dt. The new de Donder equation (v_f/v_r=exp (A_i/RT)) was used to calculate the changes in Gibbs free energy (dG_i=-RT·ln(d[ONPG]/dt)/(d[ONP]/dt)) (Fig.7.). The results suggested that the interaction of substrate with enzyme in binding process is controlled by the active sites on the surface of enzyme and the conversion of substrate to product is an irreversible reaction. Combining current experimental findings with the previous studies, a new model termed as an irreversible-catalysis model was proposed for glycosidase-catalyzed reaction, written as (?),in which the regenerated enzyme from the dissociation ofES-complex will re-bind with substrate and then the catalysis is going again. The product formed plays key roles as a completive inhibitor for catalysis process.
本论文针对上述存在的问题，通过对E.coli CVCC249群体瞬时生长速率的研究，分析了不同生长条件下菌群生长的动力学特性，表明用菌群瞬时生长速率，可揭示出菌群异质性形成的机理。通过对生长动力学过程的分析，指出了经典方程在限制性培养条件下表征生长动力学过程的局限性，建立了新的区分生长阶段、估算生长参数，以及准确表征菌群异质性对生长动态过程的影响的新方法。进一步用Fibonacci数列分析，以及连续培养法与流式细胞技术相结合的研究策略，提出了细菌群体增殖的新模式。测定了均质性和非均质性细菌群体对抗菌药物的敏感性。并在此基础上，将数值微分法成功应用于菌群生长过程中酶催化的研究。与此领域中广为认可的研究结果相比较，本论文在以下5个方面取得了具有一定原始创新性的结果：
1．提出了分批培养条件下细菌群体生长阶段的区分及确定生长参数的新方法
为减少光吸收测定数据的误差，应用3次样条插值法(Spline Interpolation Method)将观测数据的离散点转化为连续光滑曲线，以真实反映动态过程，再用数值微分法(Numerical Differential Method)直接求得瞬时生长速率(instantaneous growth rate，v_(inst))。然后用Gaussian分布函数拟合过程曲线，求解出具有明确生物学含义的参数，简称该方法为SNG法(Spline-Numerical-Gaussian)。由此提出了区分群体生长阶段的新方法：即依据瞬时生长速度和瞬时生长加速度规律性的变化，把细菌群体生长过程区分为加速生长期、恒速生长期、减速生长期和衰亡期四个阶段(图1)，并得到新的生长参数。测定了群体生长过程中DNA、蛋白质含量，以及脱氢酶比活力的变化与瞬时生长速率的相关性。
在上述研究过程发现，在经典的表征群体生长的Logistic方程中，通常都假定群体中每个个体都是同质的，因此，解方程时的初值仅限定为N(t_0)=N_0，这样，对N_0相同而生理状态不同的接种物的生长动力学就不能区分，这一影响生长方程正确求解的思路在广泛应用的生物生长方程中都存在。事实上，在菌群生长的任一时刻，只有一定比例的细胞在进行分裂(设为θ)，这样，dN/dt不仅由N，而且实际上应由N和θ共同决定。因此，解方程的初值条件还应加入θ(t_0)=θ_0，而这一重要和必需条件又被在构建生长方程时所忽略，但这一附加条件使Logistic方程的求解甚为复杂。基于细菌生长过程的瞬时生长速率v_(inst)的时间过程曲线都呈Gauss分布形状这一特点，在用瞬时生长速率成功地表征限制性条件下区分群体生长阶段的基础上，进一步用Gaussian分布近似函数表达式(Y_i=αe~([-0.5((x_i-x_0)/b)~2]))以求得异质性群体的生长参数，这些参数可真实地表征不同生理状态细菌群体的生长动态，为微生物学基础研究和在生物工程等方面的应用提供了一个新方法。将新的表征群体生长的建模方法(SNG法)与依据Logistic方程划分生长阶段的传统方法作了比较，表明此方法可克服Logistic方程中存在的问题，而能准确表征E．coli CVCC249在分批培养下的生长动力学过程。
2．根据在分批和连续培养条件下菌群对药敏试验的不同影响，提出了一个准确进行药敏试验的新方法
