On-line parameter and state estimation of continuous cultivation by extended Kalman filter

Summary The on-line estimation of biomass concentration and of three variable parameters of the non-linear model of continuous cultivation by an extended Kalman filter is demonstrated. Yeast growth in aerobic conditions on an ethanol substrate is represented by an unstructured non-linear stochastic t-variant dynamic model. The filter algorithm uses easily accessible data concerning the input substrate concentration, its concentration in the fermentor and dilution rate, and estimates the biomass concentration, maximum specific growth rate, saturation constant and substrate yield coefficient. The microorganismCandida utilis, strain Vratimov, was cultivated on the ethanol substrate. The filter results obtained with the real data from one cultivation experiment are presented. The practical possibility of using this method for on-line estimation of biomass concentration, which is difficult to measure, is discussed.Nomenclature D dilution rate (h^-1) - DO₂ dissolved oxygen concentration (%) - E identity matrix - F Jacobi matrix of the deterministic part of the system equations g - g continuousn-vector non-linear real function - h m-vector non-linear real function - K Kalman filter gain matrix - K _S saturation constant (kgm^-3) - K_S expectation of the saturation constant estimate - M Jacobi matrix of the deterministic part of the measurement equations h - P(t₀) co-variance matrix of the initial values of the state - P(t_k/t_k) c-variance matrix of the error in (t _k|t _k) - P(t_k+1/t_k) co-variance matrix of the error in (t _k+1|t _k - Q co-variance matrix of the state noise - R co-variance matrix of the output noise - S substrate concentration (kgm^-3) - S _i input substrate concentration - t time - t _k discrete time instant with indexk=0, 1, 2,... - u(t) input vector - v(t_k) measurement (output) noise sequence - w(t) n-vector white Gaussian random process - x(t₀) initial state of the system - (t₀) expectation of the initial state values - x(t) n-dimensional state vector - x(t_k) state vector at the time instantt _k - (t_k|t_k) expectation of the state estimate at timet _k when measurements are known to the timet _k - (t_k+1|t_k) expectation of the state prediction - X biomass concentration (kgm^-3) - expectation of the biomass concentration estimate - y(t_k) m-dimensional output vector at the time instantt _k - Y _XIS substrate yield coefficient - _X|S expectation of the substrate yield coefficient estimate - specific growth rate (h^-1) - _M maximum specific growth rate (h^-1) - expectation of the maximum specific growth rate estimate - state transition matrix     

The problem of nonstationary ion fluxes in excitable membranes

Time-dependent electrodiffusion through a membrane is analysed within a simple model treating the boundary-layers in a consistent manner. It is shown that time-independent reversal potentials for the ion fluxes exist only under steady-state conditions. We argue that this result holds very generally. Therefore nonstationary effects like ion storage and depletion inside the membrane should not contribute to the phenomena of excitability.Glossary of Symbols A _mv [V] functional cf. Equation (3) - C membrane capacitance - d one half the thickness of the membrane - F[V] functional cf. Equation (1) - g _i electrochemical potential inside membrane - g _i electrochemical potentials outside membrane at x ±d, respectively - i (index) refers to i-th ionic species - J electric current across membrane - j = j ^} = j ^< current density measured by external electrodes - j _i (x) current density inside membrane in x-direction - j _i ^inst(x) instantaneous current density - J _i ^stat steady-state current density - k Boltzmann constant - m (index) is used in Sec. 2 to denote the independent diffusion currents - n ^< ionic strength of electrolyte at x = - - n _i density of ions inside membrane - n _i density of ions outside membrane at x = ±, respectively - Q charge per unit area of boundary layers at x ± d, respectively - Q ₀ fixed charge per unit area of membrane - q elementary charge - q _i ionic charges - T temperature - it time - V membrane potential (= (-)-()) - V _i Nernst potential - V potential drops inside boundary layers (can be neglected, see Appendix II) - V ^± potential steps at x = ± d, cf. Equation (29) - V ₀ = V ^–-V ⁺ - w _i activation energy inside membrane - x spatial coordinate perpendicular to membrane - y, z spatial coordinates parallel to membrane - dielecric constant - ₀ dielectric constant of electrolyte solution ( 80) - _m dielectric constant of membrane ( 5) - (x) electrostatic potential - charge density of boundary layers - ⁰ fixed charge density inside membrane - spatial average, cf. Equation (12)     

Tolerance regions with segmented fermentation processes

Many microbial fermentation processes exhibit different phases (e.g. adaption phase, main growth phase, main production phase). The process variables e.g. the biomass vary randomly about their mean. The experimentalist is interested to know the break points of the different phases, and a tolerance region, i.e. a range of possible values of the process variable that can be considered as normal. This paper deals with statistical methods for determining break points and tolerance regions.List of Symbols a _i intercept in phasei - b _i specific growth rate in phasei - e _t deviation of a measurement in timet - tEX expectation of variableX - r number of phases of fermentation - T _i break point of phaseit - t _ij time of measurementj in phasei - t _n–2.1–/2 quantile oft distribution - Y(t) logarithm of measurement at timet Greek Letters 1 – cover probability of tolerance region - 1 – part covered by the tolerance region - ² variance ofe _t - (·) standard normal distribution - quantile of chisquare distribution     

