3.
            a. m = growth rate (or nutrient uptake rate)
            b. S = substrate concentration
            c. K_s = substrate concentration where growth rate is half of maximum

    F. Factors affecting uptake rate
        1. Cell size – amount of surface area relative to volume; surface area/volume gets lower as cell gets bigger
            (4Pr2 = area of a sphere; 4/3Pr3 = volume; so A/V = 3/r)
        2. Nutritional state of cell
            a. Luxury uptake – cells take up more than they need
            b. Inhibition by internal stores
        3. Transport limitation

           a. sinking speed or swimming speed
            b. turbulence
        4. Inducible enzyme systems affect Ks
        5. Toxicity effects (if nutrient abundance too high)
    G. Determining the limiting nutrient – How do we determine the limiting nutrient?
        1. Liebig’s law of the minimum – only 1 thing limits growth at any one time (something else may be close)
                    nutrient in shortest supply relative to needs
        2. Stoichiometries – again, deviations from the expected Redfield ratio.
        3. Bioassay techniques 
        4. Co-limitation
            
    H. Other nutrient factors
        1. Organic nutrients and mixotrophy
            a. Happens most often under several conditions:
                1) DOM at high concentrations
                2) Under the ice when light is low
                3) If the algae can produce specific enzymes to assist taking up organic nutrients – e.g., alkaline phosphatase & DOP
            b. Problems: bacteria have a lower Ks than do algae
        2. Vitamins – B12 is essential for cyanobacteria, diatoms, greens and dinoflagellates
        3. Organic compounds – antibiosis – chemical warfare between algae or between some macrophytes and algae;

• autoinhibition (same species) or inhibition of other species of algae
• indirect inhibition by releasing chelators that take up nutrients.

    B. Incorporation of loss/death rates
        But you also have death rates (natural loss, sinking, grazing, viruses) – if algae are growing below the death rate level,
                then the populations will not persist

            N^*_B and N^*_A are the concentrations of nutrients needed to get growth rates equal to the death rates for the two species –
                    break-even nutrient points.
            If nutrient value is greater than the N*, then that algal species will increase.
            If not, then growth rate, m, will be less than the death rate

    C. What happens if you have two nutrients?
        1. Example with two diatom species

        Now add a second nutrient:

        At low Si, Cyclotella will win.

            At low Si, Cyclotella will win.

            The outcome of competition between these diatoms depends on the relative abundance (ratio) of the two nutrients

            Easy way to view – isoclines of the two nutrients:

            P/Si increases, then Cyclotella will win
            P/Si decreases, then Asterionella will win

            Predictions from growth curves match the competition results in lab and in the field

      2. Diatoms versus blue-greens

           blue-greens have little N requirement (or Si requirement), but a high P requirement – low N:P ratios favor blue-greens

VI. Interaction of Factors Affecting Growth Control (temperate lake examples)
    A. Temperature cycle

C. Nutrient supply

D. Integration of growth factors

    How can we integrate these three major growth factors?
             Best times for growth when these factors match
             Right after ST ends and before FT begins – temp ok, light high, nutrients ok
             Second best time – mid summer
             Worst at winter time – low light and temperature
             Sometimes there is a bloom right after the fall turnover due to increased nutrients

VII. Attrition control
    A. Sinking rate
        - Under the ice there is often dominance of phytoflagellates that can control their position in the water column

    D. Parasitism
        a. Fungi – chytrids (can infect many cells – range <1-70% of cells)
        b. Viruses – can have large periodic effects; are often species specific

VIII. General seasonal succession
        Can now make generalizations about which algae will dominate when
            (These events also happen in tropical lakes, but are prompted by wet/dry seasons or turnover events – different periodicity)

            SPRING               SUMMER                                                LATE SUMMER
           Diatoms              Greens                                                    Blue-greens
            High nutrients        Good competitors at low nutrients               Lowest nutrients (N fix.)
            Low grazing          Moderate grazing                                        High grazing – resist by being unpalatable (sheath/toxin)
            Low sinking          High sinking rates (many flagellated)             Moderate sinking

            WINTER – small phytoflagellates; sometimes motile dinoflagellates

        -Each major group’s abundance curve is made up of individual species curves
        -Hundreds of species of algae live in any one lake over the course of a year
        -To predict each you need to know nutrient requirements, responses to temperature, light, grazing, sinking rates

PRIMARY PRODUCTION

I. Fate of Energy

        NPP = GPP – E – R
        The whole process is 0.03-2% energy efficient

        Standing crop/biomass vs. production

II. Measurement
    A. General equation and units
        1. units of carbon produced or oxygen emitted (sometimes calories; 1 mg C~10 cal energy,
                depending on storage material – fat, starch…)
        2. per volume or surface area of lake
        3. per time
    B. oxygen change method
        1. light-dark bottles

            Measure initial and incubate the others for a period of time
            R=I-D
            NPP=L-I (assumes the same respiration in the light and dark)
            GPP=L-D

            Problems with this method:
                (1) Enclosure/bottle effects
                (2) Sensitivity

        2. whole environment
                measure oxygen change in a lake or stream over a day
                avoid enclosure effects
                must compensate for invasion and evasion of oxygen to the lake

    C. C change -- ¹⁴C method
        Add radioisotope of carbon (¹⁴C) as bicarbonate, H¹⁴CO₃^-, and it is converted to labeled carbon by the algae
        Incubate in light and dark bottles
        Measure of roughly NPP (how much ¹⁴C is incorporated into the algae)
        Is more sensitive than oxygen method

        Problems with this method
            (1) ¹⁴C and ¹²C don’t have the same reactivity
            (2) Doesn’t measure ¹⁴C that entered the cell and then left by excretion or respiration before the end of the experiment

    D. yield method
        Look at the change in algal biomass over time
        No bottle effects
        Only used with lots of growth so that there is no sensitivity problem
        What is the problem with this measure?  Doesn't account for attrition – gives an underestimate of production
        Also a problem with moving water masses – spatial heterogeneity – may be sampling different water masses

III. Patterns of productivity
    A. Productivity versus latitude

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