根据菌群瞬时生长速率的变化，将E．coli CVCC249的分批培养过程区分为加速生长期、恒速生长期、减速生长期和衰亡期四个阶段，各阶段中菌群的核酸、蛋白质含量和脱氢酶活力都有明显差别；分别取样，测定各阶段菌群的浓度-杀菌曲线(Concentration-killing Curve，CKC)，检测结果表明这种生理异质性会导致各阶段菌群对药物敏感性的不同，同时，菌群内个体间药物敏感性也存在差异，呈现正态分布。以加速生长期菌群的CKC曲线为参照，对于庆大霉素，恒速生长期和衰亡期菌群药物敏感性增强；对于恩诺沙星，恒速生长期反而减弱，其它阶段差别不大。这是常规药敏试验方法难以显示的，表明对不同生理状态菌群的药敏试验测定结果可反映出药物的不同杀菌机理，但这同时也表明常规的药敏检测不能给出一个稳定的药物敏感性结果(图2)。而在连续培养下得到的“同步生长”状态的菌群，则可得到稳定的药敏试验结果(图3)，这有助于常规药敏试验方法的改进，并可为抗菌药物的合理使用和抗菌药物的筛选提供新方法。
3．证实了连续培养下细菌菌群以1→1+1'…模式进行增殖
用Fibonacci数列分析方法对在分批和连续培养下E．coli CVCC 249菌株的群体增殖过程进行了分析。Fibonacci数列模式是一经典递增函数，已广泛应用于表征生物生长发育过程。其数列分析属于数列的差分分析方法，它可把一个复杂的动力学、动态学过程简化求解。本文首次由数列的递推系数(recurrent coefficient for sequence)、递增系数(incremental recurrent coefficient for sequence)和净增率(net increment％)(或增长百分比)表征群体的增殖规律。分析结果表明，E．coli CVCC 249在连续培养下，群体中的细胞以1_(-M)(母细胞)→1_(-M)(母细胞)+1_(-D)(子细胞)→…的分裂模式进行增殖，即子细胞形成后，母细胞仍可继续进行同样的分裂增殖，简称为1→1+1'…(1…母细胞，1'…子细胞)模式。在分批培养条件下，菌群的净增率逐渐下降，最终成为不再进行增殖的状态，这是在营养供给受限和产物抑制双重作用下，代时逐渐延长的结果。在分批培养下，菌群的瞬时生长率，DNA复制和蛋白质合成的瞬时增长率，以及表征代谢还原能力的脱氢酶的比活力，都呈现正态分布，其数列分析结果也与活细胞数的分析结果一致。在相同的分批培养条件下，低密度接种可得到高的细胞净增率，而高密度接种则可迅速达到最高生长量。稳态接种材料的生长动力学类似于低密度接种。在连续培养条件下，稀释率的增加可加速淘汰最快和最慢增殖的个体，更快达到同步生长状态，从而使群体增殖率分布更趋于正态化，这与以流式细胞技术检测的不同稀释率下菌群内细胞尺度分布的结果相一致。数列分析结果表明，稳态下群体净增率为零，存在于培养罐中菌群的递增系数恒定在1．0(图4)，表明在开放系统条件下E．coli CVCC 249群体以1→1+1'…模式增殖。用Fibonacci数列分析方法对大肠杆菌以1→1+1'…(即类似于DNA的半复制模式)这一增殖模式的确定，涉及细菌基本增殖规律性的认识。对细菌突变率的测定、药敏测定中不确定性等难题的解决，以及生物工程中条件优化设计等，都会提供新的策略。
4．用瞬时速率可准确表征β-半乳糖苷酶的催化效率并证实了活性中间体的存在
通常依Michaelis-Menten方程进行测试时，酶的最大反应速率v_(max)和最初反应速率v_0决定着酶催化反应速率的常数k_(cat)。但由于v_0和v_(max)测试和拟合中的不确定性，从而导致k_(cat)结果的不确定性。因此本文采用一种改进的方法即数值微分法来准确的估算β-半乳糖苷酶的催化反应速率常数。依照IUPAC(International Union of Pure andApplied Chemistry)用ξ来定义反应速率的建议和国标的类似规定，dξ／dt定义为瞬时反应速率(v_(inst))。用实验数据直接得到瞬时反应速率，并以其作为目标函数，来确定β-半乳糖苷酶水解邻硝基苯酚-β-D-半乳糖苷(ONPG)的最适反应条件(如最适温度、时间、pH值和[S]／[E]等)。β-半乳糖苷酶的活性可以由反应过程中不断生成的邻硝基苯酚(ONP)的UV-Vis吸收光谱来表征。因此，在最适反应条件下，得到的瞬时反应速率的最大值(v_(inst-max))与达到瞬时反应最大值时的酶浓度([E]_(inst-max))之比，可准确的估算β-半乳糖苷酶的催化反应速率常数。未经纯化的样品中的β-半乳糖苷酶也可以通过这种方法估测。由这种改进方法与广为应用的Michaelis-Menten方程(v=v_(max)[s]/k_m+[s])得到的v_(max)和k_m求k_(cat)进行了比较，进一步剖析了Michaelis-Menten测试方法中不能得到稳定的k_(cat)等问题的原因。
通常用固定温度下求不同时间过程对反应速率影响的方法，或固定时间而改变温度的方法，来表征温度和时间对反应速率的影响。此方法属于求偏导数方法，因而无法全面表征温度和时间对速率的复合作用。而这2个变量的复合函数c-v_(inst)=[(?)v/(?)t]_T+[(?)v/(?)T]_t则可表征2者的共同影响。β-半乳糖苷酶降解ONPG过程中，ONPG的减少及ONP的增加可以用UV动态光谱图来测定，此过程的净转化速率(d[ONPG]／dt-d[ONP]／dt)即可表示活性中间产物(ES-复合物)的存在(图5)。温度和时间对ES-复合物形成的v_(inc)的复合影响，可用等高线作图法进行展示(图6)。通过这种途径，可以得出ONPG转化成ONP酶解过程中一些新的信息，即活性中间物并不是动力学曲线中的一个点，而是在三维空间上从反应物到产物酶解过程中的一个运动聚集体，并且受温度和时间的综合影响。活性中间物的动力学行为也可以在三维图上直观地表征出来。研究结果可为酶促反应动力学研究的深入开展提供一种新思路。