Intensity of microcarrier collisions in turbulent flow

A model is developed, allowing estimation of the share of inelastic interparticle collisions in total energy dissipation for stirred suspensions. The model is restricted to equal-sized, rigid, spherical particles of the same density as the surrounding Newtonian fluid. A number of simplifying assumptions had to be made in developing the model. According to the developed model, the share of collisions in energy dissipation is small.List of Symbols b parameter in velocity distribution function (Eq. (28)) - c _K factor in Kolmogoroff spectrum law (Eq. (20)) - D _t(r _p ) m²/s characteristic dispersivity at particle radius scale (Eq. (13)) - E(k, t) m³/s² energy spectrum as function of k and t (Eq. (16)) - E _K (k) m³/s² energy spectrum as function of k in Kolmogoroff-region (Eq. (20)) - E _p dimensionless mean kinetic energy of a colliding particle (Eq. (36)) - E _cp dimensionless kinetic energy exchange in a collision (Eq. (37)) - G(x, s) dimensionless energy spectrum as function of x and s (Eq. (16)) - G _B(x) dimensionless energy spectrum as function of x for boundary region (Eq. (29)) - G _K(x) dimensionless energy spectrum as function of x for Kolmogoroff-region (Eq. (21)) - g m/s² gravitational acceleration - I _cp dimensionless collision intensity per particle (Eq. (38)) - I _cv dimensionless volumetric collision intensity (Eq. (39)) - k l/m reciprocal of length scale of velocity fluctuations (Eq. (17)) - K dimensionless viscosity (Eq. (13)) - n(2) dimensionless particle collision rate (Eq. (12)) - n(r) l/s particle exchange rate as function of distance from observatory particle center (Eq. (7)) - r m vector describing position relative to observatory particle center (Eq. (2)) - r m scalar distance to observatory particle center (Eq. (3)) - r _pm particle radius (Eq. (1)) - s dimensionless time (Eq. (10)) - SC kg/ms³ Severity of collision (Eq. (1)) - t s time (Eq. (2)) - u(r, t) m/s velocity vector as function of position vector and time (Eq. (2)) - u(r, t) m/s magnitude of velocity vector as function of position vector and time (Eq. (3)) - u _r(r, t) m/s radial component of velocity vector as function of position vector and time (Eq. (3)) - u _r (r, t) m/s magnitude of radial component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s latitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s magnitude of latitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s longitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s magnitude of longitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u _gsm/s superficial gas velocity - u(r) m/s root mean square velocity as function of distance from observatory particle center (Eq. (3)) - u_r(r) m/s root mean square radial velocity component as function of distance from observatory particle center (Eq. (4)) - u (r) m/s root mean square latitudinal velocity component as function of distance from observatory particle center (Eq. (4)) - u (r) m/s Root mean square longitudinal velocity component as function of distance from observatory particle center (Eq. (4)) - w(x) dimensionless root mean square velocity as function of dimensionless distance from observatory particle center (Eq. (11)) - V _pm³ particle volume (Eq. (36)) - w(2) dimensionless root mean square collision velocity (Eq. (34)) - w ^* parameter in boundary layer velocity equation (Eq. (24)) - x dimensionless distance to particle center (Eq. (9)) - x ^* value of x where G _Band G _K-curves touch (Eq. (32)) - x _K dimensionless micro-scale (Kolmogoroff-scale) of turbulence (Eq. (15)) - volumetric particle hold-up - m²/s³ energy dissipation per unit of mass - m²/s kinematic viscosity - kg/m³ density - (r) m³/s fluid-exchange rate as function of distance to observatory particle center - Latitudinal co-ordinate (Eq. (5)) - Longitudinal co-ordinate (Eq. (5))     

Determination of the optimum operating time for batch isothermal performance of enzyme-catalyzed multisubstrate reactions

This communication consists of a mathematical analysis encompassing the maximization of the average rate of monomer production in a batch reactor performing an enzymatic reaction in a system consisting of a multiplicity of polymeric substrates which compete with one another for the active site of a soluble enzyme, under the assumption that the form of the rate expression is consistent with the Michaelis-Menten mechanism. The general form for the functional dependence of the various substrate concentrations on time is obtained in dimensionless form using matrix terminology; the optimum batch time is found for a simpler situation and the effect of various process and system variables thereon is discussed. The reasoning developed here emphasizes, in a quantitative fashion, the fact that the commonly used lumped substrate approaches lead to nonconservative decisions in industrial practice, and hence should be avoided when searching for trustworthy estimates of optimum operation.List of Symbols O 1/s row vector of zeros - a 1/s row vector of rate constants k _i(i = 2,...,N) - A 1/s matrix of rate constants k _i and k_–i (i=2,...,N) - b 1/s row vector of rate constant k ₂ and zeros - C mol/m³ molar concentration of S - C mol/m³ vector of molar concentrations of C _i (i=0, 1, 2, ..., N) - C ₀ mol/m³ column vector of initial molar concentrations of C _i(i=0, 1, 2,.., N) - C _–01 mol/m³ column vector of initial molar concentrations of C _i(i=2,..., N) - C _{E, tot} mol/m³ total molar concentration of enzyme molecules - C _i mol/m³ molar concentration of S _i (i=0,1,2,...,N) - C _{i, o} mol/m³ initial molar concentration of S _i(i=0, 1, 2, ..., N) - E enzyme molecule - I identity matrix - K 1/s matrix of lumped rate constants - k _i 1/s pseudo-first order lumped rate constant associated with the formation of S _i -1 (i=1, 2, ...,N) - k _{cat, i} 1/s first order rate constant associated with the formation of S _i-1 (i=1, 2, ..., N) - K _m mol/m³ Michaelis-Menten constant - L number of distinct eigenvalues - M _i multiplicity of the i-th eigenvalue - N maximum number of monomer residues in a single polymeric molecule - r ₁ mol/m³ s rate of formation of S ₀ - r _i mol/m³ s rate of release of S _i -1 - r _opt maximum average dimensionless rate of production of monomer S₀ - S lumped, pseudo substrate - S₁ inert moiety - S _i substrate containing i monomer residues, each labile to detachment as - S₀ by enzymatic action (i=1,2,...,N) - t s time elapsed since startup of batch reaction - t _lag s time interval required for cleaning, loading, and unloading the batch reactor - t _opt s time interval leading to the maximum average rate of monomer production - v _ij s^1-j eigenvectors associated with eigenvalue imi (i=1, 2, ..., L; j =1, 2, ..., M_i) Greek Symbols _ij mol/m³ arbitrary constant associated with eigenvalue _i (i=1, 2, ..., L; j=1, 2, ..., M _i ) - 1/s generic eigenvalue - _i 1/s i-th eigenvalue     

