On the average cellular volume in synchronized cell populations

As was done by Sinclair and Ross (1969(, we consider a cellular population that consists initially (at time zero) ofN ₀ newborn cells, all with the same volumev _o. It is assumed that the occurrence of cell division is determined only by a cell’s age, and not by its volume. The frequency function of interdivision times, τ, is denoted byf(τ). If cell death is negligible, the expected number of cells,N(t), will increase according to the laws of a simple age-dependent branching process. The expression forN(t) is obtained as a sum over all generations; thevth term of this sum, in turn, is a multiple convolution integral, reflecting the life history ofvth generation cells (i.e., the lengths of thev successive interdivision periods plus the age of the cell at timet). Assuming that cell volume is a given function of cell age, e.g., linear or exponential, and that cellular volume is exactly halved at each division, it is possible to calculate the volume of a cell with a given life history, and thus the average cellular volume of the whole population as a function of time. If at time zero the volumes differ from cell to cell, the final equation must be modified by averaging over initial volumes. In the case of linear volume increase with age, a very simple asymptotic expression is found for the average cellular volume ast→∞. The case of exponential volume increase with age also leads to a simple asymptotic formula, but the resulting volume distribution is unstable. The mean cellular volume at birth and the second moment of the volume distribution can be calculated in a similar manner. Work supported by the U.S. Atomic Energy Commission.     

Mathematical models for cellular systems the von Foerster equation. Part I

P. B. M. Walker (1954) and H. C. Longuet-Higgins (quoted by Walker), as well as O. Scherbaum and G. Rasch (1957), made the first attempts towards a mathematical study of the age distribution in a cellular population. It was H. Von Foerster (1959), however, who derived the complete differential equation for the age density function,n(t, a). His equation is obtained from an analysis of the infinitesimal changes occurring during a time elementdt in a group of cells with ages betweena anda+da. The behavior of the population is determined by a quantity λ which we call the loss function. In this paper a rigorous discussion of the Von Foerster equation is presented, and a solution is given for the special case when λ depends, ont (time) anda (age) but not on other variables (such asn itself). It is also shown that the age density,n(t, a), is completely known only if the birth rate,α(t), and the initial age distribution, β(a), are given as boundary conditions. In Section II the steady state solution and some plausible forms of intrinsic loss functions (depending ona only) are discussed in view of later applications.     

Topological biology: A note on Rashevsky's transformation T.

In the bio-topological transformation between graphs denoted by (T ⁽¹⁾ X) N. Rashevsky (Bull. Math. Biophysics,18, 173–88, 1956) considers the number of fundamental sets which (a) have only one specialized point as source (and no other sources), (b) have no points in common (are “disjoined”); he proves that this number is an invariant of the transformation. In this note we show that Rashevsky's Theorem can be extended as follows:The number of fundamental sets of the first category is an invariant of the transformation. We must, however, count the subsidiary points of the transformed graph as specialized points. We recall that fundamental sets of the first category are those whose sources consist of specialized points only (Trucco,Bull. Math. Biophysics,18, 65–85, 1956). But in this modified version of the Theorem the fundamental sets may have more than one source and need not be disjoined.     

Note on a combinatorial problem

In the first part of this paper we have assembled some properties of the quantitiesR _m ⁿ , whereR _m ⁿ denotes the number of distributions ofn different objects intom indifferent parcels, with no empty parcels allowed. We then discuss the following problem (N. Rashevsky, 1954, 1955 a,b, 1956): to find the total number,G _n , of graphs that can be obtained from the biotopological transformation (T ⁽¹⁾ X) for a given value of the parametern. This is related to the distribution ofn indifferent objects intom different boxes. A formula forG _n is given which, however, is not very convenient for practical computations because it involves a summation over certain “admissible partitions” of the numbermn (m is a second parameter of the transformation). Some theorems are derived; with their help we can simplify the calculation ofG _n to a small extent. The numbersG _n are calculated forn≤9 and estimated forn=10. It is found thatG ₇≈5.4×10⁴,G ₈≈8.3×10⁵,G ₉≈1.4×10⁷, andG ₁₀≈3×10⁸. These values ofn are those which might be used in connection with N. Rashevsky’s work (cf. Rashevsky, 1956).     