5．提出了β-半乳糖苷酶催化邻硝基苯酚-β-半乳糖苷(ONPG)水解机理的新模式
以固定浓度的β-半乳糖苷酶，对不同浓度的ONPG在0℃下进行结合。以紫外线光谱、荧光光谱的变化分别确定酶的饱和结合位点数SBP。结果表明，β-半乳糖苷酶的SBP数为8，即酶分子的一个亚基可结合2个ONPG分子。通过用新的de Donder方程v_f/v_r=e~(A_i/RT)和公式dG_i=-RT×ln(d[ONPG]/dt/d[ONP]/dt)计算Gibbs自由能变化，表明随着温度的升高和时间的延长，自由能由相对高的区域逐渐低移，至最低点时催化反应不再进行，体现了热力学过程和动力学过程的一致性(图7)。综合上述结果，提出了糖苷酶类催化底物水解的新机理，即(?)，表明在水解反应不能进行的条件下，结合可导致ONPG的分子构型转变为ONP分子构型，而β-半乳糖苷酶的构象变化则可循环发生。“ONPG-β-半乳糖苷酶复合物”与β-半乳糖苷酶和ONPG之间不存在可逆性的平衡过程，在酶分子不断循环进行的结合／解离过程中，由β-半乳糖苷酶构象变化提供的能量，使ONPG分子不断转化为ONP分子构型，结合复合物解离出ONP与半乳糖后，再生的β-半乳糖苷酶又可与ONPG结合，而形成的产物半乳糖作为抑制性因子在水解过程中起作用。这是与通常假定的酶催化过程为一系列可逆平衡反应组成，不可逆反应仅发生于ES复合物分离出产物P的阶段这一传统认识所不同的新观点。
Under batch culture conditions, a bacterial population may be heterogeneous in many important characteristics, such as age-specific reproduction and death rates, cell shape and size and biochemical activities; therefore, the growth of bacterial population presents a nonlinear process with very complicated dynamics. Thus, due to such complexity, the construction of the growth model has to be based on some plausible or implicit assumptions, such as introducing the "β" factor into the Malthus equation (dN/dt=rN ) asin Logistic equation (dN/dt=rN -βN~2), parameter "s" as in Gompertz equation(dN/dt=μ_0Ne~(-st)) and parameter "m" (shape factor) as in Richards equation(Vt = A(1- Be~(-kt))~(1/(1-m))). Generally, these models are constructed based on some simplisedassumptions that may not reflect the well known properties of a biological system and cannot be adapted for all circumstances.
There is a tendency in mathematics to rush into the analysis of a growth model without truly understanding the underlying biology. For example, the term CnN~i (i≥3) hasbeen introduced to the polynomial growth equation(dN/dt=F(N)=C_0+C_1N+C_2N~2+…+C_nN~i),however, the biological significance of theparameters in the model is unclear. Similarly, non-linearity equations with multi-parameters were proposed in biological studies, but did not serve the purposes. It was difficult or impossible to find a sufficient scheme to establish an adequate definition of biological parameters. An exponential polygonal model for population growth has been recently proposed, but it does not provide a sufficient basis for distinction of the growth curve. Microbiologically, it is usually assumed that bacteria divide by a process referred to as binary fission in which one cell divides to produce two cells and a bacterial population is homogeneous. Naturally, there is no complete homogeneous system for a bacterial population and it is only a hypothesized state.