Comments on the “Maintenance coefficient” changes during alcohol fermentation

Summary The concept of maintenance is discussed in terms of the biological meaning and the applicability of the maintenance coefficient, m, in bioengineering for optimization of yields in fermentation. A method of calculation is proposed for the evaluation of m in the course of fermentation in the case of a metabolite (e.g, ethanol). During alcoholic fermentation m is not constant and decreases with the growth rate.The phenomena involved in maintenance are numerous and complex and there is a semantic problem in its definition which can be generalized by the apparently non-finalized substrate consumption.Nomenclature a specific maintenance rate (defined by eq. (9)) - m maintenance coefficient - X cell mass concentration (as a dry weight) - S substrate concentration - P product concentration - r rate of reaction - r_x,r_s,r_p rate of reaction related to biomass, substrate and product - r_sm,r_sg,r_spi rate of reaction related only to the consumption by maintenance, growth and the synthesis of the ith product - r_xe maintenance rate defined by eq. (10) - q_s,q_Pi specific rate of substrate consumption and i^th product production - Y yield coefficient - Y^o, Y^o apparent yield coefficient related to the cell and i^th product - Y _xs ^xs , Y _Pis ^Pis maximum theoretical yield coefficient related to the cell and i^th - specific growth rate produce 420     

Malthusian parameters,reproductive values and change under selection in self fertilizing age-structured populations

A population, reproducing wholly by selfing, is assumed to be observed at times . Individuals between x–1 and x units of age at time t are said to be in age class x at that time. The rate of increase in the long run of individuals of type A_iAj is denoted by m_ij+1=m_ji+1. For each genotype there is also a set of reproductive values, corresponding to all age classes and genotypes of individuals having descendants of that genotype. Then, if the number of individuals of each sort of ancestor is multiplied by its reproductive value and the products are summed, the result is the total value, which is V_ij(t) for genotype A_iAj. Then V_ij(t+1)–V_ij(t) is equal to m_ijV_ij(t), where m_ij is the Malthusian parameter for A_iAj. Furthermore, if the mean and variance at time t of the m_ijs, weighted by their corresponding reproductive values, are respectively (t) and _m²(t), then m¯(t+1)–m¯(t)=_m²(t)/(1+m¯(t)).     

Photoinhibition of photosynthesis in intact kiwifruit (Actinidia deliciosa) leaves: Effect of light during growth on photoinhibition and recovery

Photoinhibition of photosynthesis was induced in intact kiwifruit (Actinidia deliciosa (A. Chev.) C. F. Liang et A. R. Ferguson) leaves grown at two photon flux densities (PFDs) of 700 and 1300 mol·m^-2·s^-1 in a controlled environment, by exposing the leaves to PFD between 1000 and 2000 mol·m^-2·s^-1 at temperatures between 10 and 25°C; recovery from photoinhibition was followed at the same range of temperatures and at a PFD between 0 and 500 mol·m^-2·s^-1. In either case the time-courses of photoinhibition and recovery were followed by measuring chlorophyll fluorescence at 692 nm and 77K and by measuring the photon yield of photosynthetic O₂ evolution. The initial rate of photoinhibition was lower in the high-light-grown plants but the long-term extent of photoinhibition was not different from that in low-light-grown plants. The rate constants for recovery after photoinhibition for the plants grown at 700 and 1300 mol·m^-2·s^-1 or for those grown in shade were similar, indicating that differences between sun and shade leaves in their susceptibility to photoinhibition could not be accounted for by differences in capacity for recovery during photoinhibition. Recovery following photoinhibition was increasingly suppressed by an increasing PFD above 20 mol·m^-2·s^-1, indicating that recovery in photoinhibitory conditions would, in any case, be very slow. Differences in photosynthetic capacity and in the capacity for dissipation of non-radiative energy seemed more likely to contribute to differences in susceptibility to photoinhibition between sun and shade leaves of kiwifruit.Abbreviations and symbols F _o , F _m , F _v instantaneous, maximum, variable fluorescence - F _v /F _m fluorescence ratio - F _i =F _v at t=0 - F F _v at t= - K _D rate constant for photochemistry - k(F _p ) first-order rate constant for photoinhibition - k(F _r ) first-order rate constant for recovery - PFD photon flux density - PSII photosystem II - _i photon yield of O₂ evolution (incident light)     

