Transcript Math Poster
EXPERIMENTAL MODELS OF MORPHOGENESIS David Moore & Audrius Meškauskas School of Biological Sciences, The University of Manchester, Manchester M13 9PT, U.K. 1. There is no reason why the “rules” which govern morphogenesis should not be established. From the rules and a few dimensions, times and rate values a mathematical expression to describe the morphogenetic process could emerge. Computer models could be derived from that to simulate the whole process. Nice idea! But where do you start? Start simple. Making a stem bend in response to a tropic stimulus is a suitably simple experimental approach. YOU can choose when to apply the stimulus, it is easily replicated and reaction and response times can be measured readily. Also, the response itself can be measured so the quantitative demands of mathematical modelling can be satisfied. The gravitational imperative ... Norm a l, upright growth of the fruit body 2. We have used the gravitropic reactions of mushroom fruit bodies to study control of morphogenesis because being the right way up is crucial to a mushroom. Changing orientation is a non-invasive stimulus. We’ve coupled video observation and image analysis to get detailed descriptions of the kinetics. We’ve made and used clinostats to vary exposure to gravity, and we’ve combined a variety of microscopic observation techniques to make quantitative observations. Disorient P hototropic response NO Meiosis complete YES Gravisensors Perception threshold exceeded The video sequence in the background shows (top) a culture placed on its side at 16:37h. The weight of the cap makes the Coprinus cinereus fruit body swing downwards, but by 19:19h it has bent upwards at 45o. Sadly, one second later the connection with the mycelium breaks and the fruit body swings round so the cap points downwards. But it doesn’t give up! By 20:37 it’s back to the horizontal; by 22:10 it’s almost upright, and by 22:58 the cap is fully expanded and releasing spores. YES Bend a pex upwa rds Bend ing signa l 6. The model we have now describes the shapes assumed by real stems of Coprinus cinereus. Bending rate is determined by the balance between signals from detectors of the direction of gravity (a function of the angle of the stem) and for curvature compensation (a function of the local amount of bending). In a straight stem displaced to the horizontal the gravitropic signal is maximal and curvature compensation signal is zero. As the stem bends the gravitropic signal weakens (as the angle of displacement of the perception system lessens) but the bending enhances the curvature compensation signal. NO 3. The first step is to summarize these observations and experiments into a flow chart (as shown at left). This concentrates attention on critical features and, in a nonmathematical way, produces a formalized description which is a good starting point for mathematical analysis. 7. In our local curvature distribution model, therefore, straightening is determined by local curvature, independently of the spatial orientation of that part of the stem. Signa l from apex migrates basipetally Ea ch successive segm ent is ca used to bend NO Tip angle > 35o from the horizontal YES Activate curva ture com pensation 4. Next, a “scheme” needs to be constructed which lends itself to mathematical expression whilst still keeping a firm footing in cell physiology. The basic assumptions of ours are that change in the angle of the apex occurs as a result of four consecutive stages: (i) the physical change which occurs when the subject is disoriented (this is called susception) (ii) conversion of the physical change into a physiological change (this is called perception) (iii) transmission of the physiological signal to the competent tissue (called transduction) (iv) the differential regulation of growth which causes the bend and change in apex angle (called response). We used this scheme to estimate and calculate numerical values for the various parameters of a combined equation that could generate apex angle kinetics which imitated the reaction of mushroom stems quite well. 8. This model is predictive and successfully describes the gravitropic reaction of stems treated with metabolic inhibitors, confirming its credibility and indicating plausible links between the equations and real physiology. Com pensa tion signa l Gravit ropic res ponse syste m Signa l from apex migrates basipetally p s d Perception Susception fs () tr r Transduction ys Response ytr fp (ys) Ea ch successive segm ent is ca used to straighten Memory ftr (yp) d fr (ytr) 5. This imitational model dealt with change in apex angle only. Observations of real stems show a complex distribution of bending and straightening. Almost 90% of the initial curvature is reversed by subsequent straightening (we call it “curvature compensation”). A realistic model of gravitropic bending would describe the process in space as well as in time. This is where “imitation” ends and “simulation” begins. 9. Where do we go from here? It’s a predictive model, so we need to make predictions and test them. We need to develop the maths into three spatial dimensions and to cope with hyphal communities. Most importantly, we need to convince somebody to fund the project! •Stoĉkus, A. & Moore, D. (1996). Comparison of plant and fungal gravitropic responses using imitational modelling. Plant, Cell & Environment 19, 787-800. •Moore, D. & Stoĉkus, A. (1998). Comparing plant and fungal gravitropism using imitational models based on reiterative computation. Advances in Space Research 21 (8/9), 1179-1182. •Meškauskas, A., Moore, D. & Novak Frazer, L. (1998). Mathematical modelling of morphogenesis in fungi: spatial organization of the gravitropic response in the mushroom stem of Coprinus cinereus. New Phytologist 140, 111-123. •Meškauskas, A., Novak Frazer, L. & Moore, D. (1999). Mathematical modelling of morphogenesis in fungi: a key role for curvature compensation (‘autotropism’) in the local curvature distribution model. New Phytologist 143, 387-399. •Meškauskas, A., Jurkoniene, S. & Moore, D. (1999). Spatial organisation of the gravitropic response in plants: applicability of the revised local curvature distribution model to Triticum aestivum coleoptiles. New Phytologist 143, 401-407. We thank the British Mycological Society and Federation of European Microbiological Societies for award of FEMS short term Fellowships to the late Alvidas Stokus and to Audrius Meškauskas which enabled this research to be initiated. A Royal Society NATO Research Fellowship to AM supported the consolidation and further development of the work to the stage described here.