Experimental design and investigation of this thesis focuses on the above mentioned problems. Using the instantaneous growth rate as an index, we analyzed the kinetics characteriactic of a bacterial population at different conditions, discussed the formation mechanism of the population kinetics, and elucidated the relationship between the population heterogeneity and the instantanneous growth rate. Based on the analysis of growth kinetics, a new method was developed to study the growth kinetics of bacterial populations and to distinguish the growth phases in batch culture. We extensively discussed many difficulties confronted by Logistic growth model when describing microbial growth. Five new discoveries were established during my Ph.D. studies, as follows:
1. A novel approach was developed for estimating growth phases and parameters of bacterial population in batch culture
Using mathematical analysis, a new method has been developed to study the growth kinetics of bacterial populations in batch culture. First, sample data were smoothed with the spline interpolation method. Second, the instantaneous rates were derived by numerical differential techniques and finally, the derived data were fitted with the Gaussian function to obtain growth parameters. We named this the Spline-Numerical-Gaussian or SNG method. This method yielded more accurate estimates of the growth rates of bacterial populations and new parameters. It was possible to divide the growth curve into four different but continuous phases based on changes in the instantaneous rates. They are the accelerating growth phase, the constant growth phase, the decelerating growth phase and the declining phase (Fig.1.). Total DNA content was measured by flow cytometry and it varied with the growth phase. The SNG system provides a very powerful tool for describing the kinetics of bacterial population growth. The SNG method avoids the unrealistic assumptions generally used in traditional growth equations.