A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue

It is proposed that distinct anatomical regions of cerebral cortex and of thalamic nuclei are functionally two-dimensional. On this view, the third (radial) dimension of cortical and thalamic structures is associated with a redundancy of circuits and functions so that reliable signal processing obtains in the presence of noisy or ambiguous stimuli.A mathematical model of simple cortical and thalamic nervous tissue is consequently developed, comprising two types of neurons (excitatory and inhibitory), homogeneously distributed in planar sheets, and interacting by way of recurrent lateral connexions. Following a discussion of certain anatomical and physiological restrictions on such interactions, numerical solutions of the relevant non-linear integro-differential equations are obtained. The results fall conveniently into three categories, each of which is postulated to correspond to a distinct type of tissue: sensory neo-cortex, archior prefrontal cortex, and thalamus.The different categories of solution are referred to as dynamical modes. The mode appropriate to thalamus involves a variety of non-linear oscillatory phenomena. That appropriate to archior prefrontal cortex is defined by the existence of spatially inhomogeneous stable steady states which retain contour information about prior stimuli. Finally, the mode appropriate to sensory neo-cortex involves active transient responses. It is shown that this particular mode reproduces some of the phenomenology of visual psychophysics, including spatial modulation transfer function determinations, certain metacontrast effects, and the spatial hysteresis phenomenon found in stereopsis.List of Symbols (t) Post-synaptic membrane potential (psp) - Maximum amplitude of psp - t Time - The neuronal membrane time constant - Threshold value of membrane potential - r Absolute refractory period - Synaptic operating delay - v Velocity of propagation of action potentil - x Cartesian coordinate - _jj (x) The probability that cells of class j are connected with cells of class j a distance x away - b _jj The mean synaptic weight of synapses of the jj-th class at x - _jj The space constant for connectivity - _e Surface density of excitatory neurons in a one-dimensional homogeneous and isotropic tissue - _i Surface density of inhibitory neurons in a one-dimensional homogeneous and isotropic tissue - E(x, t) Excitatory Activity, proportion of excitatory cells becoming active per unit time at the instant t, at the point x - I(x, t) Inhibitory Activity, proportion of inhibitory cells becoming active per unit time at the instant t, at the point x - x A small segment of tissue - t A small interval of time - P(x, t) Afferent excitation or inhibition to excitatory neurons - Q(x, t) Afferent excitation or inhibition to inhibitory neurons - N _e (x, t) Mean integrated excitation generated within excitatory neurons at x - N _i (x, t) Mean integrated excitation generated within inhibitory neurons at x - _e [N _e ] Expected proportion of excitatory neurons receiving at least threshold excitation per unit time, as a function of N _e - _i [N _i ] Expected proportion of inhibitory neurons receiving at least threshold excitation per unit time, as a function of N _i - G( _e ) Distribution function of excitatory neuronal thresholds - G( ₁ ) Distribution function of inhibitory neuronal thresholds - ₁ A fixed value of neuronal threshold - h(N _e ; ₁) Proportion per unit time of excitatory neurons at x reaching ₁ with a mean excitation N _e - 1[ ] Heaviside's step-function - R _e (x, t) Number of excitatory neurons which are sensitive at the instant t - R _i (x, t) Number of inhibitory neurons which are sensitive at the instant t - R _e Refractory period of excitatory neurons - r _i Refractory period of inhibitory neurons - E(x, t) Time coarse-grained excitatory activity - I(x, t) Time coarse-grained inhibitory activity - Spatial convolution - Threshold of a neuronal aggregate - v Sensitivity coefficient of response of a neuronal aggregate - E(t) Time coarse-grained spatially localised excitatory activity - I(t)> Time coarse-grained spatially localised inhibitory activity - L ₁,L ₂,L,Q See § 2.2.1, § 2.2.7, § 3.1 - Velocity with which retinal images are moved apart - Stimulus width - E _o, I _o Spatially homogeneous steady states of neuronal activity - k _e ,k _ij S _e S _ij See § 5.1     

Optimal control of continuous fermentation processes

Three layer control structure is proposed for optimal control of continuous fermentation processes. The start-up optimization problems are solved as a first step for optimization layer building. A steady state optimization problem is solved by a decomposition method using prediction principle. A discrete minimum time optimal control problem with state delay is formulated and a decomposition method, based on an augmented Lagrange's function is proposed to solve it. The problem is decomposed in time domain by a new coordinating vector. The obtained algorithms are used for minimum time optimal control calculation of Baker's Yeast fermentation process.List of Symbols x(t) g/l biomass concentration - s(t) g/l limiting substrate concentration - x ⁰ g/l inlet biomass concentration - s ⁰(t) g/l inlet substrate concentration - D(t) h^–1 dilution rate - (t) h^–1 specific growth rate - Y g/g yield coefficient - (t) h^–1 specific limiting substrate consumption rate - k _D h^–1 disappearing constant - w ₁, w ₂ known constant or piece-wise disturbances - _m h^–1 maximum specific growth rate - k _s g/l Michaelis-Menten's parameter - h time delay - x ₀, s ₀ g/l initial concentrations - ¯x, ¯s, ¯D optimal steady state value - V _min , V _max , v=x,s,d,t bounds of variables - t h sampling period - K number of steps in the optimization horison - Js, J _d performance indexes - L _s Lagrange's function - L _d Lagrange's functional - ₀ weighting coefficient for the amount of the limiting substrate throwing out of the fermentor - ₁, ₂ dual variables of Lagrange's function - steps in steady state coordination procedure - errors values for steady state coordination process - _v , v=x, s conjugate variables of Lagrange's functional - _v , v=x,s penalty coefficients of augmented Lagrange's functional - _v , v=x, s interconnections of the time - e _v , v=x,s, D, _x , _s gradients of Lagrange's functional - j, l indexes of calculation procedures - values of errors in calculations The researches was supported by National Scientific Research Foundation under grants No NITN428/94 and No NITN440/94     