On the information content of graphs: Compound symbols; Different states for each point

The idea of assigning an information content to a graph is extended to include the following two situations: (a) Combinations of sets of topologically equivalent points are used as symbols; (b) Points of the graph may exist in different states.     

The beta-chains of DP4 molecules from different haplotypes are encoded by the same gene

The nucleotide sequence of a complete cDNA gene from a DP4-positive HLA-homozygous cell line, PGF, has been determined. This sequence is identical to the exon sequences in a genomic clone derived from another DP4-positive cell line, Priess. In contrast, our DP cDNA sequence shares only limited homology with partial cDNA sequences obtained from clones of three DP4-negative cell lines. On the basis of these results, we conclude that the phenotypic variation of DP alleles is directly attributable to the nucleotide sequence heterogeneity of DP-beta genes. That is, each phenotypic allelic form of DP antigen corresponds to a distinctly different DP-beta gene. Furthermore, this correspondence is found to be unaffected by the markers present at the DQ and DR loci, since the haplotypes of the PGF and Priess cell lines are, respectively, DR2,DQw1,DP4 and DR4,DQw3,DP4.     

Protection of islets by in situ peptide-mediated transduction of the Ikappa B kinase inhibitor Nemo-binding domain peptide

We have previously demonstrated that adenoviral gene transfer of the NF-kappaB inhibitor IkappaB to human islets results in protection from interleukin (IL)-1beta-mediated dysfunction and apoptosis. Here we report that human and mouse islets can be efficiently transduced by a cationic peptide transduction domain (PTD-5) without impairment of islet function. PTD mediated delivery of a peptide inhibitor of the IL-1beta-induced IkappaB kinase (IKK), derived from IKKbeta (NBD; Nemo-binding domain), and completely blocked the detrimental effects of IL-1beta on islet function and NF-kappaB activity, in a similar manner to Ad-IkappaB. We also demonstrate that mouse islets can be transduced in situ by infusion of the transduction peptide through the bile duct prior to isolation, resulting in 40% peptide transduction of the beta-cells. Delivery of the IKK inhibitor transduction fusion peptide (PTD-5-NBD) in situ to mouse islets resulted in improved islet function and viability after isolation. These results demonstrate the feasibility of using PTD-mediated delivery to transiently modify islets in situ to improve their viability and function during isolation, prior to transplantation.     

Body window-enabled in vivo multicolor imaging of transplanted mouse islets expressing an insulin-Timer fusion protein

Type 1 diabetes results from the selective destruction of insulin-producing beta cells in the islets of Langerhans, and autoimmune T cells are thought to be the mediators of this destruction. T cells are also responsible for allorejection once the islets are transplanted into a patient to reduce the negative consequences of a lack of insulin. To better understand these processes, we have developed a transgenic mouse expressing proinsulin II tagged with a live-cell fluorescent reporter protein, Timer. Timer protein is unique because it changes color from green to red in the first 24 h after synthesis. With this marker, insulin synthesis can be carefully monitored through fluorescent changes over time. To complement this new biotechnological research tool, we designed a body window to allow for in vivo imaging over time of the islets transplanted under the kidney capsule. The window device, which is sutured to replace the underlying skin and body wall over the site of islet transplantation, may be used to simultaneously observe beta cells and T cells that have been labeled with a fluorochrome distinguishable from Timer. The imaging of both insulin-producing cells and T cells may be carried out repeatedly for a week or more with no need for repeated surgery, while preserving the life of the studied animal.     

Semiautomated image segmentation of bone marrow biopsies by texture features and mathematical morphology

OBJECTIVE: To present the preliminary results of a method for semiautomated detection of fat and hematopoietic cells as well as trabecular surfaces in bone marrow biopsies in order to calculate the percentage of each type of tissue or cell area in relation to the whole area. STUDY DESIGN: The results were derived from selected clinical cases. Twenty-six biopsies were used, presenting varied distributions of cellularity and trabecular topography. The approach is based on digital image processing techniques and pattern recognition methods using textural features obtained from biopsy images. The results were improved with mathematical morphology filters. RESULTS: A low computational cost algorithm is obtained that produces highly satisfactory results. The method is faster and more reproducible than conventional ones, such as region growing, edge detection, splitting and merging. CONCLUSION: The results with this computer-assisted technique were compared to those obtained by visual inspection by 2 expert pathologists, and differences of < 9% were observed.