During past 50 years, different types of the Logistic model are constructed based on a simple assumption that the microbial populations are all composed of homogeneous members and consequently, the condition of design for the initial value of these models has to be rather limited in the case of N(t_0)=N_0. Therefore, these models cannot reflect the different dynamic behavior of the populations possessing the same No from heterogeneous phases. In fact, only a certain ratio of the cells in a population is dividing at any time duringgrowth progress, termed asθ, and thus,dN/dt not only depends on N, but also onθ. SoN(t_0)=N_0 is a necessary element for the condition design of the initial value. Unfortunately, this idea has long been neglected in widely used growth models. However, combining together the two factors (N_0 andθ) into the initial value often leads to the complexity in the mathematical solution. This difficulty can be overcome by using instantaneous rates (V_(inst)) to express growth progress. Previous studies in our laboratory suggested that the V_(inst), curve of the bacterial populations all showed a Guassian function shape and thus, the different growth phases can be reasonably distinguished. In the present study, the Gaussian distribution function was transformed approximately into an analytical form(Y_i=αe~([-0.5((x_i-x_0)/b)~2]))that can be conveniently used to evaluate the growth parameters and inthis way the intrinsic growth behavior of a bacterial species growing in heterogeneous phases can be estimated. In addition, a new method has been proposed, in this case, the lag period and the doubling time for a bacterial population can also be reasonably evaluated. This approach proposed could thus be expected to reveal insight of bacterial population growth. Some aspects in modeling population growth are also discussed.
2. A new method was proposed for estimating the effect of physiological heterogeneity of E.coli population on antibiotic susceptivity test
According to the instantaneous growth rate (dN/dt) of E.coli CVCC249 growing in batch culture, the entire growth curve was distinguished into four phases: accelerating growth phase, constant growth phase, decelerating growth phase and declining phase.Each of four phases have obvious variation in physiological and biochemical properties, including total DNA content, total protein content and MTT-dehydrogenase activity, etc. that leaded to the difference in their antibiotic susceptivity. Antibiotic susceptivity of a population sampled from each phase was tested respectively by Concentration-killing Curve (CKC) approachfollowing the formula N =N_0/1+e~( r(x-BC_(50))), showing as normal distribution at individual cell levelfor an internal population, in which the median bactericidal concentration BC_(50) representedthe mean level of susceptivity, while the bactericidal span BC_(1-99)=2/r lnN_0 indicated thevariation degree of the antibiotic susceptivity. Furthermore, tested by CKC approach, the antibiotic susceptivity of E.coli CVCC249 population in each physiological phase to gentamicin or enoxacin varied: susceptivity of population in constant growth phase and declining phase all increased, compared with that in accelerating growth phase, for gentamicin but declined for enoxacin. This primary investigation revealed that physiological phase should be taken account of in the context of both antibiotic susceptivity test and research into antimicrobial mechanism and bacterial resistance (Fig.2.). However there are few reports regarding this aspect, therefore, further research using different kinds of antibiotics with synchronized continuous culture of different bacterial strains is necessary(Fig.3.).
3. A new model was proposed for expressing the reproduction of a heterogeneous bacterial population
According to the Fibonacci sequence analysis, three parameters, the recurrent coefficient, the incremental recurrent coefficient, and the net increment (%), were used to analyze the growth and reproduction behaviour of E.coli under batch and continuous culture. The analysis result suggested that under continuous culture, E. coli CVCC 249 divided as in regular model of 1_(-M) (mother)→1_(-M)(mother ) + 1_(-D) (daughter)→…termed as 1→1 + 1'…model. Under batch culture, the net increment of the population growth continuously decrease and finally a zero rate will be reached due to nutrition exhaust and the production increase as inhibitor that can lead to the prolonged generation time. All curves including the instantaneous rate, DNA and protein biosynthesis, and MTT activity appear to fit in the normal distribution. Under same batch culture, the net increment (%) will decrease with the increase in the inoculum size, but the highter biomass will be obtained at larger inoculum size. As steady state cultures were used as the inoculum, the growth dynamic behaviour was similar to that lower inoculum size. Under continuous culture, individual at low growth rate wash out with the increase in the dilution rate and the population growth curve present a normal distribution which is similar to that cell size distribution. The research results showed that the incremental recurrent coefficient for sequence is 1.0 (Fig.4.), which suggests that the bacteria reproduce in regular model of 1→1 + 1'…This new model will provide a good indication for accurately determining the mutation rate and the degree susceptivity of bacteria to antibiotic et al.