Slow conductance changes due to potassium depletion in the transverse tubules of frog muscle fibers during hyperpolarizing pulses

Summary When hyperpolarizing currents are applied between the inside and outside of a muscle fiber it is known that there is a slow transient decrease (300- to 600-msec time constant) in the measured fiber conductance sometimes referred to as creep which is maximal in K₂SO₄ Ringer's solutions and which disappears on disruption of the transverse tubular system. An approximate mathematical analysis of the situation indicates that these large, slow conductance changes are to be expected from changes in the K⁺ concentration in the tubular system and are due to differences in transport numbers between the walls and lumen of the tubules. Experiments using small constant-voltage and constant-current pulses (membrane p. d. changes 20 to 30 mV) on the same fibers followed by an approximate mathematical and more exact computed numerical analysis using the measured fiber parameters and published values of tubular system geometry factors showed close agreement between the conductance creep predicted and that observed, thus dispensing with the need for postulated changes in individual membrane conductances at least during small voltage pulses. It is further suggested that an examination of creep with constant-voltage and constant-current pulses may provide a useful tool for monitoring changes in tubular system parameters, such as those occurring during its disruption by presoaking the fibers in glycerol.Table of main symbols used R, T, F Gas constant, Temperature in °K and the Faraday - a Fiber radius - r Radial distance from the center of the fiber (cf. Fig. 2A) - t Time in sec - V ₁,V ₂ Voltages measured by electrodes 1 and 2 (cf. p. 248) - Longitudinal fiber space constant ( ²=R _m a/2R _i ) - R _m ,R _m (t) Total membrane resistance per unit surface area of fiber ( cm²) - R _m (0),R _m () As above att=0 (excluding capacity transient) and att= during a current or voltage pulse - G _m ,G _m (t) Total membrane conductance (mho·cm^–2) per unit area of fiber surface - G _m (0),G _m () As above att=0 (excluding the capacity transient) and att= during a current or voltage pulse - R _sm ,G _sm Surface membrane resistance ( cm²) and conductance (mho·cm^–2), respectively, excluding the TTS - R _T ,G _T Input resistance ( cm²) and conductance (mho·cm^–2) of the TTS referred to unit area of fiber surface - f _T Fraction of the K⁺ conductance in the TTS to the total K⁺ conductance of the fiber [cf. Eq. (7)] - R _i Internal resistivity of the fiber ( cm) - r _s Electrical access resistance of the TTS [ cm²;cf. Fig. 3 and Eq. (24)] - h Diffusional access resistance of the TTS [cf. Eq. (27)] - I ₀ Total current entering fiber (amp) - I _m ,i _m Total current per unit area of fiber surface (amp·cm^–2; considered positive in the hyperpolarizing direction) - i _sm Current going through the surface membrane alone (amp·cm^–2;cf. Fig. 3) - i ₀,i ₀(t) Total current entering the TTS referred to unit area of surface membrane (amp·cm^–2;cf. Fig. 3) - I _K,I _K(r) K⁺ current density crossing the equivalent TTS disc at radial distancer [cf. Fig. 2A and Eq. (23)] - i, i(r, t) Radial current in the lumen of the TTS at radial distancer and timet (cf. Fig. 2B) - C, C(r, t) K⁺ concentration within the TTS at radial distancer and timet (mEquiv·liter^–1) - C _o ,C _K Both refer to external solution and initial TTS K⁺ concentration (mEquiv·liter^–1) - V, V(r, t) The potential at radial distancer in the lumen of the TTS with respect to the external solution at timet (cf. Figs. 2 and 3) - V(a), V(a, t) The p.d. across the access resistance (cf. Figs. 3B and 3C) - V ₀,V ₀(t) The potential of the sarcoplasm with respect to the external solution (cf. Figs. 2 and 3) - E _K The K⁺ equilibrium potential between the sarcoplasm and the externa solution or across the tubular wall - t _K ^m ,t _K ^s The transport number for K⁺ in the TTS membranes and in the solution of the tubular lumen, respectively - The fraction of fiber volume occupied by tubules, and not implicitly including branches - As above but always including branches - A dimensionless network factor for the TTS - G _W Conductance per unit area of tubular wall (mho·cm^–2) - G _L Conductance of tubular lumen (mho·cm^–1) - Volume-to-surface ratio of the TTS - Effective wall conductance of TTS membranes per unit volume of fiber [mho·cm^–3;cf. Eq. (14)] - Effective radial conductance of the lumen of the TTS per unit volume of fiber [cf. Eq. (20)] - d The thickness of the equivalent disc representing the TTS [cf. Eq. (15)] - _T Space constant of the TTS [cf. Eq. (37).cp. Eq. (11)] - D _K The diffusion coefficient of K⁺ ions in the lumen of the TTS (cm² sec^–1) - The effective radial K⁺ diffusion coefficient in the TTS [cf. Eq. (28)] - J ₀,J ₁ Bessel functions of order 0 and 1, respectively - I ₀,I ₁ Modified Bessel functions of order 0 and 1, respectively - Time constants of slow conductance changes - _vc Time constant of slow conductance changes during a constant-voltage pulse - _cc Time constant of slow conductance changes during a constant-current pulse - , _m Roots of various Bessel function equations - g ₁,g ₂,g ₃,g ₄ Constants used to fit cubic equation for conductance-voltage curves [cf. Eq. (71)]     

Inference for an age-dependent,multitype branching-process model of mast cells

We consider an age-dependent, multitype model for the growth of mast cells in culture. After a colony of cells is established by an initiator type, the two possible types of cells are resting and proliferative. Using novel inferential procedures, we estimate the generation-time distribution and the offspring distribution of proliferative cells, and the waiting-time distribution of resting cells.List of Notations B _i cumulative distribution function for the time until branching of a cell of type i - b _i probability density function for the time until branching of a cell of type i - b _i b _i (1–D _i ) - D _i cumulative distribution function for the time until death of a cell of type i - d _i probability density function for the time until death of a cell of type i - probability density function of a gamma distribution - G _i cumulative distribution function for the lifetime of a cell of type i - G _1*2 Convolution of G ₁ and G ₂ - ¯G _i 1–G _i - g _i probability density function for the lifetime of a cell of type i - L _i likelihood of a history of type i - m average number of proliferative daughters produced by dividing cells - M _ij (t) the expected number of type-j cells in a colony at time t if that colony began at time 0 with one type-i cell - M _i+ (t) M _i0 (t) + M _i 1(t) + M _i 2(t) - p _rs probability that a dividing cell produces r proliferative and s resting daughters - t _i times defining colony histories. See IV.2.1 - T ₀ time to division of an initiator cell - T ₁, T ₂ times from birth to division of the two daughters of an initiator cell - T ₍₁₎, T ₍₂₎ order statistics of T ₁ and T ₂ - minimum value of a gamma distribution - scale parameter of a gamma distribution or of an exponential distribution - probability per unit time of death for proliferative and resting cells - _rs expected value of p _rs when there is heterogeneity - shape parameter of a gamma distribution     

Basic functions of variability of simple pre-planned movements

A model of a pre-planned single joint movements performed without feedback is considered. Modifications of this movement result from transformation of a trajectory pattern f(t) in space and time. The control system adjusts the movement to concrete external conditions specifying values of the transform parameters before the movement performance. The preplanned movement is considered to be simple one, if the transform can be approximated by an affine transform of the movement space and time. In this case, the trajectory of the movement is x(t) = Af(t/ + s) +p, were A and 1/ are space and time scales, s and p are translations. The variability of movements is described by time profiles of variances and covariances of the trajectory x(t), velocity v(t), and acceleration a(t). It is assumed that the variability is defined only by parameters variations. From this assumption follows the main finding of this work: the variability time profiles can be expanded on a special system of basic functions corresponding to established movement parameters. Particularly, basic functions of variance time profiles, reflecting spatial and temporal scaling, are x ²(t) and t ² v ²(t) for trajectory, v ²(t) and (v(t) + t · a(t))² for velocity, and a ²(t) and (2a(t) +t · j(t))², where j(t) = d³ x(t)/dt ³, for acceleration. The variability of a model of a reaching movement was studied analytically. The model predicts certain peculiarities of the form of time profiles (e.g., the variance time profile of velocity is bi-modal, the one of acceleration is tri-modal, etc.). Experimental measurements confirmed predictions. Their consistence allows them to be considered invariant properties of reaching movement. A conclusion can be made, that reaching movement belongs to the type of simple preplanned movements. For a more complex movement, time profiles of variability are also measured and explained by the model of movements of this type. Thus, a movement can be attributed to the type of simple pre-planned ones by testing its variability.     