4. The instantaneous reaction rate was used to express the catalytic efficiency ofβ-galactosidase
To overcome the difficulties in determining the initial rate (V_0) of enzyme-catalyzed reactions based on the classic Michaelis-Menten kinetic assumption and to avoid the uncertainty of calculating the catalytic rate constant (k_(cat)) by extrapolation of V_(max)and k_m data, the instantaneous reaction rate was used to estimate the k_(cat) ofβ-galactosidase. This was based on analysis of the changes in relationship between input data [the value of added O-nitrophenylβ-D-galactoside (ONPG)] and output data [the instantaneous rate of O-nitrophenol (ONP) formation] for entire process of the enzyme-catalyzed reaction. The amount of ONP was determined based on the online UV-visible dynamic spectrum during the entire course of the reaction and its instantaneous change rate (v_(inst)) was directly derived from the dynamic spectrum curve. v_(inst) was used as an objective function to determine the optimum assay conditions forβ-galactosidase activity with variables such as temperature, time, pH, and the ratio of [E] to [S]. Under these optimal assay conditions, the maximum value of v_(inst) (v_(inst-max)) and the corresponding concentration of enzyme ([E] inst-max) could be accurately determined. Then the k_(cat) of p-galactosidase was calculated with the formula: k_(cat) = v_(inst-max)/[E]_(inst-max) .
The amount ofβ-galactosidase in a sample could also be easily determined by this approach without a pre-purification step. The general effectiveness of this new approach and the problems of applying the Michaelis-Menten approximation, particularly for estimating the initial velocity, are also discussed in detail.
The disappearance of ONPG and the appearance of ONP were synchronously measured during the hydrolysis of ONPG byβ-galactosidase using UV-visible spectrum in situ on-line. The conversion process of ONPG to ONP was calculated using d[ONPG]/dt - d[ONP]/dt (Fig.5.). The combined effects of temperature and time on v_(inst) and v_(inc) were expressed as relative variability and visualized by the isograms method - contour plots. With this approach, new insights into the irreversible-continuous conversion of ONPG to ONP during hydrolysis can be clearly observed. That is, the intermediate was a moving-mass flow in three-dimensional space from substrate converting to product during hydrolysis and the temperature-time compensating effects. The dynamic behavior of the intermediate was effectively visualized in a two-dimensional plot (Fig.6.). The results of the present study provide evidence to support the isograms method as a useful tool in understanding the reaction mechanisms of enzyme catalysis.
5. A new model for enzyme-catalyzed reaction was proposed based on a case study on the hydrolysis of ONPG byβ-galactosidase
Based on the synchronously measuring the disappearance of ONPG and the appearance of ONP during hydrolysis of ONPG byβ-galactosidase, the existence of ES-complex as an intermediate can be estimated by the differences between d[ONPG]/dt and d[ONP]/dt. The new de Donder equation (v_f/v_r=exp (A_i/RT)) was used to calculate the changes in Gibbs free energy (dG_i=-RT·ln(d[ONPG]/dt)/(d[ONP]/dt)) (Fig.7.). The results suggested that the interaction of substrate with enzyme in binding process is controlled by the active sites on the surface of enzyme and the conversion of substrate to product is an irreversible reaction. Combining current experimental findings with the previous studies, a new model termed as an irreversible-catalysis model was proposed for glycosidase-catalyzed reaction, written as (?),in which the regenerated enzyme from the dissociation ofES-complex will re-bind with substrate and then the catalysis is going again. The product formed plays key roles as a completive inhibitor for catalysis process.
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