Reactor design for the enzymatic isomerization of glucose to fructose

A comprehensive methodology is presented for the design of reactors using immobilized enzymes as catalysts. The design is based on material balances and rate equations for enzyme action and decay and considers the effect of mass transfer limitations on the expression of enzyme activity. The enzymatic isomerization of glucose into fructose with a commercial immobilized glucose isomerase was selected as a case study. Results obtained are consistent with data obtained from existing high-fructose syrup plants. The methodology may be extended to other cases, provided sound expressions for enzyme action and decay are available and a simple flow pattern within the reactor might be assumed.List of Symbols C kat/kg specific activity of the catalyst - D m²/s substrate diffusivity within the catalyst particle - Dr m reactor diameter - d d operating time of each reactor - E kat initial enzyme activity - E _i kat initial enzyme activity in each reactor - F m³/s process flowrate - F _i m³/s reactor feed flowrate at a given time - F ₀ m³/s initial feed flowrate to each reactor - H number of enzyme half-lives used in the reactors - K mole/m³ equilibrium constant - K _S mole/m³ Michaelis constant for substrate - K _P mole/m³ Michaelis constant for product - K _m mole/m³ apparent Michaelis constant f(K, K _s, K_p, s₀) - k mole/s · kat reaction rate constant - k _d d^–1 first-order thermal inactivation rate constant - L m reactor height - L _r m height of catalyst bed - N _R number of reactors - P _i kg catalyst weight in each reactor - p mole/m³ product concentration - R m particle radius - R _P ratio of minimum to maximum process flowrate - r m distance to the center of the spherical particle - s mole/m³ substrate concentration - s _0i mole/m³ substrate concentration at reactor inlet - s ₀ mole/m³ bulk substrate concentration - s mole/m³ apparent substrate concentration - T K temperature - t d time - t _i d operating time for reactor i - t _s d time elapsed between two successive charges of each reactor - V m³ reactor volumen - V _m mole/m³ s maximum apparent reaction rate - V _p mole/m³ s maximum reaction rate for product - V _R m³ actual volume of catalyst bed - V _r m³ calculated volume of catalyst bed - V _S mol/m³ s maximum reaction rate for substrate - v mol/m³ s initial reaction rate - v _i m/s linear velocity - v _m mol/m³ s apparent initial reaction rate f(K_m, s,V_m) - X substrate conversion - X _eq substrate conversion at equilibrium - =s/K dimensionless substrate concentration - ₀=s₀/K bulk dimensionless substrate concentration - _eq=s_eq/K dimensionless substrate concentration at equilibrium - local effectiveness factor - mean integrated effectiveness factor - Thiéle modulus - =r/R dimensionless radius - _s kg/m³ hydrated support density - substrate protection factor - s residence time     

Generalized stable population theory

In generalizing stable population theory we give sufficient, then necessary conditions under which a population subject to time dependent vital rates reaches an asymptotic stable exponential equilibrium (as if mortality and fertility were constant). If x ₀(t) is the positive solution of the characteristic equation associated with the linear birth process at time t, then rapid convergence of x ₀(t) to x ₀ and convergence of mortality rates produce a stable exponential equilibrium with asymptotic growth rate x ₀–1. Convergence of x ₀(t) to x ₀ and convergence of mortality rates are necessary. Therefore the two sets of conditions are very close. Various implications of these results are discussed and a conjecture is made in the continuous case.     

Modelling of alcohol fermentation in a tubular reactor with high biomass recycle

Fermentation in tubular recycle reactors with high biomass concentrations is a way to boost productivity in alcohol production. A computer model has been developed to investigate the potential as well as to establish the limits of this process from a chemical engineering point of view. The model takes into account the kinetics of the reaction, the nonideality of flow and the segregation in the bioreactor. In accordance with literature, it is shown that tubular reactors with biomass recycle can improve productivity of alcohol fermentation substantially.With the help of the computer based reactor model it was also possible to estimate the detrimental effects of cell damage due to pumping. These effects are shown to play a major role, if the biomass separation is performed by filtration units which need high flow rates, e.g. tangential flow filters.List of Symbols Bo d Bodenstein number - c kg/m³ concentration of any component - CPFR continuous plug flow reactor - CSTR continuous stirred tank reactor - d _h m hydraulic diameter - D _eff m²/s dispersion coefficient - f residence time distribution function - K _s kg/m³ monod constant for biomass production - K _s kg/m³ monod constant for alcohol production - p kg/m³ product concentration - P _i kg/m³ lower inhibition limit concentration for biomass production - p _i kg/m³ lower inhibition limit concentration for alcohol production - p _m kg/m³ maximum inhibition limit concentration for biomass production - p _m kg/m³ maximum inhibition limit concentration for alcohol production - q _p h^–1 specific production rate - q _p,max h^–1 maximum specific production rate for alcohol production - q _s h^–1 specific substrate consumption rate - Q _L m _gas ³ /m³h specific gas rate - r _p , r _s , r _x kg/(m³ · h) reaction rate for ethanol production substrate consumption and cell growth, respectively - S _F kg/m³ substrate concentration in feed stream - s kg/m³ substrate concentration - t h time - x kg/m³ biomass concentration - x _max kg/m³ maximum biomass concentration for biomass production - Y _p/s yield coefficient - h^–1 specific growth rate - _max h^–1 maximum specific growth rate - dimensionless time (t/) - h mean residence time - _s glucose conversion     

Gas-phase influence on the mixing in a fluidized bed bio-reactor

Summary The liquid and solids mixing in fluidized bed bio-reactors containing particles with a density only slightly higher than water (1100 kg/m³) is generally consistent with the results found in previous studies for reactors with particles of higher density. The liquid mixing can be described by an axial dispersion model for a large variety of conditions while the solids follow the streamlines of the liquid. In the presence of a gas phase the degree of mixing of both the liquid and the solid phase increased. This effect became larger with increasing reactor diameter. In the extrapolation of laboratory data of three phase fluidized bed bio-reactors to pilot plant systems this effect should be taken into account. The liquid and solids mixing may have a substantial effect on overall conversion rates and on possible microbial stratification in the reactor.Nomenclature Bo Bodenstein number v L/D (-) - D _r diameter of the fluidized bed reactor (m) - D ₁ Dispersion coefficient of the liquid phase (m²/s) - D _g dispersion coefficient of the solid phase (m²/s) - E(in) normalized dye concentration function entering the ideally mixed tank reactor (-) - E(t) normalized dye concentration function as measured (-) - L length of the axial dispersed reactor (m) - t time after dye injection (s) - t _m time constant for microbial selection (s) - t _s solid mixing time constant (s) - t time interval in which a particle migrates within the bed (s) - v _t superficial gas velocity (m/s) - v _g superficial liquid velocity (m/s) - z migration distance of a particle in the bed (m) - ₁ in situ growth rate of a dominant organism (s^-1) - ₂ in situ growth rate of a recessive organism (s^-1) - average residence time in the axial dispersed reactor (s) - _t average residence time in the ideally mixed tank reactor (s)     

Genetic evaluation with autosomal and X-chromosomal inheritance

Summary At present, genetic evaluation in livestock using best linear unbiased prediction (BLUP) assumes autosomal inheritance. There is evidence, however, of X-chromosomal inheritance for some traits of economic importance. BLUP can accommodate models that include X-chromosomal in addition to autosomal inheritance. To obtain BLUP with autosomal and X-chromosomal additive inheritance for a population in which allelic frequency is equal in the sexes, and that is in gametic equilibrium, we write y _i = x_i + a_i + s_i + e_i, where y _i is the phenotypic value for individual i, x_i, is a vector of constants relating y _i to fixed effects, is a vector of fixed effects, a _i is the additive genetic effect for autosomal loci, S _i is the additive genetic effect for X-chromosomal loci, and e _i is random error. The covariance matrix of a _i's is A _A ² , where A is the matrix of twice the co-ancestries between relatives for autosomal loci, and _A ² is the variance of additive genetic effects for autosomal loci. The covariance matrix of s _i's is S _F ² , where S is a matrix of functions of co-ancestries between relatives for X-chromosomal loci and _F ² is the variance of additive genetic effects for X-chromosomal loci for noninbred females. Given the covariance matrices of random effects a _i, s_i, and e _i, BLUPs of autosomal and of X-chromosomal additive effects can be obtained using mixed model equations. Recursive rules to construct S and an efficient algorithm to compute its inverse are given.Dedicated to the memory of Dr. C. R. Henderson, whose encouraging comments stimulated the research in this paper. Supported in part by the Illinois Agricultural Experiment Station, Hatch Project 35-0367, Estimation of Genetic Parameters.     

Stability of stationary distributions in a space-dependent population growth process

We consider a spatial population growth process which is described by a reaction-diffusion equation c(x)u _t = (a ²(x)u _x ) _x +f(u), c(x) >0, a(x) > 0, defined on an interval [0, 1] of the spatial variable x. First we study the stability of nonconstant stationary solutions of this equation under Neumann boundary conditions. It is shown that any nonconstant stationary solution (if it exists) is unstable if a _xx0 for all x[0, 1], and conversely ifa _xx>0 for some x[0, 1], there exists a stable nonconstant stationary solution. Next we study the stability of stationary solutions under Dirichlet boundary conditions. We consider two types of stationary solutions, i.e., a solution u ₀(x) which satisfies u ₀ x0 for all x[0, 1] (type I) and a solution u ₀(x) which satisfies u _0x = 0 at two or more points in [0, 1] (type II). It is shown that any stationary solution of type I [type II] is stable [unstable] if a _xx 0 [a _xx 0] for all x[0, 1]. Conversely, there exists an unstable [a stable] stationary solution of type I [type II] if a _xx <0 [a _xx >0] for some x[0, 1].     

The effect of micro- and macromixing on enzyme reaction in a real CSTR

The effect of micromixing and macromixing on enzyme reaction of Michaelis-Menten type in a real continuously stirred tank reactor (CSTR) is considered. The effect of bypassing of a fraction of feed stream, dead space, initial enzyme concentration and Michaelis-Menten constant on substrate conversion is evaluated. Bypass reduces the substrate conversion significantly compared with other parameters in the case of micro and macromixing. Micromixing predicts higher substrate conversions compared with macromixing. The effect of micro and macromixing on substrate conversion is negligible at low and high conversions.List of Symbols C kmol/m³ concentration of reactant - ¯C kmol/m³ average concentration of reactant - C_A kmol/m³ exit concentration of reactant A - C_Aa kmol/m³ exit concentration of reactant A from active zone - C_AO kmol/m³ initial concentration of reactant A - C_EO kmol/m³ initial enzyme concentration - C_O kmol/m³ initial concentration of reactant - E(t) 1/s exit age distribution function - k 1/s reaction rate constant - M kmol/m³ Michaelis-Menten constant - r kmol/(m³s) rate of reaction - –r_A kmol/(m³s) rate of reaction with respect to A - t s time - v m³/s volumetric feed rate - v_a m³/s volumetric feed rate entering the active zone - v_b m³/s volumetric feed rate entering the bypass stream - V m³ total volume of the vessel - V_a m³ active volume of the vessel - V_d m³ volume of dead space - X_A conversion of A Greek Letters fraction of feed stream bypassing the vessel (v_b/v) - fraction of the total volume as dead space (V_d/V) - (t) 1/s Dirac delta function, an ideal pulse occurring at time t = 0 - s life expectancy of a molecule - 1/s intensity function or escape probability function - s space time or mean residence time